Ab‑Initio Density Functional Theory from Higher‑Categorical Integrability: Spectral‑Law‑Constrained Exchange–Correlation without Fitting

Ab-Ini io Densi y Func ional Theo y om Highe -Ca ego ical In eg abili y:
Spec al-Law-Cons ained Exchange–Co ela ion wi hou Fi ing
And ei T. Pa ascu
FAST Founda ion, Des in FL, 32541, USA
email: and ei.pa [email p o ec ed]
Con en ional densi y unc ional heo y (DFT) a ains ema kable compu a ional e iciency bu
depends c i ically on exchange–co ela ion (XC) unc ionals ha a e ei he semi–empi ical o de i ed
om pe u ba i e app oxima ions. Bo h app oaches s uggle o desc ibe s ongly co ela ed, mul i–
e e ence, o opologically non i ial elec onic s a es and lack sys ema ic e o con ol. He e we
de elop a ully i s –p inciples DFT based on highe –ca ego ical Lax pai s and a gauge heo y o non–
isospec al in eg abili y. In his amewo k, he KS gene a o
L
[
n
]is p omo ed o an ope a o – alued
connec ion on a highe bundle whose cu a u e encodes quan um en anglemen and co ela ion.
Exchange–co ela ion e ec s a ise no om i ed pa ame e s bu om en o cing a ini e se o spec al
laws Φ
 k
(
L
[
n
])

=
φk
, whe e Φis an Ad–in a ian s a e on he ope a o algeb a o
L
[
n
]and
{ k}
a e esol en o p ojec o p obes ixed by exac sum ules, i ial ela ions, and in eg able e e ence
models. The esul ing XC po en ial,
xc( ) = X
k
λkΦZΓk
k(z)(z−L)−1δL
δn( )(z−L)−1dz
2πi,
is ob ained by sol ing o he mul iplie s
λk
ha en o ce hese spec al cons ain s du ing each sel –
consis en i e a ion. Highe –ca ego ical in a ian s— he Ha o i–S allings ace
HH0
and
K
– heo e ic
sec o classes—gua an ee size consis ency, o a ional in a iance, and opological s abili y wi hou
empi ical uning. G adien and ime–dependen ke nels ollow om he Whi ham hie a chy gene a ed
by he cu a u e o he highe connec ion, ensu ing exac comp essibili y and
–sum ules. The
app oach yields ab–ini io accu acy a Kohn–Sham cos : o al–ene gy e o s o o de milli–Ha ee
and exci a ion ene gies wi hin 0
.
05 eV o coupled–clus e benchma ks a compu a ional o e heads
below 30% ela i e o s anda d GGA. Because all inpu s a e algeb aic in a ian s anspo ed h ough
a ini e a las o in eg able e e ence iles, he me hod elimina es semi–empi ical i ing and p o ides
explici cu a u e–based e o es ima es. This es ablishes a pa h owa d genuinely i s –p inciples,
quan um–consis en DFT go e ned by ca ego ical in eg abili y a he han pa ame iza ion.
0. EXECUTIVE SUMMARY
Con ex and goal.
Kohn–Sham densi y unc ional heo y (DFT) deli e s ema kable scale and e iciency,
bu i s accu acy hinges on he exchange–co ela ion (XC) unc ional. Exis ing XC amilies a e ei he
semi–empi ical, wi h pa ame e s i ed o da a, o de i ed om pe u ba i e expansions; bo h su e om
limi ed ans e abili y and may ail in s ongly co ela ed, mul i- e e ence, o opologically non i ial
egimes. Founda ional esul s [
1
,
2
,
425
,
426
] ensu e ha an exac unc ional exis s, bu hey do no
p esc ibe how o cons uc i .
Co e idea.
We eplace empi ical XC design wi h a cons ained, ab-ini io cons uc ion g ounded in
highe -ca ego ical in eg abili y. Conc e ely, we p omo e he KS gene a o
L
[
n
] o an ope a o - alued objec
equipped wi h a highe (2-)connec ion (
A,B
)and o mula e spec al laws Φ
 k
(
L
[
n
])

=
φk
as algeb aic
in a ian s o be en o ced du ing he sel -consis en ield (SCF) i e a ion. He e Φis an Ad-in a ian
s a e on he ope a o algeb a gene a ed by
L
(e.g. a no malized ace o KMS s a e [
5
]), and
{ k}
a e
esol en o p ojec o p obes ixed by exac sum ules [
6
], i ial ela ions, homogeneous-elec on-gas
(HEG) esponses [
271
], and sol able e e ence limi s (e.g. SCE/op imal- anspo [
8
,
9
]). The highe -gauge
iewpoin builds on Lax/monod omy s uc u es [
10
] and hei quan um-g oup gene aliza ions [
11
,
12
],
ex ended o non-isospec al lows ia he 2-connec ion (A,B)in he spi i o highe gauge heo y [13].
XC om spec al-law cons ain s.
Fo a ixed densi y
n
, we o mula e a Le y–Lieb- ype cons ained
sea ch
F[n] = in
Ψ→nnhΨ|ˆ
T+ˆ
W|ΨiΦ k(L[n])=φk, k = 1, . . . , mo,(1)
and iden i y he XC po en ial
xc
=
δExc/δn
om he s a iona i y o he associa ed Lag angian wi h
mul iplie s λk. Using esol en unc ional calculus, we ob ain he explici o mula
xc( ) =
m
X
k=1
λkΦZΓk
k(z)(z−L)−1δL
δn( )(z−L)−1dz
2πi,(2)
2
whe e each con ou Γ
k
encloses he spec al egion ele an o
k
. Equa ion
(2)
is e alua ed wi hin
each SCF mac o-i e a ion by upda ing
{λk}
so ha he spec al laws Φ(
k
(
L
[
n
])) =
φk
a e sa is ied o
ole ance.
Why his is ab ini io.
The a ge s
{φk}
a e no i o da a; hey a e de e mined a p io i om exac
iden i ies ( -sum and comp essibili y [
6
], i ial, scaling), analy ic HEG esponses [
271
], sol able limi s
(Be he-ansa z/TBA, SCE [
8
,
9
]), o anspo ed along exac legs ( anishing cen al cu a u e) o he
highe connec ion. The in a ian s a e ope a o -le el quan i ies (expec a ion alues unde Φ) and hus
e ain quan um-phase and en anglemen in o ma ion absen om pu ely densi y-based i s.
Buil -in s uc u al gua an ees.
(i) Size consis ency and o a ion in a iance ollow by p ojec ing
o he Ha o i–S allings quo ien
HH0
, which kills commu a o s and ende s Φinsensi i e o o bi al
o a ions [
436
,
437
]. (ii) Topological/sec o s abili y is en o ced by including Riesz p ojec o s
PI
(
L
)among
he p obes
k
: hei
K
- heo y classes [
PI
]
∈K0
emain in a ian un il a ue gap closu e, p e en ing
spu ious s a e swapping; s anda d e e ences on
C∗
-algeb a
K
- heo y apply [
351
]. (iii) G adien and
dynamic ke nels a e de i ed om he momen /Whi ham hie a chy gene a ed by he cen al cu a u e,
ensu ing exac comp essibili y and
-sum ules; ime-dependen esponse is handled by Runge–G oss
TDDFT ounda ions [
237
], wi h nonlocal/b aided co ec ions cons ained by he non-cen al cu a u e
(Yang–Bax e /quan um-g oup consis ency [11, 12]).
The a las.
We assemble a ini e a las o uni e sal iles, each ile being an in eg able (o nea -in eg able)
backbone speci ying: a gene a o empla e
L
[
n
], a p obe se
{ k}
, and exac a ge s
{φk}
o a physical
egime (HEG/me allic, co alen molecula , SCE/Mo , 1D Lu inge -like, mul i-o bi al TM). Once
compu ed once, a ile’s
{φk}
a e anspo ed ac oss geome y/composi ion/ ields ia he highe connec ion;
cu a u e supplies quan i a i e e o ba s. This geome ic cons uc ion eplaces empi ical e- i ing wi h
algeb aic anspo .
Compu a ional p o ile.
The me hod p ese es KS complexi y: diagonaliza ions (o linea sol es)
domina e; en o cing
m.
6spec al laws adds a hand ul o shi ed linea sol es ( esol en s), commonly
a ailable in DFPT/TDDFT s acks [
237
]. Typical o e head is
∼
10
–
30% o e GGA pe SCF while enabling
accu acies app oaching coupled-clus e /GW benchma ks—wi hou in oking hose high-cos many-body
sol e s.
Con ibu ions and claims.
•
A cons ained- a ia ional, i - ee de i a ion o
xc
gi en by Eq.
(2)
, wi h spec al p obes ixed by
exac physics [6, 8, 9, 271].
•
S uc u al gua an ees om
HH0
and
K0
(mixing obus ness and sec o p o ec ion) g ounded in
ope a o algeb a/K- heo y [5, 351, 436, 437].
•
A highe -gauge o mula ion connec ing Lax/in eg abili y [
10
–
12
] o TDDFT-consis en ke nels [
237
],
including non-isospec al co ec ions con olled by cu a u e [13].
•
A ini e a las o iles eplacing e- i ing by algeb aic anspo , wi h cu a u e-based e o quan i i-
ca ion.
Impac .
The amewo k p o ides a pa h o ab-ini io-accu a e DFT a KS cos by ele a ing XC
cons uc ion om pa ame e i ing o en o cemen o algeb aic in a ian s ha a e quan um-consis en ,
opologically s able, and ans e able ac oss chemical space.
I. INTRODUCTION
A. DFT oday: successes and limi a ions
Scope.
Densi y unc ional heo y (DFT) has become he de aul i s -p inciples wo kho se in condensed
ma e physics, chemis y, and ma e ials science because i o e s a ac able mapping o an in e ac ing
many-elec on sys em o an auxilia y single-pa icle p oblem. In p ac ical Kohn–Sham (KS) implemen-
a ions, he compu a ional cos scales as in a gene alized eigen alue p oblem while e aining access o
g ound-s a e ene gies, o ces, and (by linea esponse) many spec oscopic obse ables ac oss sys em sizes
anging om small molecules o housands o a oms in pe iodic solids. In his subsec ion we e iew,
a a echnical le el, (i) why con empo a y exchange–co ela ion (XC) app oxima ions succeed ac oss
3
as applica ion domains and (ii) whe e hey ail: ans e abili y ac oss egimes, s ong co ela ion, and
exci ed-s a e/ esponse p ope ies.
The Jacob’s ladde o XC app oxima ions.
Mode n p ac ice o ganizes app oxima e XC unc ionals
by hei ing edien dependence, o en dubbed “Jacob’s ladde ”:
•
Local densi y app oxima ion (LDA):
ELDA
xc
[
n
] =
Rn
(
)
εUEG
xc n
(
)
d ,
whe e
εUEG
xc
(
n
)is aken om
he uni o m elec on gas. Despi e i s simplici y, LDA deli e s obus s uc u al p ope ies and
cohesi e ene gies in many solids.
•
Gene alized g adien app oxima ion (GGA): adds educed-g adien dependence
s
=
|∇n|/
(2
kFn
),
imp o ing equilib ium geome ies and eac ion ene ge ics; he PBE unc ional [
234
] is he canonical
non-empi ical GGA.
•
Me a-GGA: inco po a es kine ic-ene gy densi y
τ
(
) =
Pi|∇ψi|2
o he Laplacian
∇2n
o dis-
inguish one- s. many-o bi al egions and weak s. s ong bonds; he SCAN unc ional [
236
]
exempli ies a cons ain -based me a-GGA wi h subs an ially imp o ed he mochemis y and la ice
cons an s.
•
Global and ange-sepa a ed hyb ids: mix a ac ion o exac (Ha ee–Fock) exchange,
Ehyb
xc
=
a EHF
x
+(1
−a
)
EDFT
x
+
EDFT
c,
o educe sel -in e ac ion and delocaliza ion e o s. B3LYP [
20
,
21
] is
widely used o molecules; sc eened hyb ids such as HSE [
22
,
23
] apply sho - ange exac exchange
o imp o e band gaps and de ec le els in solids. Long- ange-co ec ed hyb ids (e.g. CAM-B3LYP
[24]) spli 1/ by an e o - unc ion ke nel o cap u e cha ge- ans e exci a ions mo e ai h ully.
•
Nonlocal dispe sion co ec ions: pai wise and many-body an de Waals e ms augmen semilocal
unc ionals o eco e asymp o ic
C6/R6
beha io and collec i e dispe sion; examples include
DFT-D [25], Tka chenko–Sche le [26], and many-body dispe sion [27].
Pe dew and Schmid ’s aming [
28
] emphasizes ha each ung sys ema ically adds exac cons ain s/in-
g edien s o app oach he “hea en” o chemical accu acy.
B ead h o applica ions and scale.
Plane-wa e pseudopo en ial and p ojec o -augmen ed-wa e
implemen a ions [
325
] and mode n open-sou ce pla o ms [
30
] ha e enabled eliable geome y op imiza ion,
phonons, he mal p ope ies, su aces, and in e aces o hund eds o housands o a oms. In ca alysis
and su ace science, semilocal and hyb id DFT combine wi h d-band and scaling ela ions o a ionalize
adso p ion and ac i i y ends ac oss ansi ion-me al se ies [
31
]. Fo la ge o ganic and biomolecula
sys ems, dispe sion-co ec ed DFT cap u es nonco alen in e ac ions wi h quan i a i e accu acy [
25
–
27
].
These successes a ise because common app oxima ions ai h ully encode key cons ain s (spin scaling,
uni o m-densi y limi , exchange hole no maliza ion) and exploi e o cancella ions ha pe sis ac oss
chemical amilies.
Known e o channels and hei ma hema ical o m.
Despi e b oad u ili y, s anda d XC app oxi-
ma ions exhibi sys ema ic, physically in e p e able e o s:
(i) Sel -in e ac ion and delocaliza ion e o s. Fo any one-elec on densi y
n1
, he exac condi ion is
EH
[
n1
]+
Exc
[
n1
] = 0; semilocal unc ionals gene ally ail o cancel he Ha ee sel - epulsion [
32
], p oducing
oo-low ba ie s and spu iously s abilized ac ional cha ges. Ene ge ically, exac DFT is piecewise linea
in elec on numbe Nbe ween in ege s; de ia ions (con ex cu a u e) quan i y delocaliza ion e o and
unde lie cha ge- ans e bias and o e binding [33].
(ii) De i a i e discon inui y and band gaps. The undamen al gap
Eg
=
I−A
di e s om he KS
eigen alue gap by he XC discon inui y ∆
xc
,
Eg
=
εLUMO −εHOMO
+ ∆
xc,
wi h ∆
xc
a ising om he
discon inuous jump o
xc
a in ege
N
[
34
,
35
]. Semilocal unc ionals lack his ea u e, explaining
band-gap unde es ima ion and he misplacemen o de ec le els; sc eened hyb ids pa ially es o e i .
(iii) S a ic (s ong) co ela ion. Nea -degene acy (e.g. bond dissocia ion, Mo insula ing s a es,
ansi ion-me al complexes) equi es a mul i-de e minan desc ip ion; common XC o ms a e biased
owa d a single Sla e de e minan and hus incu “ ac ional spin” e o s ha pe sis e en when he
densi y is quali a i ely co ec [
33
]. Remedies (DFT+U, hyb ids) imp o e ene ge ics bu equi e empi ical
choices (double coun ing, ac ion o exac exchange) and can iola e size consis ency in challenging
egimes.
(i ) Long- ange and nonlocal co ela ion. Semilocal app oxima ions miss asymp o ic dispe sion and
collec i e sc eening. Empi ical and physics-based dispe sion co ec ions [
25
–
27
] epai long- ange ails;
4
adiaba ic-connec ion luc ua ion–dissipa ion (ACFD) me hods (e.g. RPA) cap u e collec i e sc eening
bu a inc eased cos .
( ) Exci ed s a es and esponse. Linea - esponse ime-dependen DFT (TDDFT) ecas s op ical
spec a and exci a ion ene gies as eigenmodes o a esponse ke nel
xc
(
, 0, ω
). Adiaba ic local/semi-local
ke nels miss he co ec 1
/R
asymp o ics o dono –accep o cha ge ans e and unde es ima e Rydbe g
exci a ions; ange sepa a ion and uned nonlocal ke nels a e needed [
24
,
36
,
38
,
414
]. In solids, exci ons
and plasmons equi e nonlocal, equency-dependen ke nels ha a e absen in adiaba ic semilocal
TDDFT.
Summa y o he s a us quo.
Con empo a y DFT achie es eliable, scalable p edic ions ac oss an
imp essi e swa h o chemis y and ma e ials science, especially when enhanced by cons ain -based
me a-GGAs [
236
], sc eened/long- ange hyb ids [
22
–
24
], and dispe sion co ec ions [
25
–
27
]. Ne e heless,
limi a ions ied o exac condi ions (one-elec on sel -in e ac ion, piecewise linea i y), nonlocal co ela ion,
de i a i e discon inui y, and mul i- e e ence cha ac e pe sis [
32
–
35
]. These issues mo i a e p incipled
al e na i es ha p ese e he KS e iciency while en o cing quan um-consis en cons ain s beyond
semi-empi ical i ing—a di ec ion de eloped in he subsequen sec ions ia highe -ca ego ical in eg abili y
and spec al-law cons ain s.
B. Why in eg abili y & ca ego y heo y? F om Lax pai s o cohe ence
Mo i a ion.
The cen al challenge o densi y unc ional heo y (DFT) is o encode many-body quan um
co ela ions in o he exchange–co ela ion (XC) unc ional wi hou eso ing o ad hoc i ing. Two
s uc u al ea u es o in e ac ing quan um sys ems s ongly sugges in eg abili y and ca ego y heo y as
he igh ma hema ical sca olding:
(i) Conse a ion laws and commu a i i y. Exac ly sol able (in eg able) models possess commu ing
amilies o ans e ma ices o cha ges ha o ganize dynamics and co ela ions in a nonpe u ba i e
manne ; hey a ise om Lax pai s, monod omy ma ices, and he Yang–Bax e equa ion [
39
,
41
–
43
,
281
].
(ii) Cohe ence and unc o iali y. Complex many-body sys ems decompose and ecombine along enso
p oduc s. The algeb aic cohe ence go e ning e-associa ion and exchange is cap u ed by monoidal and
b aided monoidal ca ego ies, whose cohe ence laws (pen agon and hexagon) ensu e ha any way o
pa en hesizing o b aiding gi es he same esul [44, 46, 47, 247].
These wo h eads mee in he heo y o quan um g oups and Hop algeb as: quasi- iangula Hop
algeb as p oduce b aided monoidal ca ego ies o ep esen a ions, R-ma ices sa is ying Yang–Bax e
ela ions, and hence commu ing ans e amilies; on he geome ic side, noncommu a i e unc ion algeb as
and ope a o algeb as supply spec al and opological in a ian s [
48
,
51
,
53
,
429
,
430
,
434
]. We now
de elop hese poin s in de ail and explain how hey u nish ab-ini io cons ain s o XC.
In eg abili y: Lax pai s, monod omy, and commu ing cha ges.
A (classical o quan um) sys em
is said o be in eg able i he e exis s a Lax pai (
L
(
u
)
, M
(
u
))—ope a o s o ma ices depending on a
spec al pa ame e u—such ha he equa ions o mo ion ake he Lax o m
d
d L(u)=[M(u), L(u)],(3)
so ha he isospec al in a ian s (eigen alues o
L
(
u
)) a e conse ed [
42
,
281
]. In he quan um in e se
sca e ing me hod (QISM), one builds he monod omy ma ix
Ta
(
u
)ac ing on an auxilia y space
a
and
he physical (quan um) space, wi h ma ix elemen s ha gene a e he model’s algeb a. The co ne s one
is he Yang–Bax e equa ion (YBE) o an R-ma ix Rab(u− )ac ing on wo auxilia y spaces:
Rab(u− )Rac(u−w)Rbc( −w) = Rbc( −w)Rac(u−w)Rab(u− ),(4)
which gua an ees he RTT ela ions
Rab(u− )Ta(u)Tb( ) = Tb( )Ta(u)Rab(u− ).(5)
Taking a ace o e he auxilia y space,
(
u
) =
aTa
(
u
)
,
one ob ains a commu ing amily [
(
u
)
,
(
)] = 0
,
whose loga i hmic de i a i es gene a e an in ini e owe o conse ed cha ges
Q1, Q2, . . .
[
39
,
41
,
42
,
281
].
Physically, his yields ac o ized sca e ing and exac sol abili y, and ma hema ically i p o ides a ich se
o in a ian s and sum ules.
Rele ance o XC. DFT needs nonpe u ba i e in o ma ion abou sho - ange exchange holes, long- ange
co ela ion, and collec i e modes. The QISM amewo k deli e s commu ing cons ain s—iden i ies among
5
co ela ion unc ions and esponse coe icien s— ha can be ansc ibed as spec al laws o an ope a o
L
[
n
]chosen as a KS-like gene a o . In p ac ice, a small se o esol en o p ojec o p obes
k
and an
Ad-in a ian s a e Φon he algeb a gene a ed by
L
[
n
]yields in a ian s Φ(
k
(
L
[
n
])) which we en o ce
du ing SCF. In eg abili y mo i a es which laws o impose (and when hey commu e), while he YBE/RTT
s uc u e explains why hese cons ain s a e simul aneously sa is iable in la ge classes o sys ems.
F om quan um g oups o b aided enso ca ego ies: cohe ence as physics.
AHop algeb a
H
is an algeb a wi h mul iplica ion
m
, uni
η
, comul iplica ion ∆ :
H→H⊗H
, couni
ε
, and an ipode
S
, obeying compa ibili y axioms [
48
]. A quasi- iangula Hop algeb a is equipped wi h an in e ible
R∈H⊗Hsa is ying
(∆ ⊗id)R=R13R23,(id ⊗∆)R=R13R12,
and he quan um Yang–Bax e equa ion
R12R13R23
=
R23R13R12
, which induces a b aiding on he
ep esen a ion ca ego y
Rep
(
H
)[
429
,
430
]. Ca ego ically,
Rep
(
H
)is a b aided monoidal ca ego y wi h
enso p oduc
⊗
, associa o
α
, and b aiding
cX,Y
:
X⊗Y→Y⊗X
sa is ying he pen agon and
hexagon cohe ence diag ams o Mac Lane and Joyal–S ee [
44
,
247
]. These cohe ence laws gua an ee
ha mul i-pa icle exchanges and e-associa ions a e unambiguous: e e y diag am buil om
α
and
c
wi h he same endpoin s commu es.
Why his ma e s o XC. The XC unc ional mus be in a ian unde a bi a y o bi al o a ions and
mus encode he co ec exchange s a is ics and collec i e en anglemen pa e ns. Cohe ence p o ides he
algeb aic gua an ee: b aidings co espond o s a is ics ( e mionic/bosonic/anyonic), while he pen agon
ensu es ha changing pa en hesiza ion (g ouping o subsys ems) does no a ec obse ables. In ou
cons uc ion, he XC cons ain s a e implemen ed as na u al ans o ma ions in he monoidal ca ego y
gene a ed by
L
[
n
]; he exis ence o a cohe en b aiding ensu es ha he small se o spec al laws can
be en o ced consis en ly ac oss enso -p oduc decomposi ions ( agmen s, embedding, supe cells). This
emo es a i icial dependence on he o bi al basis o pa i ioning scheme and eplaces i wi h ca ego ical
in a iance.
Noncommu a i e geome y and ope a o algeb as: spec al da a as geome y.
DFT adi-
ionally ea s he densi y
n
(
)as a classical ield. Howe e , exchange and co ela ion a e inhe en ly
quan um and nonlocal. Noncommu a i e geome y e ames spaces ia hei algeb as o obse ables;
Connes’ spec al- iple pa adigm (
A,H, D
)encodes geome y in he spec um o an ope a o
D
and he
ep esen a ion o
A
[
434
]. Quan um g oups p o ide noncommu a i e unc ion algeb as on “quan um
spaces” (e.g., compac ma ix pseudog oups) [
51
]. Meanwhile, ope a o algeb as (C
∗
and on Neumann
algeb as) supply s a es Φ(posi i e linea unc ionals), spec al measu es, and
K
- heo e ic in a ian s ha
a e obus unde pe u ba ions and encode opology [
53
]. In ou XC cons uc ion, we selec an ope a o
L
[
n
](KS-like gene a o ), o m he C
∗
-algeb a C
∗
(
L
), pick an Ad-in a ian s a e Φ, and en o ce ela ions o
he o m Φ(
k
(
L
[
n
])) =
φk
o esol en /p ojec o p obes
k
. These spec al laws a e geome ic in a ian s
o he noncommu a i e space ep esen ed by C
∗
(
L
)and a e s able unde cohe en anspo (see §I and
Eq. (2) in oduced ea lie ).
Topological in o ma ion and quan um s a is ics.
B aided ca ego ies and quan um g oups p oduce
link and 3-mani old in a ian s (e.g., Jones polynomial, Reshe ikhin–Tu ae in a ian s) by unc o ial
e alua ion o ibbon diag ams [
54
,
55
]. Fo elec onic s uc u e, such opology appea s as sec o labels
(e.g., gapped subspace p ojec o s) and holonomy (Be y phases) along pa ame e loops (geome y, ields).
The ca ego ical pic u e ensu es ha hese sec o labels a e na u al and anspo ed consis en ly; in ou
DFT scheme hey a e en o ced by including Riesz p ojec o s among he p obes
k
, hus gua ding ac i e
spaces and p e en ing spu ious s a e swapping.
Cohe ence ⇒commu ing cons ain s ⇒ab-ini io XC.
Pu ing he pieces oge he : he Yang–
Bax e /RTT amewo k g an s a commu ing amily o in a ian s; b aided-monoidal cohe ence ensu es
hei compa ibili y ac oss decomposi ions; noncommu a i e geome y and ope a o algeb as u n hese
in a ian s in o spec al laws compu able om
L
[
n
]and Φ. Ins ead o i ing XC, we sol e o he
mul iplie s en o cing hese laws (c . Eq.
(2)
), he eby impo ing genuinely quan um in o ma ion (s a is ics,
en anglemen , opology) in o
xc
a Kohn–Sham cos . This is p ecisely he eplacemen o eg ession by
algeb aic cons ain ha unde lies he ab-ini io cha ac e o ou app oach.

6
C. Ou con ibu ions
We summa ize he echnical con ibu ions o his wo k. Each poin is s a ed p ecisely, connec ed o
he ma hema ical s uc u es in oduced in §IB, and accompanied by ope a ional de ails ha will be
de eloped in Sec ions I– and la e heo y/algo i hm sec ions. Th oughou , we make epea ed use o
esol en unc ional calculus and con ou in eg als in he sense o Ka o and o Reed–Simon [392, 427].
•A cons ained a ia ional o mula ion o XC using spec al laws and an Ad-in a ian
s a e Φ.
We o mula e he uni e sal unc ional
F
[
n
]as a Le y–Lieb cons ained sea ch augmen ed by
spec al-law cons ain s on a KS-like gene a o
L
[
n
]. Le
{ k}m
k=1
be a ixed se o esol en /p ojec o
p obes and Φan Ad-in a ian s a e on he C∗-algeb a C∗(L)gene a ed by L. We en o ce
Φ k(L[n])=φk, k = 1, . . . , m,
whe e he a ge s
{φk}
a e de e mined a p io i by exac physics (sum ules, i ial, uni o m limi s) o
anspo ed along exac legs o he highe connec ion. The XC ene gy is hus he unique unc ional
whose po en ial and ke nel sa is y hese nonpe u ba i e ope a o iden i ies. Ma hema ically,
he cons ain s a e commu ing hanks o he b aided-monoidal cohe ence and RTT/Yang–Bax e
s uc u e e iewed in §I B, and hey a e e alua ed by con ou - esol en calculus [392, 427].
•A cons uc i e exp ession o xc (and xc) ia esol en unc ional calculus.
S a iona i y
o he Lag angian wi h mul iplie s λkgi es he ope a o exp ession o he XC po en ial
xc( ) =
m
X
k=1
λkΦZΓk
k(z)(z−L)−1δL
δn( )(z−L)−1dz
2πi,(6)
which is he same o mula in oduced schema ically in Eq.
(2)
o he Execu i e Summa y bu now
emphasized as ou main cons uc i e esul . The s a ic esponse (GGA/me a-GGA coe icien s)
is ob ained om la ge-
|z|
expansions o (
z−L
)
−1
; he dynamic ke nel
xc
(
q, ω
) ollows om
he dynamical esol en and linea - esponse iden i ies (in p ac ice, wi hin he same nume ical
in as uc u e as DFPT and TDDFT [
58
]). This cons uc ion makes di ec con ac wi h ACFD/RPA-
ype objec s (bu wi hou hei diag amma ic cos ) and can be benchma ked agains GW-le el
sc eening [59, 60].
•HH0and K0in a ian s en o cing size-consis ency, o a ion in a iance, and sec o p o-
ec ion.
To ensu e s uc u e ha is usually added heu is ically, we compu e in a ian s in he
Ha o i–S allings quo ien
HH0
(C
∗
(
L
)) (which annihila es commu a o s, gua an eeing insensi i i y
o o bi al o a ions) and ack Riesz p ojec o s
PI
(
L
) o gapped subspaces, eco ding hei
K
- heo y
classes [
PI
]
∈K0
(C
∗
(
L
)). Du ing SCF, we include hese p ojec o s among he p obes
k
o gua d
sec o s:[
PI
]canno change unless a moni o ed gap closes, p e en ing spu ious s a e swaps in bond
b eaking, conical-in e sec ion neighbo hoods, and ansi ion-me al ligand- ield eo ganiza ions. This
eplaces empi ical b anching ules by algeb aic in a ian s and yields size-consis en ene ge ics by
cons uc ion. Nume ically, he Riesz p ojec o s a e ealized by con ou in eg a ion o he esol en ,
and hei s abili y p ope ies ollow om s anda d pe u ba ion heo y [427].
•A ini e a las o uni e sal “ iles” (in eg able backbones) wi h highe -gauge anspo —
no e i s.
We assemble a ini e a las o uni e sal iles (HEG/me allic, co alen molecula , SCE/Mo ,
1D-co ela ed, mul i-o bi al TM), each ile speci ying: (i) a gene a o empla e
L
[
n
]app op ia e o
he egime, (ii) a p obe se
{ k}
, and (iii) he exac a ge s
{φk}
de i ed om analy ic seeds o
small in eg able e e ences. Once compu ed, a ile’s in a ian s a e anspo ed ac oss geome y/-
composi ion/ ields by pa allel anspo along he highe connec ion; de ia ions a e con olled by
he cu a u e magni ude. This eplaces pe -sys em e i ing wi h algeb aic anspo , in he same
spi i as anspo ing densi y- esponse da a ac oss pa ame e space in DFPT [58], bu now a he
le el o ope a o in a ian s. The a las is ini e ye co e s chemis y b oadly; i is e ined adap i ely
when cu a u e indica es he need o a new cha .
•Complexi y and pe o mance: KS-class cos ; la ge speed-ups o ab-ini io accu acy.
Ou implemen a ion s ays in he Kohn–Sham cos class: he domina ing s eps a e he same
diagonaliza ions/linea -sol es as in s anda d DFT [
61
]. En o cing
m.
6spec al laws adds a hand ul
o shi ed linea sol es ( esol en s), e icien ly handled by K ylo /GMRES wi h shi -in a iance
7
o selec ed in e sion echniques [
63
,
346
]. Hence he pe -SCF o e head is ypically 10–30% o e
GGA. In con as , s a e-o - he-a ab-ini io me hods (e.g., CCSD(T) and GW/BSE) scale as
O
(
N7
)
and
O
(
N4
)–
O
(
N5
) espec i ely [
59
,
60
,
64
]. Consequen ly, ime- o-accu acy imp o es by o de s o
magni ude: we app oach CC/GW accu acy a (nea -)DFT wall- imes by sol ing cons ain s a he
han e alua ing many-body diag ams.
Na a i e link o he es o he pape .
Sec ion I se he s age: §IA e iewed wha DFT does
well and whe e i al e s; §I B mo i a ed in eg abili y and ca ego ical cohe ence as he igh abs ac ions
o quan um cons ain s. The i e con ibu ions abo e ope a ionalize ha mo i a ion: Eq.
(6)
is he
compu a ional engine;
HH0
/
K0
in a ian s supply non-nego iable s uc u e; he a las and highe -gauge
anspo deli e ans e abili y wi hou e i s; and he complexi y discussion explains why we can each
ab-ini io accu acy a KS cos . The emainde o he manusc ip de elops each i em igo ously, hen
implemen s and benchma ks i .
II. BACKGROUND & PRELIMINARIES
A. Hohenbe g–Kohn, Le y–Lieb, and Kohn–Sham; exac cons ain s (sum ules, i ial, scaling)
O e iew.
This subsec ion ecalls he ounda ional a ia ional s uc u e o g ound–s a e DFT and
assembles he exac cons ain s ha any admissible exchange–co ela ion (XC) unc ional mus sa is y.
We p oceed in h ee s eps: (i) he Hohenbe g–Kohn (HK) mapping and a ia ional p inciple (wi h
ensemble gene aliza ions); (ii) he Le y–Lieb cons ained sea ch ha pu s he uni e sal unc ional
F
[
n
]on
a igo ous oo ing and cla i ies i s con exi y; (iii) he Kohn–Sham (KS) cons uc ion and Eule equa ion.
We hen collec exac cons ain s—coo dina e scaling, i ial iden i ies, exchange–co ela ion hole sum
ules, spin scaling, Lieb–Ox o d bounds, and s a iona i y ela ions (Janak’s heo em)—which oge he
o m he “non–nego iable” condi ions ha ou spec al–law app oach (Sec ions I B and IC) will en o ce
in ope a o o m.
Hamil onian and admissible s a es.
Fo a sys em o
N
elec ons in an ex e nal po en ial
ex
(
), he
non ela i is ic elec onic Hamil onian is
ˆ
H=ˆ
T+ˆ
W+ˆ
Vex =−1
2
N
X
i=1 ∇2
i+X
1≤i<j≤N
1
| i− j|+
N
X
i=1
ex ( i).(7)
The g ound s a e ene gy is
E0
=
in Ψh
Ψ
|ˆ
H|
Ψ
i
o e no malized an isymme ic Ψ. The g ound–s a e
one–body densi y is
n
(
) =
NPσR|
Ψ(
σ, x2,...,xN
)
|2dx2···dxN
. We w i e
7→ n
o he (HK) map
om admissible ex e nal po en ials o hei g ound–s a e densi ies; when degene acy occu s, one uses
ensemble densi ies.
HK one– o–one map and HK a ia ional p inciple.
The i s HK heo em asse s injec i i y (up
o an addi i e cons an ) o he map
7→ n
o nondegene a e g ound s a es; he second HK heo em is a
a ia ional p inciple:
E0= in
nnE [n]o, E [n] := F[n] + Z ex ( )n( )d ,(8)
whe e
F
[
n
]is a uni e sal (sys em–independen ) unc ional. While he o iginal p oo s we e gi en in [
1
,
2
],
we will ely on he mo e gene al cons ained–sea ch amewo k below o exis ence and con exi y.
Le y–Lieb cons ained sea ch.
The uni e sal unc ional can be de ined wi hou e e ence o any
as
a cons ained in imum o e wa e unc ions yielding n:
F[n] := in
Ψ→nΨˆ
T+ˆ
WΨ.(9)
The in imum anges o e all
N
– e mion Ψ(o mixed s a es in he ensemble ex ension) whose densi y
equals
n
. This de ini ion es ablishes
F
[
n
]on he se o
N
– ep esen able densi ies and yields key p ope ies:
con exi y in
n
, lowe semicon inui y, and exis ence o minimizing sequences. Toge he wi h
(8)
,
(9)
implies
he cons ained–sea ch e sion o he a ia ional p inciple:
E0
=
in n{F
[
n
] +
R n}.
In p ac ice, he KS
cons uc ion below eco e s a ac able minimiza ion by eplacing he in e ac ing kine ic ene gy wi h i s
nonin e ac ing coun e pa a ixed n.
8
Kohn–Sham decomposi ion and Eule equa ion.
De ine he nonin e ac ing kine ic ene gy
Ts
[
n
] :=
in Φ→nhΦ|ˆ
T|Φiwi h he in imum o e Sla e de e minan s Φyielding n. Decompose
F[n] = Ts[n] + EH[n] + Exc[n], EH[n] = 1
2ZZ n( )n( 0)
| − 0|d d 0.(10)
S a iona i y a ixed pa icle numbe (Rn=N) gi es he Eule –Lag ange equa ion
δTs
δn( )+ ex ( ) + H( ) + xc( ) = µ, H( ) = Zn( 0)
| − 0|d 0, xc := δExc
δn .(11)
Kohn–Sham o bi als {ψi}a e in oduced so ha n( ) = Pocc
i|ψi( )|2, wi h he o bi als sa is ying
−1
2∇2+ e ( )ψi( ) = εiψi( ), e = ex + H+ xc.(12)
The g ound s a e is ound sel –consis en ly. The o bi al s a iona i y implies Janak’s heo em (see below)
ela ing de i a i es o he o al ene gy wi h espec o o bi al occupa ions o he KS eigen alues.
Exac cons ain s on F
[
n
]
and Exc
[
n
]
.
Any candida e
Exc
used in
(10)
mus sa is y a se o exac
condi ions. We collec hose mos ele an o ou subsequen spec al–law cons uc ion.
(a) Coo dina e scaling and coupling–cons an scaling. Fo uni o m coo dina e scaling
nγ
(
) =
γ3n
(
γ
)
wi h γ > 0, one has he exac scaling laws (Coulomb sys ems)
Ts[nγ] = γ2Ts[n], Ex[nγ] = γ Ex[n], Ec[nγ] = γ2E1/γ
c[n],(13)
whe e
Eλ
c
[
n
]deno es he co ela ion ene gy a in e ac ion s eng h
λ
(adiaba ic connec ion). Di e en ia ion
o
(13)
a
γ
= 1 yields i ial– ype ela ions o
Ts
,
Ex
, and
Ec
and gi es powe ul shape cons ain s
on admissible app oxima ions [
65
]. In pa icula ,
Ex
is homogeneous o deg ee one, whe eas
Ts
is
homogeneous o deg ee wo.
(b) Adiaba ic connec ion and ACFD exp ession. In oduce he
λ
–in e ac ing sys em wi h Hamil onian
ˆ
Hλ
=
ˆ
T
+
λˆ
W
+
ˆ
Vλ
adjus ed so ha i s g ound–s a e densi y equals
n
. The adiaba ic connec ion o mula
hen eads
Exc[n] = Z1
0
dλ Wλ[n], Wλ[n] := hΨλ[n]|ˆ
W|Ψλ[n]i−EH[n].(14)
Equi alen ly, in e ms o he densi y– esponse unc ion
χλ
, one ob ains he (ze o– empe a u e) ACFD
ep esen a ion
Exc[n] = −1
2πZ1
0
dλ Z∞
0
dω T nχλ(iω)−χ0(iω) co, c( , 0) = 1
| − 0|,(15)
which exposes exac sum ules (e.g. he
–sum) and mo i a es many–body sc eening app oxima ions
[258]. Iden i ies (138)–(15) unde lie ou use o spec al esol en s in Sec ion IC.
(c) Vi ial ela ions. Fo ex e nal Coulomb po en ials, he in e ac ing i ial heo em implies an XC
i ial iden i y ha can be w i en compac ly using scaling de i a i es:
Exc[n] = d
dγ γ2E1/γ
c[n]γ=1
+d
dγ γEx[n]γ=1
.(16)
Equi alen in eg al o ms ela e
Exc
o he po en ial
xc
h ough
Rd n
(
)
·∇ xc
(
); such ela ions will
be au oma ically espec ed when xc is cons uc ed om spec al laws consis en wi h (13)–(15) [65].
(d) Exchange–co ela ion hole sum ules. De ine he pai densi y
n(2)
(
, 0
)and he exchange–
co ela ion (xc) hole
nxc
(
, 0
) ia
n(2)
(
, 0
) =
n
(
)
n
(
0
)[1 +
hxc
(
, 0
)] wi h
nxc
=
n hxc.
Then he exac
sum ule
Znxc( , 0)d 0=−1(17)
holds o all
; he exchange hole in eg a es o
−
1and he co ela ion hole in eg a es o 0. The wa e–
ec o analysis o
Exc
connec s
(17)
o he small–
q
beha io o he s uc u e ac o and o exac esponse
cons ain s [67, 254, 258]. Ou ope a o cons ain s will mi o (17) a he le el o spec al p ojec o s.
9
(e) Spin scaling o exchange. Fo spin densi ies n↑, n↓,
Ex[n↑, n↓] = 1
2Ex[2n↑] + Ex[2n↓],(18)
an exac iden i y ha any spin–pola ized unc ional mus obey. Co ela ion sa is ies addi ional (less
es ic i e) spin cons ain s. Equa ion
(18)
will be p ese ed au oma ically by ou spec al–law cons ain s
when he p obes espec spin block s uc u e [69, 70].
( ) Lieb–Ox o d lowe bound. Fo Coulomb in e ac ions one has he igo ous lowe bound
Exc[n]≥ −CLO Zn( )4/3d , CLO ≈1.68,(19)
which cons ains he magni ude o he indi ec Coulomb ene gy and p e en s pa hological o e binding
[
71
]. Ou cons uc ion can en o ce
(19)
ei he by explici checks on he spec al–law mul iplie s o by
including p obes ha con ol he high–densi y (γ→∞) scaling.
(g) S a iona i y wi h espec o o bi al occupa ions (Janak’s heo em). Fo ac ional occupancies
0≤ i≤1wi h Pi i=N, a a s a iona y solu ion one has
∂E
∂ i
=εi,(20)
which jus i ies e alua ing de i a i e gaps om KS eigen alues and unde pins p ac ical algo i hms ha
adjus occupa ions in me als and open–shell sys ems [
72
]. Ou spec al–law SCF (Sec ion IC) p ese es
(20)
because he cons ain mul iplie s en e addi i ely h ough
xc
and do no b eak he a ia ional
s uc u e.
(h) Kine ic–ene gy inequali ies. The nonin e ac ing kine ic ene gy obeys
Ts[n]≥TW[n] := 1
8Z|∇n( )|2
n( )d ,(21)
he on Weizsäcke lowe bound, sa u a ed o single–o bi al ( wo–elec on, spin–pai ed) sys ems [
73
].
While no an XC cons ain pe se,
(21)
is help ul in bounding
Ts
when we cons uc gene a o empla es
L[n]consis en wi h he KS kinema ics.
Rema ks o he ca ego ical cons uc ion.
Cons ain s
(13)
–
(21)
will eappea in ope a o o m as
spec al laws o a KS–like gene a o
L
[
n
]e alua ed unde an Ad–in a ian s a e Φ, c . Sec ion IC. In
pa icula , he ACFD iden i y
(15)
ansla es o esol en iden i ies o
L
; he hole sum ule
(17)
becomes
a ace condi ion on spec al p ojec o s; and he scaling ela ions
(13)
become homogenei y cons ain s
on he mul iplie s en o cing ou con ou p obes. This eph asing is wha allows us o compu e
xc
by
sol ing algeb aic cons ain s, a he han i ing pa ame e s.
B. Lax pai s, in eg abili y, and beyond isospec ali y
Pu pose.
We summa ize he classical and quan um in eg abili y s uc u es ha unde pin ou spec al–
law cons ain s o exchange–co ela ion (XC). We begin wi h classical Lax pai s, monod omy and ans e
ma ices, hen pass o quan um in eg abili y ia
R
–ma ices and quasi– iangula Hop algeb as. Finally,
we o mula e a beyond–isospec al ex ension using highe connec ions (
A,B
)and a cu a u e spli in o
cen al s. non–cen al pa s. This p o ides he geome ic o igin o he “exac legs” ( la di ec ions o
in a ian s) and he con olled co ec ions we la e use in cons uc ing xc.
Classical Lax pai s and monod omy.
A classical e olu ion equa ion is in eg able i i admi s ope a o s
L(u, )and M(u, )(Lax pai ) depending on a spec al pa ame e usuch ha
∂ L(u, ) = [M(u, ), L(u, )].(22)
Equa ion
(22)
implies ha he spec um o
L
is conse ed (isospec al low), so spec al in a ian s (e.g.,
Lk
o spec al cu es) a e in eg als o mo ion. The a che ypes a e he KdV and NLS hie a chies
[
75
–
77
]. In 1
+
1dimensions, one assembles he monod omy ma ix by pa h–o de ed exponen ia ion o a
Lax connec ion Lx(u;x)along he spa ial line,
T(u; ) = −→
PexpZLx(u;x, )dx, ∂ T(u; ) = L (u; +∞, )−L (u;−∞, )T(u; ),(23)
16
In ini e dimension, he no malized ace
/dim
is Ad-in a ian ; mo e gene ally, Ad-in a iance can be
en o ced by uni a y a e aging: gi en any s a e Φ
0
, se Φ(
a
) =
RU
Φ
0
(
uau†
)
dµ
(
u
)wi h
µ
he Haa measu e
on a compac uni a y subg oup ha ac s ansi i ely on he ele an subspace [
359
]. Ad-in a iance
gua an ees ha spec al da a Φ(
(
L
)) a e insensi i e o o bi al o a ions and in e nal mixing, which is
essen ial o size consis ency and o a ional in a iance o he XC cons uc ion.
Canonical choices o Φ.We use h ee p ac ical inca na ions.
(a) No malized ( ibe ) ace a
T=
0.Fo ini e clus e s o ini e-dimensional KS subspaces one se s
Φ
(
A
) =
1
NT A
, wi h
N
he one-pa icle dimension. Fo pe iodic sys ems wi h Bloch ibe s
L
(
k
), we
ake he ace pe cell ( ibe ace):
Φpe (A) = 1
|BZ|ZBZ
1
No b
A(k)dk,(54)
whe e
No b
coun s bands in he wo king window and
is he ini e ma ix ace on he ibe . Bo h
(54)
and he ini e ace a e Ad-in a ian by cyclici y.
(b) KMS s a es a ini e empe a u e. Le
α
(
a
) =
ei Hae−i H
be he Heisenbe g dynamics o an
e ec i e one-body Hamil onian
H
gene a ing ime e olu ion on
A
( ypically he cu en KS gene a o ).
A s a e Φ
β
sa is ies he KMS condi ion a in e se empe a u e
β>
0i , o all
a, b ∈A
, he unc ion
Fa,b
(
) = Φ
βa α
(
b
)

ex ends o a s ip analy ic unc ion obeying
Fa,b
(
+i
β
) = Φ
βα
(
b
)
a
[
127
,
128
,
406
].
In ini e dimension,
Φβ(a) = T (e−βH a)
T (e−βH ),(55)
which is Ad-in a ian when
α
is inne . G and-canonical a ian s include a chemical po en ial
µ
ia
H→H−µN
. KMS s a es le us ex end spec al laws and hei anspo o ini e-
T
and open-sys em
con ex s while main aining posi i i y and no maliza ion.
(c) Singula aces (Dixmie ace) o pseudodi e en ial ails. When semi-classical/pseudodi e en ial
asymp o ics p oduce bo de line (loga i hmic) di e gences, he Dixmie ace
T ω
can se e as a gene alized,
Ad-in a ian ace on ce ain ideals, eco e ing physically ele an densi ies (e.g. Che n cha ac e s) [
129
].
While no needed in s anda d KS p ac ice, his op ion suppo s ou symbol-le el cons ain s o ough
po en ials o high-o de g adien co ec ions.
P ojec ion o HH0and killing commu a o s.
Recall
HH0
(
A
) =
A/
[
A,A
]. Any ace-like unc ional
ac o s h ough he quo ien : he e exis s a unique
Φ
:
HH0
(
A
)
→C
such ha Φ =
Φ◦π
, whe e
π:A→HH0(A)is he canonical p ojec ion. Equi alen ly,
Φ([a, b]) = 0 (∀a, b ∈A).(56)
In p ac ice we compu e in a ian s as
S
[
n
] =
Φ
[
(
L
[
n
])]

, so ha any uni a y mixing wi hin degene a e
subspaces o block o a ions in a composi e sys em lea e
S
unchanged. This is he algeb aic eason why
ou cons ained XC is size-consis en and o a ion-in a ian by cons uc ion, a he han by pos -hoc
co ec ions. The compa ibili y o
HH0
wi h
K0
ia he Ha o i–S allings pai ing will be used la e o
en o ce sec o s abili y (see §IIC) [389].
Di e en ia ion o spec al laws: double ope a o in eg als.
Fo F éche -smoo h
L7→
(
L
)
(bounded Bo el
wi h sui able smoo hness), a ia ions can be exp essed by Bi man–Solomyak’s double
ope a o in eg al (DOI) calculus:
δ (L) = ZZR2
(λ)− (µ)
λ−µdEL(λ)(δL)dEL(µ),(57)
whe e ELis he spec al measu e o L[131, 132]. Applying Φand using cyclici y (Ad-in a iance) gi es
δΦ (L)=ZZ (λ)− (µ)
λ−µΦdEL(λ)(δL)dEL(µ).(58)
Fo esol en p obes
z
(
λ
)=(
λ−z
)
−1
,
(57)
educes o he amilia con ou sandwich used in Eq.
(6)
.
Fo mulas
(57)
–
(58)
jus i y ou New on/B oyden upda es o he mul iplie s en o cing Φ(
k
(
L
)) =
φk
in
he SCF loop.

17
G ound-s a e s ini e-Tand pe iodic ealiza ions.
In a clus e a
T=
0, ake Φ=Φ
; in a
pe iodic c ys al, use he ibe ace
(54)
and he Bloch ibe decomposi ion o
L
o e alua e Φ(
(
L
)) as
B illouin-zone in eg als. A ini e empe a u e o in weak con ac wi h a ba h, choose he KMS s a e
(214)
. In all cases, Ad-in a iance and he
HH0
p ojec ion gua an ee ha
S
is ep esen a ion-independen ;
o pe iodic sys ems, his includes independence wi h espec o he choice o smoo h gauge o Bloch
s a es. These p ope ies can be aced back o ace-pe - olume cons uc ions and he gene al spec al
heo y o di ec in eg als [133].
Consequences o xc and xc.
Wi h Φ ixed ( ace, ibe ace, o KMS), he only sys em-dependen
ing edien en e ing he XC po en ial is
δL/δn
, c . Eq.
(6)
. Thus,
xc
is compu ed by sol ing he spec al
cons ain s Φ(
k
(
L
)) =
φk
a each SCF s ep; he e a e no adjus able pa ame e s. Because Φ ac o s
h ough
HH0
, he cons ain s a e immune o o bi al mixing and agmen e-pa i ioning, and because
Φis Ad-in a ian , hey a e compa ible wi h he highe -ca ego ical anspo discussed in §II B. In he
ime-dependen se ing, eplacing
by dynamical esol en s yields
xc
(
q, ω
)wi h exac sum ules and
causali y p ese ed by he DOI/con ou calculus.
C. Spec al laws: bounded Bo el unc ional calculus and S [n] = Φ (L[n])
Objec i e.
We o malize he spec al laws ha se e as algeb aic cons ain s in ou cons uc ion o he
exchange–co ela ion po en ial. Gi en he KS–like gene a o
L
[
n
]speci ied in §III A and an Ad–in a ian
s a e Φ om §IIIB, we de ine, o any bounded Bo el unc ion :R→C,
S [n] := Φ (L[n]),(59)
whe e
(
L
)is unde s ood in he sense o he bounded Bo el unc ional calculus u nished by he spec al
heo em. We collec he unc ional–analy ic ac s ensu ing ha
(59)
is well posed, s a e i s measu e–
heo e ic ep esen a ion, eco d s abili y and di e en iabili y p ope ies needed la e , and lis canonical
p obe unc ions used in p ac ice.
Bounded Bo el unc ional calculus.
Le
L
be a sel –adjoin ope a o on a Hilbe space
H
. The e
exis s a unique p ojec ion– alued measu e (PVM) EL(·)on (R,B(R)) such ha
L=ZR
λ dEL(λ).
Fo any bounded Bo el unc ion ∈ Bb(R), he bounded Bo el unc ional calculus se s
(L) := ZR
(λ)dEL(λ),(60)
and enjoys he p ope ies: (i)
k
(
L
)
k≤k k∞
; (ii) i is a
C∗
–algeb a homomo phism
7→
(
L
)modulo
equali y almos e e ywhe e; (iii) spec al mapping
σ
(
(
L
)) =
(σ(L))
; (i ) o con inuous
,
(
L
)
coincides wi h he con inuous unc ional calculus, and he assignmen is con inuous in he sup no m [
348
,
Ch. IX], [
135
, Ch. X], [
360
, Ch. 3]. Con ou /Hel e –Sjös and ep esen a ions p o ide p ac ical o mulas:
i is smoo h wi h compac suppo (o admi s an almos –analy ic ex ension), hen
(L) = 1
πZC
¯
∂e
(z)(z−L)−1d2z, (61)
wi h
¯
∂e
suppo ed nea he eal axis and
d2z
Lebesgue measu e on
C
[
137
]. Fo a ional
(e.g. esol en s),
Dun o d–Riesz con ou in eg als apply.
Spec al measu es induced by
Φ
.
Fix an Ad–in a ian s a e Φon
A
= C
∗
(
L
). De ine he ini e
posi i e Bo el measu e νnon Rby
νn(B) := ΦEL[n](B), B ∈ B(R).(62)
By cons uc ion
νn
(
R
) = Φ(1) = 1 i Φis no malized; hus
νn
is a p obabili y measu e. Then
(60)
and
linea i y yield he spec al–law ep esen a ion
S [n] = Φ (L[n])=ZR
(λ)dνn(λ),(63)
18
which is he cen al objec we en o ce as a cons ain . Equa ion
(63)
shows ha he map
7→ S
[
n
]is
a posi i e no malized linea unc ional on
Bb
(
R
), i.e. a s a e on he commu a i e
C∗
–algeb a
L∞
(
νn
)
[
138
, Ch. 7]. In pa icula , mono one sequences
k↓
imply
S k
[
n
]
↓ S
[
n
]by mono one con e gence,
and app oxima ion o indica o s by con inuous unc ions allows us o ea Riesz p ojec o s and spec al
windows as legi ima e p obes.
Basic iden i ies and in a iances. F om (63) one immedia ely ob ains:
S1[n]=1,S [n] = S [n],S| |2[n]≥0,|S [n]|≤k k∞,(64)
S +g[n] = S [n] + Sg[n],Sc [n] = cS [n], c ∈C,
(Ad–in a iance) S [n]is unchanged unde L7→ ULU†,Φ7→ Φ◦AdU†.
I
=
1I
is he indica o o a Bo el se
I
, hen
S1I
[
n
] = Φ(
PI
(
L
[
n
])) e u ns he Φ–weigh o he spec al
island
I
(e.g. an occupied band coun pe cell wi h he ibe ace). These p ope ies u nish he algeb aic
le e s used la e o sec o p o ec ion and size consis ency.
Canonical p obe amilies. In compu a ions we employ a small, uni e sal menu o p obe unc ions :
•
Resol en s (S iel jes ans o ms):
z
(
λ
) = (
λ−z
)
−1
wi h
dis
(
z, σ
(
L
))
>
0. Momen s o he esol en
con ol esponse and yield apidly con e gen quad a u e by a ional app oximan s [
361
, Ch. 1].
Fini e collec ions
{zk}
sample di e en spec al egions and connec di ec ly o g adien expansions
ia he symbol calculus in §III A.
•
Smoo hed p ojec o s:
(
λ
) =
1
2
1 +
anh
((
λ−λ0
)
/η
)

o Fe mi–Di ac
(
λ
) = (1 +
eβ(λ−µ)
)
−1
. As
η→
0(o
β→ ∞
), hese con e ge o indica o s o hal –lines; hey egula ize Riesz p ojec o s and
a e amenable o con ou in eg als (61).
•
Low–o de polynomials and a ional unc ions:
(
λ
) =
λk
o bounded spec a ( ini e windows), o
bes a ional/chebyshe app oximan s on an in e al enclosing
σ
(
L
)[
140
]. These cap u e i ial/scale
momen s and enable e icien s ochas ic ace es ima ion o S o la ge sys ems [141, 142].
•
Spec al windows:
(
λ
) =
1I
(
λ
) o a union o disjoin in e als
I
; compu ed ia Riesz in eg als o
smoo hed app oximan s; hese a e he building blocks o K0sec o gua ds.
Di e en ia ion and anspo .
Va ia ions wi h espec o he densi y en e h ough
L
=
L
[
n
]. Fo
su icien ly egula (e.g. smoo h compac ly suppo ed, o esol en s), he F éche de i a i e obeys
δS [n] = ΦZΓ
(z)(z−L)−1(δL) (z−L)−1dz
2πi,(65)
which is he specializa ion o he gene al a ia ion o mulas al eady used o de i e Eq.
(6)
. Along a
pa ame e pa h
θ7→ n
(
θ
)in he mani old Θ, he highe connec ion (
A,B
)o §II B yields he co a ian
de i a i e DθL. Wi h he cen al cu a u e anishing (an exac leg), Ad–in a iance and cyclici y imply
DθS [n(θ)] = 0 o all admissible , (66)
so spec al laws a e p ese ed exac ly; his is he ma hema ical basis o anspo ing a ge alues
φk
om
a las iles o ealis ic sys ems wi hou e i s. Fo small non–cen al cu a u e, (65) quan i ies con olled
de ia ions, which eed p incipled g adien and dynamic co ec ions o xc.
Momen p oblems and iden i iabili y.
Because
S
[
n
] =
R dνn
, selec ing a ini e se
{ k}m
k=1
amoun s o cons aining ini ely many gene alized momen s o
νn
. The classical Hambu ge /S iel jes
momen p oblems p o ide condi ions o uniqueness o a measu e om i s momen s (e.g. Ca leman’s
condi ion) [
143
]. In ou con ex , uniqueness o
νn
is unnecessa y; we only equi e ha he chosen
{ k}
ix he physically ele an low–o de in o ma ion (comp essibili ies, p ojec o weigh s, i ial momen s).
App oxima ion heo y (Chebyshe / a ional) gua an ees exponen ial con e gence o
S N
[
n
]
→ S
[
n
] o
analy ic on neighbo hoods o σ(L)[140].
Nume ical e alua ion o S .
Th ee amilies o algo i hms e alua e
S
[
n
]e icien ly a KS cos : (i)
Con ou quad a u e o
(61)
o Dun o d–Riesz wi h shi ed linea sol es (same ke nels as DFPT); (ii)
Polynomial/ a ional expansions (Chebyshe , Zolo a e ) applied o
(
L
) ollowed by ace pe cell o
19
s ochas ic ace es ima o s [
140
,
141
,
361
]; (iii) Lanczos quad a u e o bilinea o ms
u†
(
L
)
u
a e aged
o e andom ec o s o app oxima e Φ(
·
)(SLQ) [
142
]. All h ee exploi ha he o e head scales wi h he
numbe o p obes
m
( ypically
m≤
6), so en o cing spec al laws adds only a modes cons an o he KS
cos .
Summa y.
The bounded Bo el unc ional calculus assigns a ma hema ically igo ous meaning o
(
L
)
o any bounded Bo el
; pai ing wi h an Ad–in a ian s a e Φyields spec al laws
S
[
n
] ha a e linea ,
posi i e, basis–independen , and ealized as in eg als agains a Φ–induced spec al measu e
νn
. Exac –leg
anspo p ese es hese laws; con olled de ia ions quan i y cu a u e co ec ions. A small, uni e sal
se o esol en /p ojec o /smoo h–window p obes su ices o encode he exac cons ain s ele an o XC
while keeping nume ical cos in he KS class.
D. In a iance along exac legs: DΘS
= 0
when cen al cu a u e anishes; small–cu a u e d i
bounds
Goal.
We show ha he spec al laws
S
[
n
] = Φ

(
L
[
n
])

de ined in §IIIC a e pa allel– anspo
in a ian s along pa ame e pa hs whose cen al highe cu a u e anishes, and we de i e quan i a i e
d i bounds when a small non–cen al cu a u e is p esen . This o malizes he no ion o exac legs hin ed
a in §II B and p o ides he e o ba s used la e o p opaga e a las a ge s
{φk}
ac oss chemis y wi hou
e i s.
Se up: highe connec ion, cu a u e spli , and co a ian de i a i e.
Le Θbe he pa ame e
mani old (geome y, ields, composi ion, empe a u e). On he
C∗
–algeb a bundle
θ7→ Aθ
= C
∗L
(
θ
)

choose a highe (2–)connec ion (
A,B
)wi h 2–cu a u e (
F,G
); ecall he decomposi ion in o cen al s.
non–cen al pa s wi h espec o he algeb a cen e Z(Aθ),
F=Fcen +Fncen ,G=Gcen +Gncen .
Fo a pa h γ: 7→θ( )we w i e he co a ian de i a i e on obse ables a∈Aθ( )as
D a:= ∂ a+ [A , a],(67)
and we assume he Lax– ype e olu ion o he gene a o
D L= [M , L] + R ,(68)
whe e he non–isospec al e m
R
is buil om he 2– o m componen (schema ically
R
=
ι˙γB
+
highe commu a o s). We say γis an exac leg i
Fcen |γ= 0 and Gcen |γ= 0,=⇒ R ≡0.(69)
Classically,
(67)
encodes pa allel anspo in he sense o Eh esmann/Kobayashi–Nomizu [
146
,
358
]; he
holonomy gene a ed by Ais go e ned by he (1– o m) cu a u e F[371].
Main in a iance heo em.
Le Φbe an Ad–in a ian s a e ( ace, ibe ace, o KMS in he ze o–
cu a u e limi ) as in §III B. Fo any bounded Bo el , de ine S ( ) := Φ (L(θ( ))).
Theo em III.1 (In a iance along exac legs).I γis an exac leg in he sense o (69), hen
d
d S ( ) = 0 o all admissible p obes . (70)
P oo (ske ch). Fo smoo h (o esol en s), Hel e –Sjös and/Dun o d gi es
δ (L) = 1
2πi ZΓ
(z)(z−L)−1(δL) (z−L)−1dz.
Hence ˙
S ( )=Φ1
2πi RΓ (z)R(z)D L R(z)dzwi h R(z)=(z−L)−1. Using (68) and Ad–in a iance,
ΦZ (z)R(z)[M , L]R(z)dz= 0
by cyclici y, while R ≡0along an exac leg by (69). The e o e ˙
S = 0.
20
Special case: p ojec o weigh s and sec o s abili y.
Le
I⊂R
be a spec al island isola ed by a
gap and
PI
(
L
)i s Riesz p ojec o . Assume Φis acial ( ini e ace o ibe ace). Then, e en o gene al
(non–exac ) smoo h de o ma ions ha do no close he gap, ΦPI(L(θ))is cons an :
d
d ΦPI(L(θ( )))= 0 as long as he gap a ound Idoes no close. (71)
Indeed, di e en ia ing
P2
=
P
yields
˙
P
=
P˙
P
+
˙
P P
and hence
T
(
˙
P
) =
T
(
P˙
P
)+
T
(
˙
P P
) = 2
T
(
P˙
P
) = 0
by cyclici y; he same holds o ace pe cell. Equa ion
(71)
exp esses he K– heo e ic s abili y o sec o
labels unde gap–p ese ing de o ma ions (c . §IIC); a o mal discussion is gi en in s anda d spec al
app oxima ion ex s [399].
D i bounds o small non–cen al cu a u e.
When
R 6
= 0 bu small, one ob ains quan i-
a i e con ol o spec al–law d i . Le Γbe a con ou enclosing
σ
(
L
(
θ
(
))) wi h dis ance
δ
(
) =
dis
(Γ
, σ
(
L
(
θ
(
))))
>
0. By he esol en bound
kR
(
z
)
k ≤
1
/δ
(
) o
z∈
Γ(s anda d unc ional analysis;
see [148, Ch. 7], [149, Thm. 5.8]), and kΦk=1, we ge
˙
S ( )≤1
2π`(Γ) sup
z∈Γ
| (z)|
δ( )2
R 
=: C (Γ, )
R 
.(72)
In eg a ing along ∈[0,1] yields
S (1) −S (0)≤Z1
0
C (Γ, )
R 
d . (73)
I C (Γ, )≤C and kR k ≤ εuni o mly, we ob ain he concise e o ba
∆S ≤C ε L(γ),(74)
wi h
L
(
γ
) he pa ame e – ime leng h o he pa h. In coupled e olu ions whe e
R
i sel depends
Lipschi z–con inuously on
S
,
(73)
li s o a G önwall– ype bound ( a ia ion–o –cons an s es ima e) [
150
,
Ch. 5].
P ojec o case (gap ideli y). Fo
=
1I
(Riesz p ojec o wi h Γ =
∂I
),
|
(
z
)
| ≡
1and
δ
(
)is he gap
o ∂I, hence
˙
S1I( )≤`(∂I)
2πkR k
δ( )2.(75)
As long as he gap s ays open and
kR k
s ays bounded, he pe –cell p ojec o weigh is a ia ion–s able;
in he acial case i is ac ually cons an (c .
(71)
). When a gap closes,
δ
(
)
↓
0and con olled jumps can
occu —p ecisely he quan ized sec o changes discussed wi h K– heo y in §IIC.
Holonomy, loops, and mapping– o us iewpoin .
Fo a closed loop
γ
in Θ, he holonomy o he
connec ion
A
is a uni a y
Uγ
in he holonomy g oup (Amb ose–Singe heo em) [
358
,
371
]. I he cen al
cu a u e anishes along
γ
, hen
S
is s ic ly single– alued by Theo em III.1. In gene al, he de ia ion
∆
S
is bounded by
(73)
and can be in e p e ed in he mapping– o us algeb a as a class measu ed by he
non–cen al cu a u e lux (c . §II C). This gi es a opologically meaning ul accoun ing o how much a
loop can “ wis ” he spec al laws.
Consequences o he XC algo i hm.
Equa ions
(70)
–
(74)
jus i y ou wo k low: (i) choose
{φk}
a
a basepoin (an a las ile); (ii) anspo along an exac leg whene e possible so he cons ain s emain
exac wi hou e i s; (iii) when small non–cen al cu a u e is una oidable ( ini e empe a u e, weak
dissipa ion, embeddings), use
(73)
–
(74)
o a ach igo ous e o ba s o he en o ced spec al laws; (i )
moni o p ojec o gaps and use
(71)
–
(183)
o keep sec o s s able and o de ec genuine opological e en s.
Rema ks on egula i y and nume ics.
The cons an s
C
(Γ
,
)depend only on esol en sepa a ion
and on he chosen p obes; hey a e easy o es ima e in p ac ice (known con ou s, gap es ima es, and
spec al windows). Resol en no ms and pa h–leng h ac o s a e a ailable om he same shi ed–linea –
sol e in as uc u e used in DFPT; ma ix–analysis iden i ies ( esol en iden i y, submul iplica i i y)
con ol he accumula ion o e o along composi e pa hs [349, Chs. V–VI]. Spec al p ojec o cons ancy
unde gap p ese a ion is well co e ed in spec al app oxima ion heo y [399].
21
E. Sec o iza ion wi h K0: Riesz p ojec o s, gap condi ions, in a iance un il gap closu e, and
quan ized jumps
Pu pose.
We o malize sec o iza ion: he decomposi ion o he one–pa icle space in o spec ally isola ed
subspaces ( alence, ac i e, conduc ion; spin/la ice sec o s) acked by Riesz p ojec o s and s abilized by
K
– heo y. Conc e ely, gi en a KS–like gene a o
L
[
n
](see §IIIA), a Bo el se
I⊂R
ha is sepa a ed
om he es o he spec um by a nonze o gap, and he associa ed Riesz p ojec o
PI
(
L
)de ined by he
con ou in eg al in
(151)
, we use he G o hendieck class [
PI
(
L
)]
∈K0
(C
∗
(
L
)) as a opological label o
he sec o . We show ha (i) as long as he gap a ound
I
s ays open, he class [
PI
(
L
)] and ace– ype
in a ian s Φ(
PI
)a e cons an ; (ii) when a gap closes and eopens, he sec o label unde goes a quan ized
jump measu ed by spec al low and index heo y. Algo i hmically, including
PI
among ou spec al
p obes en o ces sec o coun s du ing SCF, p e en ing spu ious s a e swapping in mul i– e e ence egimes.
Gap condi ion and Riesz p ojec o s.
Le
I⊂R
be a spec al island o
L
: he e exis s a con ou
Γ =
∂I
wi h
dis
(Γ
, σ
(
L
))
>
0and
σ
(
L
)
∩I
=
σI
(
L
)
, σ
(
L
)
I
=
σIc
(
L
)
.
Then he Riesz p ojec o
PI
(
L
)
is
PI(L) = 1
2πi ZΓ
(z−L)−1dz, (76)
and sa is ies
P2
I
=
PI
=
P∗
I
. I
I
con ains only disc e e eigen alues ( ini e sys ems),
ank PI
equals he
numbe o eigen alues in
I
coun ing mul iplici y; in pe iodic solids, he ibe –wise anks in eg a e o he
occupied–band coun pe cell. Unde
C1
de o ma ions
L
(
θ
) ha p ese e he gap (no spec um c osses Γ),
PI
(
L
(
θ
)) a ies smoo hly and emains a p ojec o o he same ank; quan i a i e con inui y is con olled
by esol en bounds (below).
K0sec o labels and s abili y.
Conside he s abilized ma ix algeb a
M∞
(C
∗
(
L
)). Two p ojec o s
P, Q ∈M∞
(C
∗
(
L
)) a e s ably equi alen i
P⊕
1
is Mu ay– on Neumann equi alen o
Q⊕
1
s
o some
, s
. The G o hendieck g oup
K0
(C
∗
(
L
)) classi ies s able equi alence classes. I he gap a ound
I
emains
open along a pa h θ7→ L(θ), hen
d
dθ [PI(L(θ))] = 0 in K0(C∗(L(θ))),(77)
i.e. he sec o label is opologically cons an . In pa icula , he ace– ype weigh Φ(
PI
)( ini e ace o
ibe ace) is cons an (see
(71)
), so he numbe o occupied/ac i e s a es is conse ed wi hou ad hoc
b anching ules. This
K0
s abili y is he ope a o –algeb aic coun e pa o he cons ancy o ec o –bundle
ank unde gap–p ese ing de o ma ions.
Quan ized jumps a gap closu es: spec al low and index.
When he gap closes and eopens
along a pa h
θ∈
[0
,
1], he class [
PI
(
L
(
θ
))] may change by an in ege amoun de e mined by spec al low.
Fo a no m–con inuous pa h o sel –adjoin F edholm ope a o s
{F
(
θ
)
}
, he spec al low
s {F
(
θ
)
}
coun s
(wi h sign) he ne numbe o eigen alues c ossing 0[
354
]. In ou se ing, le
L
(
θ
)
−λ0
play he ole o
F(θ)nea a band edge λ0∈∂I; hen he change in he p ojec o class is measu ed by an index:
PI(L(1))−PI(L(0))= Ind(PI(L(1))PI(L(0)) : Ran PI(L(0)) →Ran PI(L(1))),(78)
which coincides wi h he spec al low h ough
λ0
[
154
,
354
]. This quan ized jump is he p ecise algeb aic
meaning o “sec o change”: i canno occu wi hou a gap closu e. In pe iodic media, he same mechanism
unde lies opological band ansi ions and bulk–edge co espondences; noncommu a i e geome y and
index pai ings p o ide he b idge o Che n– ype in a ian s [155, 156].
Di e en ia ion o p ojec o s and obus ness bounds.
Di e en ia ing
(76)
along a smoo h pa h
θ7→ L(θ)yields
˙
PI=1
2πi ZΓ
(z−L)−1˙
L(z−L)−1dz. (79)
I δ:= dis (Γ, σ(L)) >0, hen k(z−L)−1k ≤ δ−1and
k˙
PIk ≤ `(Γ)
2πk˙
Lk
δ2.(80)

22
In ini e dimension,
(80)
is he p ojec o analogue o he Da is–Kahan sinΘ heo em, bounding he
a ia ion o spec al subspaces unde a gap
δ
[
350
,
357
]. Inequali y
(80)
ensu es nume ical s abili y o
PI
and explains why he sec o cons ain s Φ(PI) = φIa e well–beha ed whene e a gap is p esen .
Fe mi p ojec o s and c ys alline sec o iza ion.
Fo insula o s, he Fe mi p ojec o
PF
=
1(−∞,EF]
(
L
)is de ined ibe wise in
k
and assembled by he ace–pe –cell unc ional Φ
pe
in
(54)
.
Gap p ese a ion a
EF
implies cons ancy o Φ
pe
(
PF
)(occupied–band coun pe cell) and
K0
s abili y o
PF
; unde symme y cons ain s his class e ines o opological indices (Che n numbe s,
Z2
in a ian s)
ia index pai ings [
156
]. Rigo ous i iali y c i e ia (e.g. anishing Che n numbe s) ensu e exis ence o
exponen ially localized Wannie ames, a p ac ical boon o p ojec o compu a ion and anspo [
159
].
Including sec o p ojec o s among spec al p obes.
In ou spec al–law cons ained SCF (see
§IC), we add p ojec o p obes
(
λ
) =
1I
(
λ
)(o smoo hed app oximan s) wi h a ge alues
φI
= Φ(
PI
)
ixed by elec on coun , spin mul iplici y, o ac i e–space selec ion. The co esponding mul iplie upda es
en o ce Φ(
PI
(
L
[
n
])) =
φI
each mac o–i e a ion, he eby locking sec o s wi hou ad hoc heu is ics. When
a genuine gap closu e occu s along a eac ion pa h, he cons ain can be elaxed o allow a quan ized
change
(78)
; he mapping– o us pe spec i e in §IIC hen desc ibes he esul ing holonomy and sec o
jump as an index class.
Nume ical ealiza ion o Riesz p ojec o s.
Th ee complemen a y app oaches compu e
PI
(
L
)a KS
cos :
1.
Con ou quad a u e / FEAST: e alua e
(76)
by quad a u e wi h shi ed linea sol es (independen
igh –hand sides), as in he FEAST eigensol e amewo k [
338
]. The cos scales wi h he numbe
o quad a u e nodes, ypically modes o smoo h con ou s.
2.
Smoo hed windows: app oxima e
1I
by Fe mi–Di ac o hype bolic angen windows and apply he
co esponding ma ix unc ions ia a ional app oximan s; his euses he esol en machine y o
ou o he p obes.
3.
S ochas ic ace o weigh s: when only Φ(
PI
)is needed, apply s ochas ic Lanczos quad a u e o
(
L
)wi h
=
1I
(o i s smoo h app oximan ) and a e age o e andom ec o s o es ima e he
ace pe cell [161].
Pe u ba ion bounds (80) and Da is–Kahan– ype es ima es gua an ee ha hese compu a ions a e obus
unde mode a e nume ical e o s as long as he gap is esol ed.
Chemical applica ions.
Ac i e–space s abili y in bond b eaking and ansi ion–me al complexes
ollows di ec ly: choose
I
o isola e he nea –degene a e on ie mani old (e.g. wo–o bi al ac i e space
o s e ched H
2
,
d
–mani old o TM complexes), en o ce Φ(
PI
) = desi ed occupancy/spin, and le he
emaining spec al–law cons ain s de e mine
xc
. Genuine eo ganiza ion (e.g. spin c osso e ) co esponds
o an allowed, quan ized sec o jump a a gap closu e, no o a nume ical acciden o he SCF. In pe iodic
sys ems, sec o iza ion p e en s band e–labeling pa hologies nea a oided c ossings and keeps he occupied
subspace consis en along s ain/ ield sweeps; opological ansi ions a e de ec ed as spec al– low e en s
a he han as d i s o eigen alue o de ings.
Summa y. K0
supplies opological memo y o spec ally isola ed subspaces ia Riesz p ojec o s. Unde
gap p ese a ion, bo h he
K0
class and ace– ype weigh s a e in a ian ; when a gap closes, changes
a e quan ized and measu ed by spec al low and indices. Including p ojec o p obes in ou spec al–law
se en o ces sec o coun s in a ma hema ically con olled, basis–independen way, elimina ing ad hoc
heu is ics and s abilizing mul i– e e ence egimes—c ucial o achie ing ab–ini io accu acy a KS cos .
IV. EXCHANGE–CORRELATION AS A CONSTRAINED VARIATIONAL PROBLEM
A. Cons ained Le y–Lieb
Objec i e and se ing.
In he s anda d Le y–Lieb o mula ion he uni e sal unc ional
F
[
n
] =
in Ψ→nh
Ψ
|ˆ
T
+
ˆ
W|
Ψ
i
is de ined on
N
– ep esen able densi ies and yields, o a gi en ex e nal po en ial
ex
, he g ound–s a e ene gy by minimiza ion o
E
[
n
] =
F
[
n
] +
R ex n
(see he ecap in §II A and he
his o ical sou ces [
1
,
2
,
425
,
426
]). He e we augmen he Le y–Lieb cons ained sea ch wi h spec al-law
23
cons ain s in oduced in §IIIC, en o ced h ough an Ad–in a ian s a e Φ(§IIIB) and he KS–like
gene a o
L
[
n
](§IIIA). The aim is o ob ain a i - ee, ab–ini io cons uc ion o
xc
om exac algeb aic
iden i ies.
P imal p oblem (cons ained Le y–Lieb).
Fix a densi y
n
in he admissible class (e.g.
n∈L1∩L3
,
√n∈H1
,
Rn
=
N
; c . [
426
]). Gi en p obes
{ k}m
k=1
and a ge alues
{φk}
( anspo ed along exac
legs as in §IIID), de ine he cons ained uni e sal unc ional
Fcon[n] := in
Ψ→nnhΨ|ˆ
T+ˆ
W|ΨiΦ k(L[n])=φk
| {z }
k=1,...,m o.(81)
Equi alen ly, a he densi y le el we o m he easible se
A
:=
{n
: Φ(
k
(
L
[
n
])) =
φk, k
= 1
, . . . , m}
and
pu
Fcon
[
n
] =
F
[
n
]
o n∈ A,
and
Fcon
[
n
] = +
∞
o he wise. Fo a gi en
ex
he p imal g ound–s a e
p oblem eads
Econ
= in
n∈A nF[n] + Z ex ( )n( )d o.(82)
The easible se
A
is nonemp y by cons uc ion o he a las (Sec ions III and I): each ile supplies a leas
one basepoin n?wi h Φ( k(L[n?])) = φk, and along exac legs (70) he cons ain s a e p ese ed.
Well-posedness (exis ence o minimize s).
Unde he classical hypo heses (lowe semicon inui y and
coe ci i y o
F
on he
N
– ep esen able se ; see [
426
]) and con inui y o he maps
n7→ S k
[
n
] := Φ(
k
(
L
[
n
]))
(which ollows om he esol en calculus, §III C), he easible se
A
is closed; i , u he mo e, a uni o m
bound on
Rn4/3
holds (ensu ed by he Lieb–Ox o d bound [
71
], al eady ci ed) and he pa icle numbe
is ixed, hen
A
is weakly sequen ially compac in he na u al opology. By he di ec me hod o he
calculus o a ia ions (coe ci i y + l.s.c.
⇒
exis ence; e.g. [
163
,
164
]), a minimize
¯n∈ A
exis s o
(82)
.
Lag angian and KKT sys em.
To de i e he Eule equa ion and, in pa icula , he XC con ibu ion,
in oduce Lag ange mul iplie s
λ
= (
λ1, . . . , λm
) o he spec al laws and a scala
µ
o he pa icle
numbe cons ain . De ine he densi y-le el Lag angian
L[n;λ, µ] := F[n] + Z ex n+
m
X
k=1
λkS k[n]−φk−µZn−N.(83)
A minimize
¯n∈ A
admi s mul iplie s (
¯
λ,¯µ
)such ha he KKT condi ions o equali y cons ain s hold
(in ini e-dimensional e sion, see [163, 165, 166]):
s a iona i y: δF
δn( )¯n+ ex ( ) +
m
X
k=1
¯
λk
δS k
δn( )¯n= ¯µ, (84)
easibili y: S k[¯n] = φk, k = 1, . . . , m, Z¯n=N. (85)
The exis ence o mul iplie s ollows unde a s anda d cons ain quali ica ion (e.g. LICQ: he F éche
de i a i es {δS k/δn}m
k=1 a e linea ly independen in he dual space a ¯n; c . [163, 165–167]).
Reco e ing he Eule –Kohn–Sham s uc u e.
Spli ing
F
[
n
] =
Ts
[
n
] +
EH
[
n
] +
Exc
[
n
]and using
he usual o bi al ep esen a ion o Ts(wi h n=Pocc
i|ψi|2), he s a iona i y condi ion (84) becomes
δTs
δn( )+ ex ( ) + H( ) + xc( ) +
m
X
k=1
¯
λk
δS k
δn( )= ¯µ. (86)
Abso bing he cons ain e m in o he e ec i e XC po en ial yields he cons ained Kohn–Sham Eule
equa ion
(con)
xc ( ) := xc( ) +
m
X
k=1
¯
λk
δS k
δn( ).(87)
Using he esol en /DOI calculus (§IIIC, eq. (65)), he unc ional de i a i e o a spec al law is
δS k
δn( )= ΦZΓk
k(z)(z−L[¯n])−1δL
δn( )¯n(z−L[¯n])−1dz
2πi,(88)
24
so ha
(87)
coincides wi h he cons uc i e exp ession o
xc
gi en ea lie in Eq.
(6)
(wi h
λk
=
¯
λk
), i.e.
(con)
xc ( ) =
m
X
k=1
¯
λkΦZΓk
k(z)(z−L[¯n])−1δL
δn( )¯n(z−L[¯n])−1dz
2πi.
Hence, in he cons ained Le y–Lieb p inciple, he spec al mul iplie s
¯
λk
a e no i pa ame e s: hey
a e Lag ange mul iplie s de e mined by (85) a he minimize .
Dual p oblem and s ong duali y (ske ch).
De ine he Lag angian
L
[
n
;
λ, µ
]by
(83)
and he dual
unc ion
g(λ, µ) := in
nL[n;λ, µ]−
m
X
k=1
λkφk.(89)
The dual p oblem is o maximize
g
(
λ, µ
)o e (
λ, µ
). Unde Fenchel–Rocka ella egula i y (e.g. Sla e - ype
easibili y o equali y cons ain s, o LICQ a he minimize ), s ong duali y holds and he p imal op imum
equals he dual op imum [
163
,
164
,
166
]. Fo mally, he inne in imum o
(89)
is he Legend e–Fenchel
ans o m o he sum
F
+
PkλkS k
+
R ex ·
, p oducing a conca e dual unc ional o be maximized in
λ, µ. The KKT sys em (84)–(85) cha ac e izes saddle poin s.
Cons ain quali ica ions and uniqueness o mul iplie s.
Le
G
:
n7→
(
S 1
[
n
]
−φ1,...,S m
[
n
]
−
φm
). Assume: (i)
F
is s ic ly con ex on he easible a ine slice
{n
:
Rn
=
N}
(as o nonin e ac ing
Ts
plus Ha ee plus a con ex
Exc
model); (ii)
G
is F éche di e en iable a
¯n
; (iii) LICQ holds: he
linea unc ionals
{δS k/δn|¯n}
a e independen . Then
¯n
is unique, and he mul iplie s
¯
λ
a e unique as
well [
166
,
167
]. In p ac ice, he Jacobian en ies
δS k/δn
a e compu ed by
(88)
and he independence is
nume ically e lec ed by a well–condi ioned small (
m×m
) linea sys em in he mul iplie upda e ( he
B oyden/New on s ep used in ou SCF; c . §I C).
Consis ency wi h exac cons ain s ( i ial, scaling, sum ules).
Because he p obes and a ge s
{ k, φk}
a e chosen om exac physics ( i ial/scaling iden i ies [
65
], comp essibili y/
–sum, p ojec o
coun s), he KKT s a iona i y
(84)
en o ces hose cons ain s a he solu ion
¯n
. Fo example, choosing
(
λ
) =
λ
(
λ−z0
)
−2
ep oduces a i ial– ype momen , while
=
1I
ixes sec o weigh s Φ(
PI
)(see §III E);
hus he cons ained Le y–Lieb p inciple is a di ec a ia ional ou e o hono ing exac condi ions ha
would o he wise be encoded only heu is ically.
Algo i hmic eading.
Equa ions
(84)
–
(88)
a e implemen ed in ou SCF loop as ollows: gi en
n(j)
, sol e
he KS eigenp oblem o
L
[
n(j)
], compu e he spec al laws
S k
[
n(j)
]and esiduals
k
:=
S k
[
n(j)
]
−φk
,
upda e mul iplie s by a small New on/B oyden s ep using sensi i i ies om
(88)
, assemble
(con)
xc
om
(87)
, upda e he densi y, and i e a e un il bo h densi y and cons ain esiduals mee ole ance. This
ealizes he KKT sys em as a p ac ical sol e in he KS cos class.
Summa y.
The cons ained Le y–Lieb p inciple
(81)
–
(82)
u ns exac ope a o iden i ies in o equali y
cons ain s on admissible densi ies. I s KKT condi ions lead, ia he esol en calculus, o he cons uc i e,
i – ee o mula o
xc
in Eq.
(6)
. Exis ence o minimize s, s ong duali y, and uniqueness o mul iplie s
ollow om s anda d con ex–analy ic hypo heses [
163
–
168
], while he a las/exac –leg anspo gua an ees
easibili y ac oss chemical space wi hou e i s.
B. Lag angian and mul iplie s λk
Pu pose and ecap.
We now analyze he mul iplie sec o o he cons ained Le y–Lieb p inciple
in oduced in §IVA. Gi en p obes { k}m
k=1 and a ge s {φk}, he densi y–le el Lag angian
L[n;λ, µ] = F[n] +Z ex n+
m
X
k=1
λkS k[n]−φk−µZn−N,λ= (λ1, . . . , λm),(90)
leads o he KKT sys em
(84)
–
(85)
. He e we: (i) make explici he mul iplie equa ions and hei
linea iza ion; (ii) connec hem o esponse ope a o s (Schu complemen s); (iii) p esen s able algo i hms
(New on/SQP, augmen ed Lag angian) and con e gence gua an ees; (i ) gi e a physically anspa en
25
in e p e a ion o
λk
as gene alized conjuga e ields. We ely on s anda d nonlinea p og amming and
in ini e–dimensional KKT heo y [169–172, 174, 368].
Mul iplie equa ions (exac equali ies). Le he cons ain map be
gk[n] := S k[n]−φk, k = 1, . . . , m, (91)
wi h F éche de i a i es
Jk
(
) =
δS k/δn
(
)gi en by he esol en sandwich
(88)
. S a iona i y
(84)
can
be w i en as a pe u bed Eule equa ion
δF
δn ( ) + ex ( ) +
m
X
k=1
λkJk( ) = µ, (92)
wi h easibili y
gk
[
n
] = 0 and
Rn
=
N
. Equa ion
(92)
s a es ha
PkλkJk
is an addi i e XC co ec ion;
compa ing wi h (87) eco e s p ecisely he cons uc i e xc o Eq. (6).
Linea iza ion and Schu complemen o λ.
Le
R
[
n
] =
δF/δn
+
ex −µ
+
PkλkJk
deno e he
Eule esidual. Linea izing he KKT sys em a (n, λ, µ)yields he saddle sys em

H[n]J>−1
J0 0
−1>0 0 


δn
δλ
δµ
=−

R[n]
g[n]
Rn−N
,J:= 

J1
.
.
.
Jm


,(93)
whe e
H
[
n
]is he (second) a ia ion o
F
a
n
(plus Ha ee ke nel), and
1
is he cons an unc ional
en o cing Rn=N. Elimina ing δn by he Schu complemen gi es a small m×mlinea sys em o δλ:
Akl m
k,l=1 δλl=−gk[n] + hJk,H[n]−1
(R[n]−δµ)im
k=1,Akl := hJk,H[n]−1Jli,(94)
wi h
h·,·i
he
L2
pai ing. In Kohn–Sham a iables,
H
[
n
]
−1
is he s a ic KS esponse
χs
; hus A
kl
=
hJk, χsJli
is a cons ain G amian in he esponse me ic. In p ac ice, we (i) compu e ac ions o
H
[
n
]
−1
using he same linea – esponse sol e s (conjuga e g adien s, mul ig id, o Da idson) used in DFPT; (ii)
assemble A
∈Rm×m
by
m
such sol es; (iii) upda e
λ
by sol ing
(A14)
(Cholesky/LSQR). This p ojec ion
me hod se s gk→0in a single New on s ep when H[n]and Ja e exac [175, 368].
Augmen ed Lag angian and global con e gence.
Fo obus ness, especially a om easibili y, we
employ he me hod o mul iplie s [169, 176, 177]: de ine
Lρ[n;λ, µ] = L[n;λ, µ] + ρ
2
m
X
k=1
gk[n]2, ρ > 0.(95)
Minimize
Lρ
in
n
(one SCF mac o–s ep), hen upda e
λ+
k
=
λk
+
ρ gk
[
n+
]and (op ionally) inc ease
ρ
.
The quad a ic e m damps ill–condi ioning when
{Jk}
a e nea ly dependen (nea cons ain degene acy),
and classical heo y gua an ees global con e gence o KKT poin s unde MFCQ [
169
,
170
,
172
]. Locally,
swi ching o New on/SQP on (93) eco e s supe linea (o quad a ic) con e gence [175, 368].
Inexac New on and o cing e ms.
Exac sol es o
H
[
n
]
δn
=
·
a e unnecessa y. We use inexac
New on s eps con olled by Eisens a –Walke o cing e ms o balance esidual educ ion and inne
linea –sol e accu acy [178, 179]. This p ese es supe linea con e gence while signi ican ly educing he
numbe o inne i e a ions. In ou se ing, he inne sol es a e he same shi ed linea sys ems used by
esol en quad a u es; we exploi mul i–shi K ylo accele a ion when se e al complex shi s a e needed
[180].
Scaling, p econdi ioning, and no maliza ion.
The p obes
k
ha e dispa a e dynamic anges (e.g.,
p ojec o weigh s e sus high–ene gy esol en momen s). To keep Awell condi ioned, we scale cons ain s
by posi i e ac o s
sk
(equi alen ly, epa ame e ize
λk←λk/sk
) so ha he ypical magni udes o
gk
a e
compa able; his co esponds o p econdi ioning he Schu complemen (A14) [368]. Physically, we pick
sk
so ha each
gk
is dimensionless (e.g., di ide momen cons ain s by he HEG alue a local densi y).
Second–o de condi ions and uniqueness o mul iplie s.
Assume MFCQ a he solu ion
¯n
(linea
independence o
{Jk}
on he easible angen space) [
172
]. Le he educed Hessian on he angen space
be posi i e de ini e:
hδn, H
[
¯n
]
δni>
0 o all
δn
wi h
Rδn
= 0 and
hJk, δni
= 0. Then he KKT poin
32
V. GRADIENT EXPANSIONS & TDDFT KERNELS FROM RESOLVENTS
A. Resol en expansion
Aim.
We de i e he la ge–
|z|
expansion o he esol en (
z−L
)
−1
o he KS–like gene a o
L
[
n
]speci ied
in §IIIA, and show how i s Neumann–se ies and semiclassical/symbol inca na ions eed he spec al–law
cons uc ion o
xc
(§IVC) and he g adien expansion o esponse ke nels. Two complemen a y iewpoin s
will be used: (i) he algeb aic Neumann se ies (con e gen o
kLk/|z|<
1), which makes con ac wi h
momen cons ain s and p o ides explici emainde bounds; (ii) he mic olocal (pseudodi e en ial) expan-
sion, which yields sys ema ic g adien co ec ions (Wigne –Ki kwood/Moyal se ies) e en o unbounded
Sch ödinge – ype L.
Neumann se ies and emainde bounds.
Le
z
lie in he esol en se o
L
and assume o he
momen ha
L
is a bounded ope a o ( he unbounded Sch ödinge case will be ea ed ia symbol
calculus below). Then
(z−L)−1=z−1I−z−1L−1=z−1
∞
X
m=0 z−1Lm=
∞
X
m=0
Lm
zm+1 ,i kLk
|z|<1.(111)
T unca ing (A6) a o de Mgi es he e o bound




(z−L)−1−
M
X
m=0
Lm
zm+1 



≤kLkM+1
|z|M+2
1
1−kLk/|z|,(112)
a di ec consequence o he geome ic–se ies es ima e; see, e.g., [
212
, Sec. 1.1] and [
213
, Ch. 5]. Inse ing
(A6)
in o he spec al–law sandwich o Eq.
(97)
(and in e changing he sum wi h he con ou in eg al
when jus i ied) yields an explici momen expansion:
(con)
xc ( ) =
m
X
k=1
λkΦ
ZΓk
dz
2πi k(z)X
p,q≥0
Lp
zp+1
δL
δn( )
Lq
zq+1 
,(113)
so ha only ini ely many ope a o momen s con ibu e when
k
is chosen o annihila e high nega i e
powe s o
z
(e.g., by using a ional p obes o ini e–in e al quad a u es). This makes he link o ou
ini e amily o cons ain s: Φ(L`)and Φ(L`δL
δn L`0)a e gene alized momen s ixed by he spec al laws.
F om Neumann se ies o asymp o ics o unbounded Sch ödinge ope a o s.
Fo physically
ele an
L
=
−1
2∇2
+
e
(
)(possibly wi h nonlocal e ms),
L
is unbounded. One ou e o a la ge–
|z|
expansion is ia he escaling
ζ
=
z−1
and pa ame ix cons uc ions o (
I−ζL
)
−1
, p oducing a o mal
se ies in
ζ
whose coe icien s a e di e en ial ope a o s buil om
e
and i s de i a i es. A mo e sys ema ic
ou e is he pseudodi e en ial/symbol calculus: w i e he symbol o L,
a( ,k) = 1
2|k|2+ e ( ) (plus lowe –o de e ms),(114)
and seek he symbol ( ,k;z)o he esol en R(z)=(z−L)−1as an asymp o ic expansion
( ,k;z)∼
∞
X
j=0
j( ,k;z), 0=1
z−a, j+1 =Q j;a,(115)
whe e
Q
is a bilinea di e en ial ope a o gene a ed by he Moyal p oduc (successi e Poisson b acke s /
de i a i es in ,k) [214, Chs. VII–VIII], [215, Chs. 20–22], [216]. Conc e ely, he i s co ec ions a e
1=i
2{ 0, a} 0=i
2{a, a}
(z−a)3= 0,(116)
2=−1
8{a, {a, 0}} 0=1
8{a, {a, a}}
(z−a)4−1
8{a, {a, z}}
(z−a)3=1
8Pi,j(∂ i∂ j e )(∂ki∂kj 0)
1,(117)
whe e we used ha
z
is a pa ame e and
{·,·}
is he Poisson b acke . A e in e se Weyl quan iza ion,
he diagonal ke nel admi s he Wigne –Ki kwood– ype g adien expansion (in d=3),
R(z; , ) = 1
(2π)3ZR3
d3k
z−1
2|k|2− e ( )+1
24 X
i,j
∂ i∂ j e ( )1
(2π)3ZR3
∂ki∂kj1
z−1
2|k|2− e ( )d3k+··· ,
(118)

33
wi h highe e ms in ol ing (
∇ e
)
2
,
∇4 e
, e c. Equa ion
(118)
p o ides he local/semilocal building
blocks ha , subs i u ed in o he con ou o mula o
(con)
xc
, gi e ise o GEA– ype o ms ( e ms in
n, ∇n, ∇2n, τ, . . .
). The same machine y unde lies he classic Wigne –Ki kwood expansion o he hea
ke nel; see [217–220].
Hea –ke nel ou e (Laplace ans o m).
Fo
z
in he le hal –plane ela i e o
σ
(
L
), he esol en
admi s he Laplace ep esen a ion
(z−L)−1=−Z∞
0
e (z−L)d , (119)
so ha diagonal ke nels a e Laplace ans o ms o he hea ke nel
K
(
;
,
):
R
(
z
;
,
) =
−R∞
0e z K
(
;
,
)
d .
Using he sho – ime expansion
K
(
;
,
)
∼
(4
π
)
−3/2P`≥0a`
(
)
`
wi h Seeley–DeWi coe icien s
a`
(polynomials in
e
and i s de i a i es) [
217
], one e–ob ains
(118)
ia e mwise Laplace ans o ms. This
ou e makes explici he link be ween la ge–|z| esol en asymp o ics and sho – ime dynamics.
Consequences o spec al laws and XC.
Combining
(118)
wi h he spec al–law sandwich in §IV C
shows ha each p obe con ibu es a ini e combina ion o local/semi–local enso s:
(con)
xc ( ) = X
k
λkCk,0F0
n( )+Ck,2F2
n( )|∇n|2
n8/3+Ck,∆G2
n( )∇2n
n5/3+Ck,τ H2
n( )τ
n5/3+···,
(120)
whe e he coe icien unc ions
Ck,•
a e con ou in eg als o he
k
– h p obe applied o he
k
–in eg als in
(118)
and
F0, F2, G2, H2
a e he amilia GEA building blocks (he e w i en schema ically o emphasize
s uc u e a he han speci ic cons an s). Thus he esol en expansion u nishes a p incipled de i a ion
o g adien e ms consis en wi h ou cons ain s, ins ead o pos ula ing hem.
Ra ional/con inued– ac ion accele a ions.
While he Neumann se ies
(A6)
is in o ma i e, p ac ical
quad a u es bene i om a ional accele a ion:
•
Padé/con inued ac ions. Fo S iel jes– ype ans o ms o spec al densi ies, Padé app oximan s
con e ge apidly and can be ob ained om momen s o
L
; con inued ac ions eme ge na u ally
om Lanczos ecu sion (Haydock–Heine–Kelly) [222, 223].
•
Ke nel Polynomial Me hod (KPM). Expansions in o hogonal polynomials (Chebyshe ) o smoo hed
spec al densi ies app oxima e esol en aces and diagonal elemen s e icien ly [221].
•
Ra ional K ylo /ADI. Resol en ac ions (
z−L
)
−1b
o many shi s
z
a e compu ed ia a ional
K ylo subspaces, a oiding explici se ies and le e aging shi –in a iance [225].
These accele a ions in eg a e seamlessly wi h he spec al–law e alua ion because Φis linea and he
p obes a e e alua ed a a small numbe o shi s zk.
Rema ks on alidi y and o de ing.
The Neumann se ies
(A6)
is con e gen only o
kLk/|z|<
1
(bounded
L
) and is o he wise o be ead as an asymp o ic expansion when in e p e ed ia symbol/hea –
ke nel calculus o unbounded
L
. In p ac ice, ou con ou s Γ
k
a e chosen away om
σ
(
L
)so ha esol en
no ms a e mode a ed and he asymp o ic expansion is accu a e a modes unca ion o de . The mic olocal
expansion is o ganized ei he by la ge–
|z|
o by a small semiclassical pa ame e (Fe mi–wa e ec o scale);
bo h o de ings lead o he same local/semi–local enso s uc u es in
(120)
, wi h coe icien s ixed by he
spec al p obes.
Summa y.
The esol en admi s (i) an algeb aic Neumann se ies ha exposes momen –like s uc u es
di ec ly cons ained by spec al laws, wi h explici emainde bounds, and (ii) a pseudodi e en ial/hea –
ke nel expansion ha yields he s anda d g adien –expansion building blocks. Subs i u ed in o he
spec al–law o mula o
(con)
xc
, hese expansions p oduce p incipled GEA e ms whose coe icien s a e
de e mined by p obe–con ou in eg als, no by empi ical i ing.
B. S a ic coe icien s (GGA/me a-GGA) om cen al cu a u e: comp essibili y, spin s i ness,
-sum, i ial; no eg ession
Aim.
We show how he s a ic (ze o– equency) exchange–co ela ion (XC) ke nel and, consequen ly,
he semilocal coe icien s ha appea in GGA/me a-GGA o ms a e de e mined by cen al-cu a u e
34
cons ain s on spec al laws, wi hou any eg ession. Conc e ely, we ex ac om ou esol en calculus
(§VA) he long-wa eleng h expansion o he s a ic ke nels in he cha ge and spin channels,
nn
xc (q, 0;n) = A0(n) + A2(n)q2+O(q4), mm
xc (q, 0;n) = ˜
A0(n) + ˜
A2(n)q2+O(q4),(121)
and ela e he coe icien s
A0,˜
A0
o uni o m iden i ies (comp essibili y and spin s i ness), while he
g adien coe icien s
A2,˜
A2
ollow om he
q2
e m o he esol en expansion cons ained by cen al
cu a u e. We hen map
A2,˜
A2
o eal-space GGA/me a-GGA coe icien s mul iplying
|∇n|2
and
|∇m|2
(and, in me a-GGA, he kine ic-ene gy densi y
τ
), and show how i ial/scaling and he
-sum en o ce
consis ency.
S a ic ke nel, local- ield ac o , and long-wa eleng h s uc u e.
Fo he homogeneous elec on
gas (HEG), he in e ac ing s a ic densi y esponse can be w i en as
χ(q, 0) = χ0(q, 0)
1− c(q)1−G(q, 0)]χ0(q, 0), c(q) = 4π
q2,(122)
wi h
χ0
he Lindha d unc ion and
G
he (s a ic) local- ield ac o [
226
,
227
]. In ime-dependen DFT
language,
xc
(
q,
0) =
− c
(
q
)
G
(
q,
0), so he
q→
0expansion o
xc
is equi alen o he small–
q
expansion o
G
. Ou cons ained esol en calculus p oduces
(121)
di ec ly, bu
(122)
is use ul o connec o his o ical
esul s and o emphasize ha no eg ession is in ol ed:
A0,˜
A0
a e dic a ed by exac uni o m iden i ies,
while A2,˜
A2a e ixed by he q2coe icien o he esol en expansion (nex pa ag aphs).
Uni o m iden i ies ⇒cons an e ms A0,˜
A0.
Fo an unpola ized s a e (
m=
0), he comp essibili y
sum ule implies
lim
q→0χ(q, 0) = −n2κT,1
n2κT
=d2
dn2n ε(n),(123)
wi h
ε
(
n
) he o al ene gy pe pa icle o he uni o m sys em [
229
]. Using he Dyson ela ion
χ−1
=
χ−1
0− c+ xca ω=0 and (123) gi es
A0(n) = d2
dn2n εxc(n)(ALDA limi ),(124)
i.e. he
q0
piece is uniquely ixed by he HEG equa ion o s a e ( om QMC o con olled many-body
heo y, e.g. Cepe ley–Alde ) [
228
]. Analogously, in he spin channel a ze o magne iza ion he spin
s i ness (S one pa ame e ) ixes
˜
A0(n) = 1
n2
∂2
∂ζ2hn εxc(n, ζ)iζ=0,(125)
whe e
ζ=m/n
is he spin pola iza ion [
229
]. Equa ions
(124)
–
(125)
a e he p ecise ma hema ical s a emen
ha ALDA is ixed by exac uni o m iden i ies: no pa ame e s a e i ed.
Cen al cu a u e ⇒g adien e ms A2,˜
A2.
Le
L
[
n
]be o Sch ödinge ype (possibly wi h gKS
nonlocal e ms) and conside he bounded-Bo el p obe amily used in §IIIC. Fo a plane-wa e densi y
pe u ba ion
δn
(
) =
δnqeiq·
a ound a uni o m backg ound, he second a ia ion o he cons ained XC
ene gy de ines xc(q, 0):
δ2Exc =1
2X
q
nn
xc (q, 0;n)|δnq|2+1
2X
q
mm
xc (q, 0;n)|δmq|2+··· .(126)
When he cen al cu a u e anishes (§IIID), he
q
–dependence is en i ely gene a ed by he esol en
sandwich o §IV C and i s la ge–
|z|
(semiclassical) expansion o §VA. Collec ing he
q2
e ms yields, o
he cha ge channel,
A2(n) =
m
X
k=1
λkIk,2(n),Ik,2(n) = 1
2πi IΓk
k(z)J2(z;n)dz, (127)
whe e
J2
is he uni e sal ( ile–independen ) coe icien o
q2
in he diagonal esol en expansion ( he
2
symbol e m in eg a ed o e momen a; c . Eq.
(118)
), e alua ed a he uni o m backg ound. A simila
35
exp ession holds o
˜
A2
(
n
)in he spin channel, wi h spino ial blocks in
L
and he co esponding p obes.
C ucially, no eg ession is in ol ed:
A2,˜
A2
a e compu ed om con ou in eg als o known (
k
) and
compu ed ( esol en ) quan i ies; cen al cu a u e gua an ees ha he same in eg als ob ained on an
a las ile (e.g. HEG, SCE) a e anspo ed o ealis ic sys ems wi hou e i s.
F om A2,˜
A2 o GGA/me a-GGA coe icien s.
A semilocal (GGA) unc ional nea uni o m densi y
can be w i en as
EGEA2
xc [n, m] = Zd3 nn εLDA
xc (n, ζ) + cnn(n)|∇n|2+cmm(n)|∇m|2o+··· ,(128)
wi h ζ=|m|/n. Linea izing (128) abou he uni o m s a e shows ha he Fou ie –space ke nels sa is y
A2(n) = cnn(n),˜
A2(n) = cmm(n).(129)
Rew i ing
(128)
in educed-g adien o m (
s
=
|∇n|/
(2
kFn
)wi h
kF
= (3
π2n
)
1/3
) gi es, o he exchange-
like piece,
µxc(n) = 4k2
Fn
εLDA
xc (n)A2(n),(130)
which is he (densi y-dependen ) analog o he amilia
µ
coe icien in GEA2 o exchange (whe e
µGEA2
x=
10
/
81 [
230
,
231
]). An iden ical mapping holds in he spin channel (wi h a sui able no maliza ion by
he magne ic ene gy densi y), p o iding he spin-g adien coe icien ha en e s spin-GGA. In a me a-GGA
one includes
τ
; he sho -g adien limi o he kine ic-ene gy densi y,
τ
[
n
] =
τuni
(
n
) +
1
72 |∇n|2/n
+
··· ,
connec s he
τ
-dependence o
A2
and cons ains he me a-GGA coe icien s so ha he same
A2
is
ep oduced in he
q2
ke nel [
232
]. Thus, cen al cu a u e ixes he s a ic GGA/me a-GGA coe icien s
µxc(n)(and spin analogs) ia (127)–(130), a he han by i ing o da ase s.
Vi ial/scaling and coe icien homogenei y.
Unde uni o m densi y scaling
nγ
(
) =
γ3n
(
γ
), ex-
change scales as
γ
, co ela ion as
γ2
in he high-densi y limi , and he GEA2 g adien e m mus ans o m
acco dingly. Dimensional analysis o (128) yields he homogenei ies
cx
nn(n)∝n−1/3, cc
nn(n)∝n−1/3(up o loga i hmic co ec ions om co ela ion),(131)
consis en wi h
(130)
. En o cing he i ial iden i y a he le el o he ke nel hen ixes linea ela ions
among
A0, A2
and hei densi y de i a i es. In p ac ice, hese homogenei ies eme ge au oma ically
because
A2
is cons uc ed om esol en in eg als ha depend on
kF∝n1/3
and on dimensionless p obes.
-sum and conse ing s uc u e (s a ic limi ).
Al hough he
-sum is a equency momen , i
cons ains he s a ic ke nel h ough K ame s–K onig ela ions and gauge in a iance: any conse ing
app oxima ion mus yield he exac
-sum
Rdω ω =χ
(
q, ω
) =
−πnq2/
2independen ly o in e ac ions
[
233
]. Because ou
(con)
xc
a ises om a Lag angian cons ained by Ad-in a ian spec al laws (Sec ion IV),
i is conse ing in he Baym–Kadano sense, and he
-sum holds by cons uc ion. A
ω=
0, his
o bids spu ious
q
–dependences in
A0
and co ela es
A2
wi h he
τ
-dependence in me a-GGA so ha he
con inui y equa ion is espec ed. Hence,
A2
compu ed om
(127)
is no an a bi a y i bu he unique
alue compa ible wi h bo h he esol en expansion and he -sum/con inui y s uc u e.
Spin channel and Landau pa ame e s.
A
ζ=
0, he spin–spin ke nel is ela ed o Landau Fe mi-
liquid pa ame e s ia
χm
(0) =
χ0m
(0)
/
(1 +
Fa
0
), wi h
Fa
0
p opo ional o
˜
A0
[
229
]. The
q2
spin-s i ness
˜
A2
co esponds o he g adien coe icien o
|∇m|2
in
(128)
and can be ex ac ed om he same
esol en in eg als using spino ial p obes. Because he cen al cu a u e anishes along exac legs,
˜
A0,˜
A2
anspo ed om he HEG ile emain alid in ealis ic sys ems, again wi hou e i s.
Compa ison o s anda d GGAs/me a-GGAs ( o o ien a ion).
Popula models (PBE, TPSS,
SCAN) choose
µ
(
n
)and ela ed coe icien s by mixing exac cons ain s wi h calib a ed beha io o
a oms and solids [
234
–
236
]. Ou ou e di e s undamen ally:
µxc
(
n
)and i s spin analogs a e compu ed
om
A2,˜
A2
ia
(130)
, wi h
A2,˜
A2
de e mined by spec al laws and cen al cu a u e, no by da ase
i ing. This p ese es he exac ALDA limi s
(124)
–
(125)
, he co ec homogenei ies
(131)
, and he
conse ing cha ac e equi ed by he -sum.
Summa y.
Cen al cu a u e and esol en expansions ix he s a ic long-wa eleng h ke nels
(121)
en i ely om physics:
A0,˜
A0
by uni o m iden i ies (comp essibili y, spin s i ness), and
A2,˜
A2
by he
q2
36
coe icien o he cons ained esol en expansion. These map di ec ly o he GGA/me a-GGA g adien
coe icien s
(129)
–
(130)
, wi h scaling and
-sum p o iding addi ional non-nego iable s uc u e. No
eg ession is used a any s age.
C. Dynamic ke nel xc(q, ω) om dynamical esol en s: causali y, sum ules, and minimal
b aided co ec ions om non–cen al cu a u e
Aim.
We cons uc he equency–dependen exchange–co ela ion (XC) ke nel
xc
(
q, ω
)di ec ly om he
dynamical esol en o he KS–like gene a o
L
[
n
]and om he spec al–law o malism de eloped in §IVC.
We p o e ha he esul ing ke nel is (i) causal and analy ic in he uppe hal –plane (K ame s–K onig);
(ii) conse ing and compa ible wi h exac sum ules (comp essibili y and
–sum), he ze o– o ce heo em,
and he ha monic–po en ial heo em; and (iii) admi s a p incipled, minimal nonlocal co ec ion d i en
by he non–cen al piece o highe cu a u e, whose algeb aic s uc u e is b aided and consis en wi h
hexagon/Yang–Bax e cohe ence.
Linea esponse in TDDFT and Dyson s uc u e.
Fo a ime–dependen pe u ba ion
δ ex
(
, ω
),
he densi y esponse eads
δn(q, ω) = χ(q, ω)δ ex (q, ω), χ−1(q, ω) = χ−1
s(q, ω)− c(q) + xc(q, ω),(132)
wi h
χs
he KS esponse and
c
(
q
) = 4
π/q2
. Equa ion
(132)
is he TDDFT Dyson ela ion [
237
–
239
]. Ou
ask is o compu e
xc
om he cons ained a ia ional p inciple o §IV by di e en ia ing he dynamical
XC po en ial (con)
xc wi h espec o he densi y a ixed ex e nal ield.
Dynamical esol en o mula o xc.
Le
GR
(
z
;
)deno e he e a ded esol en o
L
[
n
(
)],
GR
(
z
;
) =
(
z
+
i
0
+−L
[
n
(
)])
−1.
The cons ained XC po en ial om §IV C gene alizes o he ime domain ia he
Keldysh con ou : o i s o de in a small ime–dependen densi y a ia ion
δn
(
, ω
), he induced XC ield
is linea and de ines
xc
as a e a ded ke nel. Using he double–ope a o –in eg al calculus and linea izing
L[n]in ime, one ob ains he causal ep esen a ion
xc( , 0;ω) =
m
X
k=1
λkΦZΓk
dz
2πi k(z)GR(z+ω)δL
δn( )GR(z)δL
δn( 0)s
,(133)
whe e he subsc ip
s
deno es symme iza ion unde (
, 0
) o en o ce ecip oci y, and
GR
(
z
+
ω
)is
unde s ood wi h
=ω >
0 o analy ici y. De i a ion ske ch: linea ize he spec al–law po en ial
(97)
, w i e
he ime–dependen pe u ba ion as a con olu ion in ime o esol en s, pass o equency, and e ain he
e a ded componen by he Keldysh p esc ip ion [
246
]. The bilinea DOI ke nel na u ally p oduces he
wo– esol en s uc u e in
(133)
. Equa ion
(133)
p o ides a closed o mula o
xc
once
L
[
n
], i s F éche
de i a i e δL/δn, and he p obe se { k}a e speci ied.
Causali y and K ame s–K onig.
Because
GR
(
·
)is analy ic in he uppe hal –plane and Γ
k
enci cles
σ
(
L
)a a ixed dis ance, he in eg and o
(133)
is analy ic in
ω
o
=ω >
0, and
xc
(
ω
)sa is ies he
He glo z p ope y equi ed by linea – esponse causali y:
ω= xc
(
q, ω
)
≤C
(
q
) o
ω >
0wi h
C
(
q
)bounded
(comes om he esol en no m bound). The e o e
xc
obeys he K ame s–K onig dispe sion ela ions and
i s eal and imagina y pa s a e Hilbe ans o ms o each o he [
242
]. This immedia ely ensu es a causal
TDDFT Dyson equa ion (132) and excludes ad hoc equency ansä ze ha would iola e analy ici y.
S a ic limi , comp essibili y, and ze o– o ce heo em.
Se ing
ω
=0 in
(133)
collapses i o he
s a ic sandwich used in §V B; hence he comp essibili y iden i y (cha ge channel) and spin s i ness
(spin channel) a e eco e ed wi hou addi ional assump ions. Mo eo e , he XC o ce
Rd n
(
,
)
∇ (con)
xc
anishes o any uni o m accele a ion (ha monic–po en ial heo em), a di ec consequence o ansla ional
co a iance o
L
[
n
]and he Ad–in a ian s a e Φ; he co esponding ze o– o ce heo em o
xc
ollows by
di e en ia ion [244].
–sum and high– equency momen .
F om he Dyson ela ion, he spec al ep esen a ion o
χ
implies he
–sum:
R∞
0dω ω =χ
(
q, ω
) =
−πnq2/
2
.
The asymp o ics o
xc
needed o sa is y he sum ule
is ob ained om
(133)
by using he la ge–
ω
expansion o
GR
(
z
+
ω
),
GR
(
z
+
ω
)
∼ω−1
+
ω−2
(
L−z
)+
··· ,
which yields
xc
(
q, ω
) =
ω−2α2
(
q
) +
O
(
ω−3
), wi h
α2
ixed by equal– ime commu a o s o he densi y
37
wi h
L
(sho – ime expansion) [
243
]. This gua an ees ha inse ing
xc
in o
(132)
p ese es he
–sum
independen ly o he speci ic p obe se .
Long–wa eleng h/ equency s uc u e and known limi s.
Fo he HEG, he small–
q
expansion
o (133) eads
xc(q, ω) = A0(n) + A2(n)q2+i ω B1(n) + O(q4, ω2, ωq2),(134)
whe e
A0, A2
a e he s a ic coe icien s o §VB. The linea –in–
ω
e m
B1
a ises om he imagina y pa
o he wo– esol en p oduc and encodes low– equency dissipa ion;
(133)
ensu es ha
B1≥
0( o
ω >
0)
so ha
= xc ≤
0and
=χ≤
0. In he adiaba ic local densi y (ALDA) limi ,
GR
(
z
+
ω
)
→GR
(
z
)and
B1→
0,
eco e ing he G oss–Kohn ke nel as a equency–independen
A0
(
n
)[
240
]. Beyond ALDA, he s uc u e
(134) in e pola es smoo hly o memo y ke nels used in p ac ical TDDFT and elec on liquids [245].
Minimal b aided/nonlocal co ec ion om non–cen al cu a u e.
When he highe 2–cu a u e
(
F,G
)along he esponse pa h has a non–cen al componen (§III D), he in a ian s
S k
acqui e con olled
d i s. A he le el o xc his induces a minimal nonlocal co ec ion ha we w i e schema ically as
∆ xc( , 0;ω) = εR"X
k
λkΦZΓk
dz
2πi k(z)GR(z+ω)δL
δn( )GR(z)δL
δn( 0)#,(135)
whe e
ε
=
O
(
kRk
)measu es non–cen al cu a u e, and Ris a b aiding ope a o ac ing on he enso
s uc u e o he wo–poin ke nel (spin, subla ice, alley). Algeb aically, Rimplemen s he b aid
cV,W
on
he ele an ep esen a ion ca ego y and sa is ies he hexagon iden i ies; in momen um–space his amoun s
o a symme y–consis en eshu ling o channels (e.g., longi udinal/ ans e se spin) ha p ese es
causali y and sum ules. Cohe ence o R(hexagon) p e en s ambigui y in iple–channel composi ions
and is he ca ego ical shadow o he Yang–Bax e equa ion; see [
247
,
430
]. The co ec ion
(135)
is he
lowes nonlocal e m compa ible wi h he measu ed cu a u e and wi h (i) analy ici y (since Ris algeb aic
and commu es wi h he bounda y alues o
GR
), (ii) cha ge conse a ion (i s longi udinal p ojec ion is
di e gence– ee), and (iii) he
–sum (high– equency decay unchanged). In p ac ice,
ε
is in e ed om
he d i bounds o §III D, and Ris chosen by symme y (e.g., SU(2) spin b aiding o la ice b aid).
Algo i hmic ealiza ion.
Equa ion
(133)
is e alua ed wi h he same esol en machine y used o
(con)
xc
:
a ional quad a u e on Γ
k
and shi ed linea sol es o
GR
(
z
+
ω
)and
GR
(
z
)a a small se o equencies.
The small–
q
coe icien s
A0, A2, B1
ollow om ei he (i) ini e–di e ence sampling in ecip ocal space, o
(ii) di ec use o he symbol/hea –ke nel expansion in §V A. When non–cen al cu a u e is de ec ed, he
b aided co ec ion
(135)
is u ned on wi h
ε
se by he a p io i e o ba and R ixed by he sys em’s
symme y class.
Summa y.
S a ing om he spec al–law esol en calculus, we ob ained a causal and conse ing
o mula
(133)
o he dynamical ke nel
xc
(
q, ω
) ha educes o he s a ic coe icien s o §VB a
ω
=0,
sa is ies K ame s–K onig and exac sum ules, and admi s a minimal nonlocal b aided co ec ion
(135)
con olled by non–cen al cu a u e. No eg ession en e s: he equency dependence ollows uniquely
om he e a ded esol en s o L[n]and he chosen spec al p obes.
D. Consis ency: wa d iden i ies, causali y, ene gy–ke nel equi alence, and symme y
cons ain s
Aim.
We e i y ha he s a ic and dynamic objec s cons uc ed in §V A–§VC o m a ma hema ically
and physically consis en se : (i) he dynamical ke nel
xc
(
q, ω
)o Eq.
(133)
is causal/analy ic, obeys
wa d iden i ies (con inui y, ze o o ce/ o que), and sa is ies he comp essibili y and
–sum ules; (ii) i s
ω→
0and
q→
0limi s ma ch he s a ic coe icien s o §V B; (iii) inse ing
xc
in o he TDDFT Dyson
equa ion ep oduces he adiaba ic–connec ion ene gy
(103)
— he ene gy–ke nel equi alence; (i ) symme y
cons ain s (gauge/Galilean in a iance, ime– e e sal, spin SU(2)) and sec o iza ion a e espec ed; and
( ) small non–cen al cu a u e induces only he minimal b aided co ec ion o §VC, p ese ing hese
p ope ies.
Causali y, analy ici y, and ecip oci y.
The esol en o mula
(133)
is w i en in e ms o e a ded
G een ope a o s
GR
, hence he ke nel
xc
(
, 0
;
ω
)is a bounda y alue o a unc ion analy ic o
=ω >
0
and hus obeys K ame s–K onig dispe sion ela ions. Mo eo e , since
xc
is ob ained as a second F éche

38
de i a i e o he (causal) ac ion unc ional behind he cons ained Lag angian (Keldysh con ou e sion
o §IVA), he ecip oci y (Schwa z symme y)
xc( , 0;ω) = xc( 0, ;ω)(136)
holds iden ically (second de i a i e o a scala unc ional), while
= xc
(
ω
)
≤
0 o
ω >
0(He glo z
p ope y), ensu ing passi i y o linea esponse [
249
–
251
]. Time– e e sal in a iance is en o ced by
xc( , 0;−ω) = xc( , 0;ω) o eal L.
Wa d iden i ies: con inui y, ze o o ce/ o que, and Galilean in a iance.
The con inui y equa-
ion in linea esponse eads
ω δn
(
q, ω
) =
q·δj
(
q, ω
). In TD(C)DFT i is equi alen o a longi udinal Wa d
iden i y ying he scala ke nel o he cu en ke nel. In ou cons uc ion, ansla ional co a iance o
L
[
n
]
and Ad–in a iance o Φimply ha a uni o m boos gene a es a pu e gauge shi o
(con)
xc
; di e en ia ing
w. . . he densi y yields he longi udinal p ojec ion o
xc
consis en wi h cha ge conse a ion. The
ze o– o ce and ze o– o que heo ems ollow because uni o m accele a ions/ o a ions ac as inne de i a ions
on
A
= C
∗
(
L
), hence hei expec a ion unde Φ anishes; he co esponding linea ized iden i ies o
xc
hold au oma ically. In he cu en –densi y language, hese a e he TD(C)DFT wa d iden i ies de i ed in
quan um con inuum mechanics and de o ma ion unc ional heo y [252–254].
S a ic–dynamic ma ching and sum ules.
Se ing
ω
=0 in
(133)
educes he wo– esol en sandwich
o he s a ic o m used in §VB, hence
lim
ω→0 xc(q, ω) = xc(q, 0) = A0(n) + A2(n)q2+O(q4),(137)
wi h
A0
ixed by he comp essibili y iden i y and
˜
A0
by spin s i ness;
A2,˜
A2
a e gi en by
(127)
. Fo he
–sum, he la ge–
ω
expansion o
GR
(
z
+
ω
)shows ha
xc
(
q, ω
) =
O
(
ω−2
)uni o mly in
q
; inse ed in he
Dyson equa ion
(132)
, his gua an ees ha he i s equency momen o
=χ
equals
−πnq2/
2exac ly,
independen o he p obe se , in ag eemen wi h s anda d many–body esul s [
254
–
256
]. Highe equency
momen s ( hi d momen ) educe o equal– ime commu a o s o Land a e likewise sa is ied because he
sho – ime expansion o he esol en s is exac a hese o de s.
Ene gy–ke nel equi alence (ACFD consis ency).
Le
χ
be he in e ac ing esponse ob ained om
(132) wi h he ke nel (133). De ine he ACFD unc ional
EACFD
xc [n] = −1
2πZ1
0
ds Z∞
0
dω T χs(iω)−χ0(iω) c.(138)
Using he Dyson ela ion a coupling
s
and he iden i y
∂sχs
=
χs∂s
(
χ−1
s
)
χs
, wi h
∂s
(
χ−1
s
) =
−∂s xc,s
(since
c
is
s
–independen ), one e i ies ha
(138)
educes by in eg a ion o e
s
o he adiaba ic–
connec ion o mula
(103)
; he needed equali y
∂s xc,s
=
Pk
(
∂sλk
)
∂S k/∂n
ollows om di e en ia ing
he cons ained Lag angian and using he en elope heo em (c . §IV D). The e o e,
EACFD
xc [n] = Econ
xc [n] o he ke nel xc de ined by (133).(139)
This es ablishes he ene gy–ke nel consis ency: he same mul iplie s
{λk}
ha en e
(con)
xc
also ende
he ACFD ene gy iden ical o he cons ained adiaba ic–connec ion ene gy, in he spi i o Φ–de i able
(conse ing) cons uc ions [257, 258].
Exchange–only and exac –exchange checks.
Fo exchange–only (gKS wi h Fock exchange and
OEP), choose p obes and
L
[
n
]such ha
δL/δn
picks he exchange–only a ia ion. Then
(133)
educes
o he Pe e silka–Gossmann–G oss exac –exchange ke nel in he small–ma ix o mula ion, p o iding
a s ingen sani y check [
259
]. The s a ic limi ep oduces he well–known exchange GEA coe icien s,
consis en wi h (130).
Symme y classes, sec o iza ion, and b aided co ec ions.
Spin SU(2), la ice poin –g oup, and
ime– e e sal symme ies ac on he ep esen a ion ca ego y o
A
= C
∗
(
L
);
xc
ans o ms co a ian ly
unde hese ac ions due o Ad–in a iance o Φand he unc o iali y o he anspo (c . §II C). Sec o iza ion
by
K0
(§IIIE) implies ha
xc
has no ma ix elemen s coupling p o ec ed spec al sec o s as long as he
gap is open— his ollows om inse ing he co esponding Riesz p ojec o s in o
(133)
. When non–cen al
cu a u e is small bu nonze o, he minimal b aided co ec ion
(135)
p ese es hese symme ies: he
39
b aiding ope a o Rsa is ies he hexagon cohe ence and ac s only wi hin symme y–compa ible channels,
so causali y, sum ules, and ecip oci y emain in ac .
Nume ical diagnos ics and a pos e io i es s.
In compu a ions, we moni o : (i) K ame s–K onig
consis ency by compa ing
< xc
ob ained di ec ly wi h he Hilbe ans o m o
= xc
; (ii) comp essibili y
(s a ic
q→
0limi ) s. LDA second de i a i e checks
(124)
; (iii) he
–sum ia nume ical in eg a ion o
ω=χ
(
q, ω
)a mul iple
q
; (i ) ecip oci y ia
k xc − >
xck
; and ( ) sec o leakage ia
k
(1
−PI
)
xc PIk
o
p o ec ed
I
. All es s a e sa is ied o wi hin he unca ion e o o con ou and a ional quad a u es and
he mul iplie s’ con e gence ole ance.
Summa y.
The dynamical ke nel om esol en s is causal/analy ic, conse ing, and symme ic; i s
s a ic limi s ma ch he g adien –expansion coe icien s, and i ep oduces p ecisely he cons ained
adiaba ic–connec ion ene gy. Symme y, sec o iza ion, and (when needed) b aided co ec ions a e buil in.
These consis ency p ope ies ollow om he combina ion o Ad–in a ian s a es, spec al–law cons ain s,
and highe –connec ion anspo , a he han om empi ical calib a ion.
VI. THE ATLAS OF UNIVERSAL TILES
A. Wha is he a las? Fini e se o iles (in eg able backbones) and ansi ion da a
Pu pose.
This subsec ion o malizes he a las we use o p opaga e exac spec al cons ain s ac oss
chemis y and ma e ials wi hou e i s (c . §III D, §IVE). An a las consis s o a ini e se o iles—
in eg able o exac ly sol able backbones—each equipped wi h spec al in a ian s, esponse coe icien s,
and sec o labels; plus ansi ion da a ha cohe en ly glues iles on hei o e laps and anspo s in a ian s
along exac legs. Ma hema ically i mi o s a
C∞
a las on a mani old: cha s
Uα
co e ing a pa ame e
mani old Θ(geome y, composi ion, ields), wi h ansi ion maps obeying cocycle iden i ies [
260
–
262
].
Physically, each ile cap u es a uni e sal egime (me allic sc eening, low-dimensional Lu inge physics,
s ong-coupling SCE/Mo , Di ac quasipa icles, e c.) wi h exac spec al da a, and he ansi ion
unc o s implemen highe -gauge pa allel anspo wi h anishing (cen al) cu a u e on o e laps.
Pa ame e mani old and co e age.
Le Θbe he ( ypically high-dimensional) space o ex e nal and
in e nal pa ame e s:
Θ3θ=n, ∇n, τ, jp,m;la ice, geome y, ields, T, µ, . . . ,
as in §IIIA. We co e Θby ini ely many open se s
{Uα}α∈A
(cha s), chosen so ha on each
Uα
he e
exis s a e e ence in eg able model ( ile) whose exac spec al laws a e anspo ed o any
θ∈Uα
along
exac legs ( anishing cen al cu a u e; §IIID). Fini e co e age is jus i ied ope a ionally by compac ness
o he ele an educed pa ame e se s in p ac ice (bounded densi y/g adien anges, disc e e symme y
classes, gap/semi-me al dicho omy) and by he ac ha low-o de spec al momen s and sec o labels
span a ini e se o in a ian s ha de e mine he cons ain s we en o ce (c . §IIIC).
De ini ion ( ile). A ile Tαis a uple
Tα=Aα, Lα,Φα,{ k}m
k=1,φα,{PI,α},Dα,(140)
wi h:
•Aα
= C
∗
(
Lα
)a
C∗
–algeb a gene a ed by an in eg able o exac ly sol able Lax- ype ope a o
Lα
(Be he-ansa z sol able, ee/Di ac, SCE/OT, e c.) [263–265, 268].
•
Φ
α
an Ad-in a ian s a e ( ace, ibe ace, o KMS) ixing he e alua ion o spec al laws on he
ile.
• { k} he (global) p obe se ( esol en s, windows) ixed once o he a las (c . §IIIC).
•φα= (φα,k)m
k=1 he exac a ge alues o Φα
 k(Lα)on he ile.
• {PI,α}
a se o Riesz p ojec o s isola ing p o ec ed spec al sec o s ( alence/ac i e, spin blocks),
wi h K0classes used o sec o iza ion (§IIIE).
40
• Dα
auxilia y da a: (i) symbol/hea -ke nel coe icien s o
Lα
( o p oduce GEA/me a-GGA cons an s,
§VB); (ii) dynamic ke nel coe icien s (small-
q, ω
) o he ile (§VC); (iii) ile-speci ic pa ame e s
(e.g., HEG densi y–dependen coe icien s [
267
]; Lu inge
K
-pa ame e s [
265
]; SCE co-mo ion maps
[266]).
On a cha
Uα
, he a las en o ces Φ(
k
(
L
[
n
])) =
φα,k
(exac ly on la legs; wi hin con olled bounds
o he wise), and uses {PI,α} o gua d sec o s du ing SCF (§IIIE).
Rep esen a i e iles (non-exhaus i e).
1.
HEG ile (me allic sc eening).
Lα
=
−1
2∇2
+
HEG
wi h uni o m
n
;Φ
α
is he ace pe olume;
φα
encodes comp essibili y and –sum consis en wi h he PW92 EOS [267].
2.
1D Lu inge ile (quasi-1D). In eg able Lu inge liquid/XXZ chain backbone wi h pa ame e s
( , K); dynamic exponen s and powe -law ke nels ixed by Be he da a [264, 265].
3.
SCE/OT ile (s ong-coupling). S ic ly co ela ed elec ons ealized ia mul ima ginal op imal
anspo wi h Coulomb cos ; co-mo ion unc ions and s ong-coupling coe icien s
W∞, W0
∞
in
Dα
[266].
4.
Di ac/Weyl ile (semime al).
Lα
=
Fσ·p
(plus symme y-allowed pe u ba ions);
Dα
con ains
low-ene gy ke nel s uc u e o linea dispe sion [268].
5.
Su ace/edge ile. Hal -space jellium o ibbon models wi h explici image-sc eening/local- ield
s uc u e (used o he e ogeneous in e aces).
The a las con ains only a ini e numbe o such iles— ypically 6–12 o co e common egimes in
molecules/solids; selec ion is guided by co e age o in a ian s and by applica ion domains (e.g., high-
h oughpu c ys als s. molecula chemis y) [269, 270].
T ansi ion da a and cocycle condi ions.
On an o e lap
Uαβ
:=
Uα∩Uβ6
=
∅
, he a las supplies
ansi ion da a
gαβ =Fαβ, Tαβ,Rαβ,
consis ing o :
•
a
∗
–isomo phism ( anspo unc o )
Fαβ
:
Aβ→Aα
ha in e wines
Lβ
wi h
Lα
along exac legs
(cen al cu a u e ze o), and pushes o wa d he s a e Φβ o Φαon o e laps (Ad–in a iance);
•
a linea map
Tαβ ∈GL
(
m, R
) ha ela es he p obe coe icien s and momen no maliza ions so ha
Φ
α
(
k
(
L
)) =
P`
(
Tαβ
)
k`
Φ
β
(
`
(
L
)) on
Uαβ
; when cen al cu a u e anishes,
Tαβ
is cons an and
p ese es he Schu me ic induced by Ain §IVB;
•
an op ional b aiding R
αβ
(only when non-cen al cu a u e is small bu nonze o) ac ing in enso
channels (spin, alley), consis en wi h hexagon cohe ence (c . §VC).
On iple o e laps Uαβγ, he cocycle ela ions hold:
Fαβ ◦Fβγ =Fαγ, Tαβ Tβγ =Tαγ ,Rαβ Rβγ =Rαγ,(141)
ensu ing consis en gluing and pa h-independen anspo o in a ian s ( la cen al connec ion). When
gaps close, K0sec o labels change by in ege jumps, acked consis en ly ac oss iles (§III E).
How a eal sys em uses he a las (ope a ional p o ocol).
Gi en (
G,composi ion, T, B, . . .
), we
build a local desc ip o
ϑ
(
θ
)
∈Rp
om low-o de spec al summa ies (e.g., HEG densi y, educed g adien s,
band gap, e ec i e mass, Di acness measu e). We choose a co e ing cha
Uα
by nea es -neighbo o
ule-based assignmen and ead om T
α
he a ge s
φα
and p ojec o s
{PI,α}
. We hen sol e he
cons ained KS p oblem wi h he mul iplie s
λ
(Sec ions IV, V), en o cing he spec al laws and sec o
weigh s. I
θ
app oaches an o e lap
Uαβ
, we p e-map he cons ain s ia
Tαβ
; i non-cen al cu a u e is
de ec ed, we include he minimal b aided co ec ion de e mined by R
αβ
wi hin he e o ba o §IIID.
Along pa hs in Θ, in a ian s a e anspo ed (exac legs) and moni o ed (cu a u e bounds), wi h sec o
s abili y gua an eed by K0un il a ue gap closu e occu s.
41
Why a ini e numbe o iles su ices (p agma ic a gumen ).
The cons ain s we en o ce a e
de e mined by ini ely many momen s o he spec al measu e
νn
( esol en samples and p ojec o
weigh s); by Tchakalo /Ca a héodo y- ype esul s, a ini e disc e e measu e ep oduces hese momen s
exac ly, hence a ini e epe oi e o uni e sal momen pa e ns su ices ac oss Θonce symme y classes
and sec o iza ion a e accoun ed o . In eg able backbones p o ide hese pa e ns in closed o m: HEG o
me allic sc eening, Lu inge o 1D co ela ions, SCE/OT o s ong coupling, Di ac o linea dispe sions,
e c. The ansi ion da a hen ealize descen o hese pa e ns ac oss o e laps in he sense o shea heo y
[261, 262].
S o ed da a pe ile ( o ep oducibili y).
Each T
α
ships wi h: (i)
φα
and hei sensi i i ies
∂φα/∂n
; (ii) ecommended con ou nodes
{zk}
and quad a u e weigh s; (iii) GEA/me a-GGA s a ic
coe icien s (
A0, A2,˜
A0,˜
A2
) and dynamic small-
ω
cons an s (
B1
) compu ed on he ile; (i ) p ojec o
de ini ions and
K0
classes; ( ) ansi ion maps
Tαβ
o neighbo ing iles and he associa ed e o moni o s;
( i) admissible anges o desc ip o s
ϑ
de ining
Uα
. This enables po able use o he a las ac oss codes
and domains (molecules, solids, 2D ma e ials), and acili a es high- h oughpu compu a ions [269, 270].
Summa y.
An a las is a ini e, cohe en collec ion o in eg able iles plus ansi ion da a ha make
spec al-law cons ain s po able ac oss chemical space. On each cha , exac in a ian s om he ile a e
en o ced; on o e laps, ansi ion maps (and, i necessa y, minimal b aided co ec ions) ensu e consis en
anspo wi hou e i s. Sec o labels a e s abilized by
K0
, and pe o mance emains a KS cos because
all compu a ions educe o e alua ing a small numbe o esol en /p obe quan i ies wi h Ad–in a ian
s a es. This s uc u e is he engine behind ou “no- i ing” ab-ini io DFT: global accu acy assembled
om exac , local building blocks.
B. Rep esen a i e iles & in a ian s ( able)
Aim.
We ins an ia e he abs ac no ion o a ile (§VIA) wi h a ini e lis o ep esen a i e, in eg able
(o exac ly sol able) backbones ha co e he egimes mos o en encoun e ed in quan um chemis y and
ma e ials. Fo each ile we lis : (i) a KS–like gene a o empla e
L
[
n
] ha se s he ope a o class; (ii)
a minimal p obe se
{ k}
( esol en samples and Riesz p ojec o s) su icien o pin down he spec al
in a ian s; (iii) he associa ed a ge alues
{φk}
( ixed by exac iden i ies on he ile); and (i ) quali a i e
no es abou he egime o alidi y. Th oughou ,
Pocc
deno es he Riesz p ojec o on o he occupied
subspace a he chosen chemical po en ial, and
Pgap
he p ojec o on o a spec al window isola ing he
undamen al gap (see §III E).
Summa y able.
48
I EN≤ε(use ole ance), s op. O he wise, choose a maximize
ˆ
θ∈a g max
θ∈K
min
αEα(θ),(148)
ins an ia e a new ile T
N+1
a (o nea )
ˆ
θ
, popula e i once (§VI C), and add o he a las. This is he
s anda d weak g eedy s ep amilia om educed–basis and adap i e FEM algo i hms, he e wi h he
cu a u e–weigh ed e o es ima o
(146)
. Sampling he
sup
in
(147)
may be done on a spa se Smolyak
g id adap ed o he desc ip o coo dina es o con ol dimension [311].
Te mina ion and co e age gua an ee (ske ch).
Endow
K
wi h he geodesic dis ance
dΛ
induced
by he line elemen Λ
α
(
θ
)
ds
on o e laps ( he choice o cha is imma e ial on exac legs and p oduces
equi alen dis ances when cu a u e is bounded). Assume: (i)
K
is compac ; (ii) Λ
α
(
θ
)is bounded abo e
and below on
K
; (iii) he ansi ion maps sa is y he cocycle law (Eq.
(141)
) so ha
dΛ
is well de ined.
Then he g eedy cons uc ion p oduces, a e ini ely many s eps, an
ε
–co e o
K
in he me ic
dΛ
(hence
in
dφ
), and he e o e
EN≤ε
o some ini e
N
. This is a di ec applica ion o co e ing–numbe /en opy
a gumen s (any compac me ic space admi s a ini e
ε
–ne ) oge he wi h he con e gence o weak g eedy
co e ings [
312
,
313
]. No i ing occu s: each new ile is compu ed once and glued by unc o ial anspo .
Re inemen in p ac ice: ma ke s and selec ion.
We use h ee ma ke s o decide ha a egion
needs e inemen :
1.
Cu a u e ma ke . I
Eα
(
θ
)compu ed along he cu en pa h exceeds a local ole ance
εloc
, ma k
θ
o e inemen (Dö le – ype ma king) [314].
2.
Sec o leakage ma ke . I a p o ec ed p ojec o iola es idempo ency beyond ole ance (
k
(
Pglue
)
2−
Pgluek>δ), ma k he neighbo hood o a new ile ocused on he leaking sec o .
3.
Dynamical ma ke . I he small–
ω
coe icien s o
xc
(
q, ω
)(e.g.,
B1
) in e ed om esol en sampling
de ia e beyond hei ce i ied bounds on he wo king ile, ma k o a dynamic–ke nel–awa e ile.
Candida e cen e s
ˆ
θ
a e hen selec ed by a co e – ee sea ch ha maximizes he local e o es ima e
while ensu ing spa ial sepa a ion among new cen e s; co e ees gua an ee
O
(
log N
)inse ion and que y
in doubling me ics [
315
]. I he desc ip o mani old is cu ed, new cen e s a e e ac ed o he mani old
by a Riemannian mean (Ka che a e age) o keep cha s well condi ioned [316, 317].
Gluing a new ile ( ansi ion upda es).
A e popula ing T
N+1
, we compu e: (i) he ansi ion
∗
–isomo phisms
Fα, N+1
by pa allel anspo along sho geodesics in
dΛ
; (ii) he p obe–coe icien map
Tα, N+1
by leas –squa es on he o e lapping sample o spec al–law alues (cons ained o p ese e he
Schu me ic o §IVB); and (iii) he (op ional) b aided ope a o R
α, N+1
de e mined by symme y class i
non–cen al cu a u e is nonze o on he o e lap. We hen e i y he cocycle iden i ies on iple o e laps
and s o e he ce i ica es.
Complexi y and amo iza ion.
The dominan cos s a e: (a) compu ing
Eα
(
θ
)along he cu en
SCF pa h (cheap: i euses esol en ac ions al eady e alua ed o he cons ain s); (b) occasional ile
ins an ia ion (one– ime ED/TBA/OT mic o–sol e; §VIC); and (c) building he ansi ion maps (small
linea sol es on o e lap samples). The amo ized o e head is negligible ela i e o he KS cos , and he
numbe o iles emains modes because he g eedy ule a ge s only egions whe e cu a u e p e en s
ai h ul anspo .
Example: Mo c osso e e inemen .
Along a p essu e–d i en pa h, he SCE/Mo ile co e s he
insula ing side and he HEG ile he me allic side. Nea he c osso e , he cu a u e ma ke spikes as
p ojec o weigh s d i (cha ge localiza ion weakens). A new in e media e ile is added, seeded by a ini e
Hubba d slice wi h pa ame e s chosen om he local desc ip o (
U/W, s
); i s in a ian s (pai co ela ions,
gap p ojec o , comp essibili y) a e compu ed once by ED and glued in. A e inse ion,
Eα
d ops below
ole ance and he SCF pa h p oceeds wi hou sec o leakage o empi ical uning.
Why no i ing is in oduced.
Re inemen ne e adjus s coe icien s o da a; i only decides whe e an
addi ional exac in eg able backbone is needed so ha anspo e o emains below ole ance. Ta ge s
{φk}
o he new ile a e compu ed om i s p inciples (analy ic o small exac sol e s), s o ed wi h
ce i ica es, and hen en o ced by mul iplie s in p ecisely he same way as exis ing iles. Thus adap i e
e inemen gua an ees co e age while p ese ing he ab–ini io, i – ee cha ac e o he me hod.

49
Summa y.
We endowed he a las wi h a cu a u e–d i en, g eedy e inemen mechanism. A cu a u e–
weigh ed me ic
dΛ
p o ides an a p io i e o es ima o ; i he es ima o exceeds ole ance, a new ile
is c ea ed, popula ed once, and glued cohe en ly. On compac egions, he p ocedu e e mina es a e
ini ely many s eps, yielding a ini e
ε
–co e o he pa ame e space in he sense o he spec al–law
me ic, wi h o mal gua an ees bo owed om co e ing/en opy and g eedy educed–basis heo y. No
i ing is in oduced a any s age.
VII. ALGORITHMS
A. SCF wi h spec al-law cons ain s (pseudocode)
Aim.
We p esen a p ac ical sel –consis en – ield (SCF) algo i hm ha implemen s he cons ained
a ia ional p inciple o §IV A, he mul iplie equa ions o §IV B, and he cons uc i e XC o mula
(97)
.
A each mac o–i e a ion he me hod (i) builds he KS–like gene a o
L
[
n
], (ii) sol es he KS p oblem (o
applies a densi y-upda e map), (iii) e alua es he spec al laws
S k
[
n
] = Φ(
k
(
L
[
n
])) and hei esiduals,
(i ) upda es he mul iplie s
{λk}
om a small New on/B oyden sys em, ( ) assembles he cons ained
XC po en ial
(con)
xc
=
PkλkVk
wi h
Vk
gi en by esol en sandwiches, and ( i) mixes densi ies using
Ande son/DIIS wi h Ke ke – ype p econdi ioning. Con e gence is decla ed when bo h he densi y and
he spec al-law esiduals a e small:

n(j+1) −n(j)
w≤εn,max
k| (j+1)
k| ≤ εS.
No a ion.
Le
k
be he ixed p obe se (§IIIC) wi h con ou s Γ
k
. W i e
R
(
z
;
L
)=(
z−L
)
−1
and
Jk
(
) =
δS k/δn
(
)( he sensi i i y dis ibu ion). The cons ain esiduals a e
k
=
S k
[
n
]
−φglue
k
,
whe e
φglue
k
a e he a las/glued a ge s a he cu en s a e (§VID). The s a ic KS esponse ope a o
(S e nheime /DFPT) is deno ed χs[319].
Algo i hmic ou line (pseudocode).
50
Algo i hm 1: SCF wi h spec al-law cons ain s
Inpu :
geome y/basis o g id; ini ial densi y
n(0)
; a las a ge s
{φglue
k}
a he ini ial s a e; ole ances
(εn, εS); mixing his o y size mDIIS.
Ou pu : con e ged densi y n?, mul iplie s λ?, and (con)
xc .
0.
Ini ializa ion. Se
j←
0, choose ini ial mul iplie s
λ(0)
k
=0, and p ecompu e a las/ ansi ion da a
on he wo king cha (Sec. VI D). Ini ialize DIIS/Ande son his o y H=∅.
1. Build KS–like gene a o . Assemble
L(j)≡L[n(j)] = T+ ex + H[n(j)] + (con)
xc [n(j);λ(j)].
2.
KS sol e / densi y upda e. Sol e he KS eigenp oblem (o apply a subspace FEAST/con ou
s ep) o ob ain he ou pu densi y n(j)
ou . De ine he SCF esidual g(j):= n(j)
ou −n(j).
3. E alua e spec al laws and esiduals. Fo each p obe k:
S k[n(j)] = ΦZΓk
dz
2πi k(z)R(z;L(j)), (j)
k:= S k[n(j)]−φglue
k.
Op ionally compu e he cu a u e moni o
kRk
(Sec. VID) and upda e glued a ge s i a ile
swi ch is indica ed.
4. Assemble sensi i i y ope a o s and Schu ma ix. Fo each k:
J(j)
k( )=ΦZΓk
dz
2πi k(z)R(z;L(j))δL
δn( )R(z;L(j)),
hen o m he m×mma ix (inne p oduc s in he DFPT me ic)
A(j)
kl =hJ(j)
k, χ(j)
sJ(j)
li,b(j)
k=− (j)
k−hJ(j)
k, χ(j)
sg(j)i.
5.
Mul iplie upda e (New on/B oyden). Sol e he small linea sys em A
(j)
∆
λ(j)
=
b(j)
and se
λ(j+1
2)
=
λ(j)
+ ∆
λ(j).
I A
(j)
is ill-condi ioned, apply a modi ied B oyden/DIIS upda e on
λ
using pas pai s λ(`), (`),`<j[322–324, 330].
6. Assemble cons ained XC po en ial. Compu e
V(j)
k( )=ΦZΓk
dz
2πi k(z)R(z;L(j))δL
δn( )R(z;L(j)),
and se (con)
xc [n(j)]( ) = Pm
k=1 λ(j+1
2)
kV(j)
k( ).
7.
Densi y mixing (Ande son/DIIS + Ke ke ). Fo m he p econdi ioned esidual
˜g(j)
=
Kg(j)
wi h
Ke ke il e
K
(
q
) =
α q2/
(
q2
+
q2
0
)in ecip ocal space o mi iga e cha ge sloshing [
326
]. Upda e
he Ande son/DIIS his o y
H
wi h pai s
˜g(j), n(j)
( unca e o size
mDIIS
). Compu e he
accele a ed s ep
n(j+1)
DIIS
sol ing he leas -squa es combina ion o pas esiduals [
327
–
330
]. Apply a
sa egua ded line sea ch on he mixing pa ame e o en o ce mono one dec ease o a me i unc ion
(e.g. kgk2
w+βk k2
2) [331], and se n(j+1) ←n(j+1)
DIIS .
8.
Con e gence es . I
kn(j+1) −n(j)kw≤εn
and
maxk| (j)
k| ≤ εS,s op
; else op ionally upda e he
augmen ed-Lag angian penal y
ρ
o he cons ain esiduals (i enabled), se
λ(j+1) ←λ(j+1
2)
,
j←j+ 1, and go o 1.
Rema ks on each block.
(i) KS sol e and esponse ope a o . The KS s ep in 2 can be pe o med by any s anda d i e a i e
diagonaliza ion (block–CG, LOBPCG, subspace i e a ion) o by a densi y–ma ix/con ou me hod; i s
ou pu is he densi y and (i needed) an implici ep esen a ion o
χs
ia he S e nheime equa ion [
319
].
51
Applying
χs
o a sou ce
h
(
)amoun s o sol ing a linea equa ion o he o m (
ˆ
Hs−εi
)
δψi
=
−ˆ
Qih ψi
o occupied o bi als, which we do by K ylo me hods (GMRES/BiCG) wi h diagonal o mul ig id
p econdi ione s [320, 321]. Only msuch sol es a e needed pe mac o–i e a ion o assemble A(j).
(ii) Mul iplie upda e. The New on s ep in 5 is small (dimension
m≤
6 ypically) and well–posed
p o ided he Jacobians
{Jk}
a e independen . When A
(j)
is noisy o nea ly singula we swi ch o modi ied
B oyden o o Johnson’s a ian used in plane–wa e codes [
322
–
325
], which builds a secan app oxima ion
o he in e se map 7→ λand is obus in p ac ice.
(iii) XC po en ial assembly. S ep 6 implemen s exac ly he cons uc i e o mula
(97)
; a ional quad a-
u es on Γk educe he in eg als o a ew shi ed linea sol es sha ed wi h he KS and DFPT s eps. Fo
local
L
[
n
],
Vk
(
)is mul iplica i e; o gKS ope a o s i becomes OEP-like, s ill handled a KS cos because
we ne e in e la ge ma ices explici ly.
(i ) Mixing and p econdi ioning. We use Ande son/DIIS o accele a e he ixed-poin i e a ion
n7→ nou
,
wi h Ke ke p econdi ioning in ecip ocal space o damp long-wa eleng h cha ge oscilla ions in me als
and small-gap sys ems [
326
–
330
]. A back acking A mijo line sea ch gua an ees global con e gence e en
when he DIIS ex apola ion is oo agg essi e [
331
]. Fo s ongly co ela ed o ill-condi ioned cases we
adop Ma ks–Luke’s obus mixing a ian [332].
( ) Cons ain penal ies (op ional). An augmen ed-Lag angian e m
ρ
2Pk 2
k
may be added o s abilize
ea ly i e a ions; we upda e
λk←λk
+
ρ k
e e y ew s eps and inc ease
ρ
i esiduals s agna e [
333
]. Nea
con e gence, we e e o he pu e New on/B oyden upda e o supe linea speed.
Complexi y and pa allelism.
Pe mac o–i e a ion he dominan cos s a e: (a) one KS cycle (iden ical
o s anda d DFT); (b)
m
DFPT applica ions o
χs
o assemble Aand
b
; (c)
m
quad a u e e alua ions
o
Vk
(sha ed shi s wi h (a)/(b)). Hence he o e head is
O
(
m
)KS-like linea sol es wi h
mNbands
,
keeping he wall ime close o con en ional SCF. All shi ed-sol e asks pa allelize o e
k
and quad a u e
nodes, and Ande son/DIIS uses only small dense linea algeb a.
Con e gence and sa egua ds.
The combined Ande son–New on scheme con e ges locally supe linea ly
when
χs
is Lipschi z and he Schu ma ix is nonsingula ; globally, A mijo back acking and (op ional)
augmen ed-Lag angian penal ies ensu e descen o he me i unc ion. A las cu a u e moni o s (Sec. VID)
p o ide a p io i bounds on a ainable esiduals: i he bound exceeds ole ance, a ile e inemen /swi ch
is igge ed be o e SCF s agna es (Sec. VI E).
Minimal wo king de aul s.
We ind he ollowing choices e ec i e ac oss ma e ials and molecules:
Ke ke (
α, q0
) = (0
.
8
,
0
.
5
boh −1
) o me als, (0
.
5
,
1
.
0) o insula o s;
mDIIS ∈
[6
,
12]; wo o ou
quad a u e shi s pe p obe; DFPT ole ance ma ched o
εS
so ha he mul iplie sol e is no o e -
esol ed.
Summa y.
Algo i hm 1 ealizes he cons ained Le y–Lieb amewo k a KS cos . The only new
ope a ions ela i e o s anda d SCF a e a hand ul o esol en e alua ions (al eady p esen in many
mode n eigensol e s) and a small mul iplie sol e. Con e gence is moni o ed bo h in he densi y and in
he spec al-law esiduals, gua an eeing ha he en o ced in a ian s hold a he ixed poin wi hou any
semiempi ical i ing.
B. E icien esol en s: shi ed linea sol es, selec ed in e sion, euse o DFPT/TDDFT
in as uc u e, and GPU accele a ion
Aim. All spec al–law objec s in his wo k educe o con ou –in eg al esol en sandwiches
Q [n;B] = ΦIΓ
(z)
2πi R(z;L[n])B[n]R(z;L[n]) dz, R(z;L) := (z−L)−1,(149)
o a bounded p obe
, a con ou Γenclosing he ele an spec um, and an ope a o
B
(e.g.
∂L/∂n
(
)).
This subsec ion de ails how o e alua e
(149)
wi h KS-class cos by combining: (i) shi ed spa se linea
sol es a a ew quad a u e nodes; (ii) selec ed in e sion o ex ac diagonals/ aces o esol en s wi hou
o ming dense in e ses; (iii) eusing DFPT/TDDFT sol e s and p econdi ione s; and (i ) GPU ba ched
ke nels. The design a ge s linea o nea -linea wall ime scaling in p ac ice, while p ese ing nume ical
s abili y and ep oducibili y.
Ra ional quad a u e and shi selec ion.
Disc e izing he con ou in eg al in
(149)
by
Nq
nodes
52
{zj, wj} u ns he ask in o Nqindependen shi ed linea sol es:
Q [n;B]≈
Nq
X
j=1
wj (zj)ΦR(zj;L)BR(zj;L).(150)
We adop quad a u es whose nodes/weigh s a e op imized o esol en in eg als: (i) o in e als,
Zolo a e – ype a ional g ids minimize he max e o o S iel jes in eg ands; (ii) o gene al spec a,
ellip ical/ci cula con ou s wi h apezoidal ules yield exponen ial con e gence (spec al analy ici y)
[334]. Nea -op imal nodes educe Nq o O(4–8) pe p obe in p oduc ion.
Shi ed spa se sol es and (block/ a ional) K ylo .
A each node
zj
we mus apply
R
(
zj
;
L
) o
one o a ew igh –hand sides. Two ou es a e used:
1.
I e a i e (ma ix– ee). Cons uc a single K ylo space
Km
(
L, b
)and euse i o all shi s
zj
(mul i–shi echnique), upda ing solu ions by sho ecu ences; block igh –hand sides handle
mul iple sou ces a once, wi h block a ional K ylo imp o ing con e gence o clus e ed spec a
[
335
]. P econdi ione s a e inhe i ed om DFPT: e.g. diagonal kine ic p econdi ioning in plane
wa es, mul ig id in eal–space g ids. We also exploi seed ecycling be ween SCF i e a ions o
wa m-s a K ylo spaces.
2.
Di ec spa se ac o iza ion (one ac o , many sol es). Fo m a spa se
LU
o
LDL>
ac o iza ion
o (
ζ−L
)a an ancho shi
ζ
(close o he ba ycen e o
{zj}
). Then sol e he shi ed sys ems
wi h pa ial e ac o iza ions o wi h She man–Mo ison–Woodbu y upda es i
{zj}
clus e igh ly
a ound ζ; accu acy is es o ed by i e a i e e inemen [336, 337].
The choice is backend-d i en: i e a i e is memo y–ligh and GPU- iendly; di ec excels when many
igh –hand sides a e needed (e.g. p ojec o windows).
Con ou -based subspace i e a ion (p ojec o s).
When
B
is a p ojec o o when we need a spec al
window (e.g.
Pocc
,
Pgap
), we di ec ly use con ou -based subspace i e a ion: FEAST/SS me hods compu e
he p ojec ed subspace by epea ed applica ions o
PjwjR
(
zj
;
L
) o andom blocks, ollowed by Rayleigh–
Ri z [338, 339]. This a oids explici eigenpai compu a ions and in eg a es seamlessly wi h (150).
Selec ed in e sion o aces/diagonals.
Many obse ables need diagonal o ace elemen s o
R
(
zj
;
L
)o o
R
(
zj
;
L
)
BR
(
zj
;
L
)unde Φ. The selec ed in e sion algo i hm e alua es only speci ied
en ies (e.g. he diagonal) o (
zj−L
)
−1
using he spa se
LU/LDL>
ac o s and he elimina ion ee—no
dense in e se is o med. In Kohn–Sham DFT his is he co e o PEXSI (pole expansion + selec ed
in e sion), which achie es
O
(
N1.5
)scaling in quasi-2D and
O
(
N2
)in 3D while e aining nea -KS accu acy
[
340
,
341
]. We adop he same idea o
(150)
: once ac o s a e a ailable (possibly a an ancho
ζ
), each
node
zj
yields diagonals/ aces a low inc emen al cos ; poles and weigh s coincide wi h he quad a u e
{zj, wj}.
Reusing DFPT/TDDFT in as uc u e.
The S e nheime /DFPT machine y al eady sol es linea
sys ems o he o m (
Hs−εi
)
δψi
=
−ˆ
Qih ψi
wi h obus p econdi ione s. Ou esol en ac ions a
zj
use
he same i e a i e ke nels, p econdi ione s, and pa allel layou , di e ing only by he complex shi and by
igh –hand sides. Likewise, TDDFT linea – esponse backends ( equency-domain S e nheime ) can be
epu posed o deli e
R
(
zj
;
L
)ac ions a imagina y equencies (bene icial o condi ioning), which a e
hen analy ically con inued by he con ou disc e iza ion.
Scheduling and euse ac oss SCF.
Le indices be
{node j}×{p obe k}×{spin}×{k-poin }
. We
schedule his 4D ba ch ac oss MPI anks (ou e ) and GPU s eams/ h eads (inne ), exploi ing ha all
asks a e emba assingly pa allel. Be ween SCF s eps we euse: (i) ac o iza ions a he ancho shi ; (ii)
K ylo subspaces ( ecycled/augmen ed a ional K ylo ); (iii) quad a u e nodes and
{wj}
; and (i ) he
spa si y pa e n/o de ings (elimina ion ee) o he ac o s. Empi ically his cu s he inc emen al cos o
esol en e alua ion o .1.3×a s anda d DFPT s ep.
GPU accele a ion. We o load h ee ho loops o GPUs:
•
Spa se ma ecs and shi ed iangula sol es (CSR/ELL o ma s wi h le el-scheduled o wa d/back
subs i u ion).
•Ba ched small dense linea algeb a (Rayleigh–Ri z on subspaces om con ou me hods).
53
•Block do -p oduc s/ educ ions o Φ(·)e alua ions and DIIS/Ande son ke nels.
Hyb id MAGMA-s yle algo i hms o e lap panel ac o iza ions on CPUs wi h ailing upda es on GPUs
[
342
]. On mul i-GPU nodes, asks a e agg ega ed by node index
j
o maximize da a euse o (
zj−L
)
ac o s; communica ion is o e lapped wi h compu e using CUDA s eams.
P econdi ioning and obus ness.
Shi ed sys ems inhe i he ill-condi ioning o (
H−σ
)nea he
spec um. We employ: (i) physics-awa e p econdi ione s (kine ic
∝ −∇2
in PW, mul ig id in RS, localized
basis diagonal in AO); (ii) spa se app oxima e in e ses o shi ed ope a o s (compu ed once a he ancho
shi and e-used); and (iii) lexible K ylo wi h igh –p econdi ione s upda ed e e y ew i e a ions. A
sho ou e New on co ec ion on he shi (upda ing ζ) main ains a small esidual ac oss all zjnodes.
P ecision and s abili y.
We use mixed p ecision: FP32 ma ecs inside K ylo sweeps and FP64
co ec ion s eps ( eliable upda es), wi h esidual con ol based on backwa d e o
kb−
(
z−L
)
xk/kbk
.
Fo di ec ou es, i e a i e e inemen in complex a i hme ic es o es FP64 accu acy a e FP32 spa se
ac o iza ions. All educ ions in Φ(·)use Kahan–Babuška compensa ed summa ion.
Complexi y model ( ule o humb).
Le
N
be he numbe o basis unc ions/g id poin s and
Nq
he
numbe o quad a u e nodes pe p obe ( ypically 4–8).
•
I e a i e ou e: cos
≈Nq
ma ec-equi alen s wi h he shi ed ope a o ; i e a ion coun is es-
sen ially ha o one DFPT sol e hanks o mul i-shi euse and good p econdi ioning; memo y
=O(mK ylo N).
•
Di ec + selec ed in e sion: one symbolic analysis + ac o iza ion (amo ized o e SCF) plus
Nq
selec ed in e sions wi h O(N1.5)(2D) / O(N2)(3D) complexi y [337, 340, 341].
In bo h cases, pa allel e iciency is high because nodes and p obes a e independen .
Minimal implemen a ion ecipe.
1. Fix a quad a u e (Zolo a e o ellipse) o each p obe amily; p ecompu e {zj, wj}.
2.
P o ide an in e ace
apply_R(z, hs)
ha euses DFPT p econdi ione s and suppo s ba ches
(mul i-shi /block).
3. I aces/diagonals a e equen ly needed, wi e a selec ed-in e sion backend (dis ibu ed memo y).
4.
Expose a
esol e_V_k
p imi i e ha e u ns
Vk
(
)by looping o e
j
and applying
R
(
zj
) wice o
he S e nheime sou ces ha ealize ∂L/∂n( ).
5. Ba ch asks o e {j, k, spin,k}; enable GPU ma ecs/ iangula sol es.
Summa y.
E icien e alua ion o esol en sandwiches uses a small se o shi ed spa se sol es o ganized
by a ional quad a u e. Con ou -based subspace i e a ion deli e s p ojec o s; selec ed in e sion compu es
diagonals/ aces a low ma ginal cos ; DFPT/TDDFT sol e s and p econdi ione s a e eused e ba im;
and GPUs handle he ba ched ke nels. The esul is a po able, high- h oughpu backend ha keeps he
addi ional o e head o spec al-law cons ain s wi hin a modes cons an ac o o s anda d SCF.
C. P ojec o cons ain s & K0 acking: Riesz in eg als, gap moni o ing, quan ized jumps on
closu e
Aim.
We de ail how he p ojec o cons ain s used o sec o p o ec ion (§III E) a e implemen ed and
moni o ed du ing SCF, and how he associa ed
K0
in a ian s a e acked. The ing edien s a e: (i)
Riesz–Dun o d spec al p ojec o s o de ine sec o weigh s, (ii) ce i ied gap moni o ing o gua an ee (o
de ec he loss o ) homo opy in a iance o p ojec o s, and (iii) a spec al– low/index mechanism ha
egis e s quan ized jumps o sec o coun s when a gap closes and eopens. Toge he hey deli e obus
sec o iza ion wi hou i s and wi h ma hema ically con olled upda es.
Riesz p ojec o s and sec o cons ain s.
Le
L
[
n
]be he KS–like gene a o a a gi en SCF i e a e
and le
I⊂R
be a spec al window (e.g., a gap window o an occupied band block). Fo a posi i ely

54
o ien ed con ou ΓIenclosing Iand no o he pa o σ(L[n]), he Riesz p ojec o is
PI(L[n]) = 1
2πi IΓI
(z−L[n])−1dz, (151)
see, e.g., [348, Ch. VII]. Ou sec o cons ain s ix he weigh s
wI[n] = ΦPI(L[n])!
=φI,(152)
wi h φIsupplied by he a las ile (§VI A). The F éche a ia ion o (151) along δL is
δPI=1
2πi IΓI
(z−L)−1(δL)(z−L)−1dz, (153)
so he sensi i i y en e ing he mul iplie upda e (§VII A) is
JI( ) = δwI
δn( )= Φ1
2πi IΓI
(z−L)−1δL
δn( )(z−L)−1dz.(154)
Equa ions
(152)
–
(154)
a e exac whene e Γ
I
encloses an open gap; all nume ical app oxima ions below
ac only on he esol en and he con ou in eg a ion (no ad hoc modeling).
Nume ical ealiza ion by con ou quad a u e.
We e alua e
(151)
and
(154)
wi h he same a ional–
quad a u e machine y used o (con)
xc (§VII B): choose nodes {zj, wj}Nq
j=1 on ΓIand o m
PI≈
Nq
X
j=1
wjR(zj;L), JI( )≈
Nq
X
j=1
wjΦR(zj;L)δL
δn( )R(zj;L).(155)
Accu acy is go e ned by (i) he dis ance
dis
(Γ
I, σ
(
L
)) and (ii) he smoo hness o he in eg and; exponen-
ially con e gen apezoid ules on ci cles/ellipses deli e machine p ecision wi h
Nq.
8in p ac ice. Fo
diagnos ics and subspace ex ac ion inside
I
, we op ionally pe o m one FEAST/SS sweep o ob ain a
Rayleigh–Ri z basis o Ran PI.
Gap moni o ing and p ojec o s abili y.
Le
δI
deno e he gap ma gin, i.e., he minimal dis ance
om ΓI o σ(L). Two complemen a y moni o s a e used:
1.
Resol en no m gua d:
δ−1
I≈maxz∈ΓIkR
(
z
;
L
)
k
(uppe and lowe bounds ollow om s anda d
esol en es ima es).
2.
Angle/dis ance gua d: when a Ri z subspace
U
inside
I
is a ailable, he Da is–Kahan
sin
Θ heo em
gi es
kPI(L)−PI(˜
L)k ≤ kL−˜
Lk
δI
,(156)
so small upda es
kL−˜
Lk
canno mo e he p ojec o much while
δI
s ays bounded away om 0
[349, Sec. VII.3], [350].
We decla e he p ojec o s able i
δI≥δmin
and he idempo ency de ec
k
(
Pnum
I
)
2−Pnum
Ik
s ays below
a ole ance ( he la e is
O
(
e−cNq
) o exac a i hme ic). I ei he es ails, we lag an imminen gap
closu e e en .
K0classes and homo opy in a iance.
Fo a uni al
C∗
–algeb a
A
= C
∗
(
L
), he G o hendieck g oup
K0
(
A
)is gene a ed by Mu ay– on Neumann classes o p ojec o s, modulo s able equi alence [
351
,
Chs. 5–6], [
352
, Chs. 1–2]. I
δI>
0along a con inuous pa h
7→ L
(
), he class [
PI
(
L
(
))]
∈K0
(
A
)is
cons an : p ojec o cons ain s p o ec sec o s opologically. When a gap closes (
δI↓
0), [
PI
]may jump
by an elemen o K0; he jump is quan ized and compu able by an index.
Spec al low and index o a pai o p ojec ions.
Le
7→ PI
(
)be he Riesz p ojec o s along a
pa h and le
Q
be a e e ence p ojec o (e.g., he ini ial
PI
(0)). The index o a pai o p ojec ions is
de ined by
ind(P, Q) = dim ke QP|RanP−dim ke PQ|RanQ,(157)
55
and is an in ege in a ian unde compac pe u ba ions [
353
]. When a gap closes and eopens ac oss he
con ou , eigen alues c oss Γ
I
and he spec al low equals
ind
(
PI
(1)
, PI
(0)), gi ing he quan ized change
o he ank/ ace o
PI
; see also [
354
]. Algo i hmically, in a ini e disc e iza ion one e alua es
(157)
om
he singula alues o
QP
and
PQ
es ic ed o he ele an subspaces (s able unde small pe u ba ions
pe [357]).
P ojec o cons ain s and SCF coupling (pseudocode).
Algo i hm 2: P ojec o cons ain s & K0 acking (pe mac o–i e a ion)
1. Build L(j)=L[n(j)]and (i needed) a FEAST/SS subspace wi hin I.
2. Riesz p ojec o : e alua e P(j)
I≈PjwjR(zj;L(j))and he weigh w(j)
I= Φ(P(j)
I).
3. Residual: (j)
I=w(j)
I−φglue
I; add i o he spec al–law esidual ec o (§VIIA).
4.
Sensi i i y: compu e
JI
(
)by
(154)
and include i in he Schu sys em o he mul iplie upda e.
5.
Gap gua d: es ima e
δI
( esol en no m and/o Ri z dis ances); i
δI< δmin
o
k
(
P(j)
I
)
2−P(j)
Ik>
P, igge GapClosu eE en .
6.
Index upda e (on
GapClosu eE en
): compu e
ind
(
P(j)
I, P(j0)
I
)wi h
j0
he las sa e i e a e; i he
index is nonze o, egis e a quan ized jump and upda e he a las sec o a ge
φglue
I←φglue
I
+
ind
(o swi ch iles i dic a ed by he a las).
7. P oceed wi h he SCF mul iplie upda e and densi y mixing (Algo i hm 1).
A pos e io i e o bounds o wI
= Φ(
PI
)
.
Two complemen a y (cheap) es ima o s accompany each
wI:
•
Quad a u e esidual: di e ence be ween
Nq
and 2
Nq
–node e alua ions on he same Γ
I
, exploi ing
exponen ial con e gence on analy ic con ou s.
•
Subspace esidual: when a Ri z basis
U
is a ailable, he Pa le bound connec s he esidual
k
(
L−λ
)
Uk
o an eigen alue enclosu e, which in u n uppe –bounds he possible change o
wI
[
355
,
Ch. 13]; see also [356].
We expose he la ge o he wo as an e o ba on
wI
and ca y i in o he augmen ed–Lag angian
penal y (i enabled) o a oid o e – esol ing cons ain s beyond hei nume ical accu acy.
Dis ance and leakage diagnos ics.
To moni o sec o ideli y we ack: (i) he dis ance
kP(j)
I−
P(j−1)
Ik
(p incipal angles), (ii) he idempo ency de ec
k
(
P(j)
I
)
2−P(j)
Ik
, and (iii) he leakage no m
k
(1
−P(j)
I
)
P(j−1)
Ik
. Da is–Kahan bounds ela e (i) o he a io
kL(j)−L(j−1)k/δI
[
349
,
350
]. I he
leakage exceeds a ole ance while
δI
is s ill la ge, he a las selec ion is e isi ed (§VI D); i i occu s
oge he wi h a small δI, a ue gap–closu e e en is decla ed.
Wha happens a a quan ized jump.
When
ind 6
= 0 a s ep 6 o Algo i hm 2, wo consis en ac ions
a e possible:
1.
Allowed sec o change. I he a las pe mi s a sec o ansi ion (e.g., spin c osso e , band in e sion),
we upda e φglue
Iby he in ege index and con inue; he K0class o PIchanges acco dingly.
2.
P o ec ed sec o . I he sec o is p o ec ed by physics (e.g., elec on numbe in a ixed subspace) o
by use in en , we ei he (i) inc ease he penal y on
I
and sh ink he s ep (line sea ch) o s ay on
he sa e side o he gap, o (ii) igge adap i e e inemen /swi ch o a ile ha keeps he gap open
(§VIE).
In all cases he change is in ege and hus immune o nume ical d i ; he index compu a ion is s able by
i ue o (157) and s anda d pe u ba ion heo y [357].
Summa y.
Sec o p ojec o cons ain s a e implemen ed exac ly by Riesz in eg als, hei g adien s ollow
om he same esol en calculus, and hei s abili y is ce i ied by gap moni o s ( esol en no ms and
56
Da is–Kahan angles). As long as he gap s ays open, he
K0
class is in a ian ; upon genuine closu e, a
quan ized index de ec s and measu es he jump, enabling cohe en a las upda es o p o ec ed e ac ion.
This machine y gua an ees obus sec o iza ion and consis en bookkeeping o opological da a in ou
cons ained SCF.
D. Cu a u e e alua ion: cen al s. non-cen al es ima o s; s opping c i e ia; e o ba s
Aim.
We gi e conc e e, compu able es ima o s o he cen al and non-cen al pa s o he highe
cu a u e ha go e ns anspo o spec al laws (§III D, §VID). These es ima o s se e h ee pu poses:
(i) ce i y ha a pa h in pa ame e space lies on an exac leg ( anishing cen al cu a u e), (ii) igge
adap i e e inemen when non-cen al cu a u e is non-negligible (§VI E), and (iii) p o ide igo ous e o
ba s on anspo ed in a ian s and on
Exc
( ia §IVD, Eq.
(109)
). We also s a e p ac ical s opping c i e ia
o mac o-i e a ions and pa h s eps.
Geome ic se up and cen al p ojec ion.
Le Θdeno e he pa ame e mani old (geome y, ields,
densi y pa ame iza ion, e c.). On he i ial bundle Θ
×A
(wi h ibe he
C∗
-algeb a
A
= C
∗
(
L
)
gene a ed by he KS-like L[n]), a ma ix- alued connec ion 1- o m A=PiAidθiinduces cu a u e
F=dA+A∧A =1
2X
i,j Fij dθi∧dθj,Fij =∂iAj−∂jAi+ [Ai,Aj],(158)
as in s anda d di e en ial geome y [
358
, Ch. II]. We spli
F
in o cen al and non-cen al pa s by
p ojec ing on o he cen e
Z
(
A
)and i s o hogonal complemen . Gi en an Ad-in a ian s a e Φ(§III B),
he Φ–cen al p ojec ion is
ΠΦ
c(X) := Φ(X)
Φ(1)1∈Z(A),ΠΦ
nc(X) := X−ΠΦ
c(X),(159)
a condi ional expec a ion on o
Z
(
A
)in he GNS ep esen a ion [
359
, Ch. 2], [
360
, Ch. 4]. An exac leg is
a pa h wi h ΠΦ
c(F)≡0; hen he spec al laws a e pa allel anspo ed wi hou d i (Theo em III.1).
In ini esimal (DOI) cu a u e es ima o om esol en s.
Le
S
= Φ

(
L
)

be a spec al law
o a bounded Bo el
. Using he Daleckii–K e˘ın / double-ope a o -in eg al o mula o he i s F éche
de i a i e o
(
L
)and di e en ia ing wice, he an isymme ic mixed second de i a i e a a poin
θ∈
Θis
F( )
ij (θ) := ∂i∂jS −∂j∂iS
= Φ1
2πi IΓ
(z)hR(z)(∂iL)R(z) (∂jL)R(z)−R(z)(∂jL)R(z) (∂iL)R(z)idz,(160)
wi h
R
(
z
) = (
z−L
)
−1
and Γenclosing
σ
(
L
). Equa ion
(160)
is a cen al (scala ) cu a u e p oxy:
i i anishes o a de e mining amily
{ k}m
k=1
, hen he Φ–cen al cu a u e is ze o o leading o de
along (
∂i, ∂j
). Fo implemen a ion we disc e ize Γas in §VIIB;
∂iL
is deli e ed by he same DFPT
in as uc u e used o esponse, e alua ed wi h ini e di e ences along
θi
when an analy ic o m is no
a ailable. The esol en /DOI se ing p o ides a di ec b idge o ou cons ain s and admi s obus e o
con ol; backg ound on ma ix unc ions and esol en bounds can be ound in [361, Chs. 1–3].
Holonomy (Wilson-loop) cu a u e es ima o om small plaque es.
Fo a small o ien ed
pa allelog am Σij(θ;hi, hj)spanned by s eps hieiand hjej, he holonomy a ound i s bounda y is
Uij(θ;hi, hj) := PexpI∂Σij A=1+Fij(θ)hihj+O(hi, hj)3,(161)
by he non-Abelian S okes heo em and he Magnus expansion [
362
,
363
]. We es ima e
Uij
by anspo ing
he cons ained s a e (o sec o p ojec o ) a ound he ou e ices o he plaque e using Algo i hm 1,
hus ob aining a nume ically exac disc e e holonomy. P ojec ing wi h (159) we de ine
bκc
ij(θ) := Φ(log Uij)
hihjΦ(1),bκnc
ij (θ) := kΠΦ
nc(log Uij)kΦ
hihj
,(162)
whe e
k·kΦ
is he Φ-induced Hilbe –Schmid no m
kXk2
Φ
:= Φ(
XX
)(uni a ily in a ian [
366
]). The
cen al es ima o
bκc
measu es a ea-no malized scala d i ; he non-cen al es ima o
bκnc
measu es b aided
57
(non-Abelian) con en le a e emo ing he cen al pa . In he Abelian case (commu ing cons ain s)
bκnc
= 0. Holonomy es ima o s a e s anda d in la ice gauge heo y (Wilson loops) [
364
,
365
] and adap
na u ally he e because ou anspo is ealized by KS-class maps.
Di e en ial (connec ion-on-mul iplie s) es ima o .
Le
gk
(
n, θ
) =
S k
[
n
]
−φglue
k
(
θ
)deno e he
cons ain s and le A=
hJk, χsJ`i
be he Schu ma ix om §VIIA. Di e en ia ing
gk
= 0 along
θi
gi es
∂iλ=−A−1∂ig,A(λ)
i:= A−1(∂ig),(163)
which de ines a connec ion on he
λ
–bundle o e Θ. I s cu a u e
F(λ)
ij
=
∂iA(λ)
j−∂jA(λ)
i
+ [
A(λ)
i,A(λ)
j
]
is e alua ed wi h cen al di e ences in
θ
, using he same linea ized quan i ies al eady compu ed in
Algo i hm 1. A nea -ze o
kF(λ)
ij k
co obo a es small d i o he in a ian s; a non-ze o alue co ela es
wi h b aided co ec ions needed in §VC. This es ima o is pu ely algeb aic and cheap.
F om cu a u e o e o ba s and s ep con ol.
Le
bκc
ij
and
bκnc
ij
be es ima es om
(162)
(o he
DOI p oxy
(160)
). Fo a pa h
γ
: [0
,
1]
→
Θwi h angen
˙γ
=
Pi˙
θiei
, de ine he cen al cu a u e densi y
b
Λc(γ( )) := Pi<j bκc
ij |˙
θi˙
θj|.Then he a p io i d i o a spec al law along a s ep o size ∆ssa is ies
∆S .C Zs0+∆s
s0b
Λc(γ(s))ds, (164)
wi h
C
he esol en -gap cons an s o ed pe p obe (c . §VI D). Simila ly, he non-cen al es ima o
yields a bound on he size o b aided co ec ions needed in
xc
(§VC). P opaga ing
(164)
in o
Exc
ia
he sensi i i y ac o s αk(s)o §IV D gi es
δExc .X
kZ1
0αk(s)Zγs
C kb
Λcdsds, (165)
a compu able, i s -p inciples e o ba consis en wi h Eq. (109).
S opping c i e ia and s ep size selec ion.
We adop he ollowing simple, obus ules (wi h use
ole ances εleg o in a ian s, εE o ene gy):
(Exac -leg accep ance) Zγb
Λcds ≤εleg =⇒accep s ep; do no e ine ile,(166)
(Ene gy gua d) δExc om (165) ≤εE,(167)
(Non-cen al gua d) Zγb
Λnc ds ≤εb aid =⇒no b aided co ec ion needed.(168)
I any gua d ails, he s ep is educed (back acking), o a new ile is ins an ia ed (§VIE). We se
hi
by a
s anda d PI-con olle on he cu a u e es ima o s, akin o a iable-s ep ODE sol e s (Magnus-based
in eg a ion) [362, Ch. II]; s opping es s ollow s anda d KKT/me i - unc ion logic [368, Chs. 3–4].
Algo i hmic synopsis (pe mac o-i e a ion o pa h poin ).
Algo i hm 3: Cu a u e e alua ion and s ep con ol
1.
Choose di ec ions. Pick wo o mo e coo dina e di ec ions
{ei}
ele an o he cu en e olu ion
(e.g., geome y mode, densi y mixing di ec ion).
2.
Plaque e holonomy. Fo each (
i, j
), o m a small plaque e Σ
ij
, anspo he cons ained s a e
a ound i (Algo i hm 1), and compu e Uij and log Uij.
3.
Cu a u e spli . E alua e
bκc
ij
and
bκnc
ij
ia
(162)
; op ionally co obo a e wi h he DOI p oxy
(160)
.
4. E o ba s. Accumula e Rb
Λcds and Rb
Λnc ds along he planned s ep; es ima e δExc om (165).
5.
Decide. I
(166)
holds, accep ; else educe s ep o igge adap i e e inemen (§VI E). I
bκnc
exceeds ole ance bu cen al cu a u e is small, enable he minimal b aided co ec ion in
xc
(§VC).
64
explici ly e i iable bounds, closing he loop be ween abs ac
K
- heo y and p ac ical p ojec o -cons ained
SCF.
C. Sum ules & Wa d iden i ies o xc
Aim.
We show ha he dynamical exchange–co ela ion ke nel
xc
(
q, ω
)cons uc ed by ou esol en /spec al–
law o malism (§V C) necessa ily sa is ies he s anda d conse a ion cons ain s: (i) he
-sum ule,
(ii) he comp essibili y (and spin–s i ness) sum ules, (iii) Wa d iden i ies associa ed wi h gauge and
Galilean in a iance (con inui y, ze o o ce/ o que, and Kohn’s heo em in ha monic con inemen ). The
p oo uses only (a) he ac ha ou
xc
is ob ained as second unc ional de i a i e o a Φ-de i able
ac ion (Baym–Kadano conse ing logic), (b) analy ici y/causali y o e a ded esol en s, and (c) he
Ad-in a iance o he s a e Φ(cyclici y). We wo k in Fou ie space wi h q=|q|and equency ω.
Linea esponse, Dyson s uc u e, and no a ions.
Le
χ
(
q, ω
)and
χs
(
q, ω
)be he in e ac ing and
KS densi y esponses o a scala pe u ba ion δ ex . In TDDFT one has he Dyson ela ion
χ−1(q, ω) = χ−1
s(q, ω)− c(q) + xc(q, ω), c(q) = 4π
q2.(184)
Equi alen ly, in cu en –densi y o m, he longi udinal cu en –cu en esponse Π
LL
obeys he Wa d
iden i y (con inui y)
ω χnn(q, ω) = qΠLn(q, ω), ω ΠnL(q, ω) = qΠLL(q, ω),(185)
and
χ
can be eco e ed om Π
LL
and he scala / ec o ke nels (see below). Iden i ies o he o m
(185)
o igina e om he Wa d–Takahashi ela ion o gauge heo ies [
404
,
405
] applied o he elec omagne ic
ou –cu en o he elec on gas ia he Kubo o malism [406].
Conse ing cons uc ion ⇒Wa d iden i ies.
Ou
xc
is ob ained om a conse ing (Φ-de i able)
ac ion in he sense o Baym–Kadano : he cons ained po en ial
(con)
xc
is he i s a ia ional de i a i e
o a scala unc ional and
xc
i s second one ( e a ded b anch), c . Eq.
(133)
. By Noe he ’s heo em
applied o he gauge symme y o he ac ion, he esul ing esponse unc ions sa is y he Wa d iden i ies
associa ed wi h pa icle–numbe conse a ion [
407
,
408
]. Conc e ely, le
δA
[
ϕ
]be he ac ion a ia ion
gene a ed by a ime–dependen gauge
ϕ
(
,
); gauge in a iance (
δA
= 0) implies
(185)
o he exac
esponse, and he same implica ion holds o any Φ-de i able app oxima ion [
407
,
408
]. The e o e, he
xc appea ing in (184) mus sa is y he longi udinal Wa d iden i y
lim
q→0
ω2
q2χ(q, ω)=ΠLL(q→0, ω),(186)
and he co esponding ela ion be ween i s scala and cu en ke nels (see he TDCDFT pa ag aph
below).
Causali y and dispe sion ela ions.
Since
xc
is buil om e a ded esol en s
GR
(
z
) = (
z
+
i
0
+−L
)
−1
(§VC), i is analy ic o
=ω >
0; i s bounda y alues on he eal axis obey he K ame s–K onig dispe sion
ela ions inhe i ed om he Kubo o malism [406]. In pa icula ,
< xc(q, ω) = 1
πPZ∞
−∞
= xc(q, ω0)
ω0−ωdω0,= xc(q, ω)≤0 (ω > 0).(187)
These p ope ies will be used momen a ily o he sum ules.
-sum ule. The exac longi udinal -sum s a es
Z∞
0
dω ω =χ(q, ω) = −π
2n q2,(188)
independen ly o in e ac ions. To see ha
(188)
holds o ou cons uc ion, expand he esol en in
(133)
a la ge equency,
xc(q, ω) = α2(q)
ω2+O(ω−3),(189)

65
whe e
α2
(
q
)is bounded as
q→
0(i a ises om equal– ime commu a o s o
L
wi h he densi y ope a o ).
Inse ing
(189)
in he Dyson equa ion
(184)
shows ha he
ω−2
ail o
xc
does no al e he coe icien o
he
ω−2
ail o
χ−1
ha ixes he i s equency momen ; he e o e
(188)
holds (Adle –Wise ’s dielec ic
esponse eaches he same conclusion om gauge in a iance [
410
,
411
]). The a gumen is he s anda d
Baym–Kadano p oo o conse ing app oxima ions, now applied o he TDDFT Dyson s uc u e
[407, 409].
Comp essibili y (and spin–s i ness) sum ules. A ω=0 and q→0,
lim
q→0χ(q, 0) = −n2κT,1
n2κT
=d2
dn2n ε(n),(190)
wi h
ε
he ene gy pe pa icle o he uni o m sys em. Using
(184)
and he known s a ic KS esponse
χs(q→0,0), one in e s he exac s a ic limi o xc,
xc(q→0,0) = d2
dn2n εxc(n)≡A0(n),(191)
which is p ecisely he ALDA limi we eco e ed in §V B. In he spin channel (ze o magne iza ion) he
same easoning wi h he longi udinal spin esponse gi es
mm
xc (q→0,0) = 1
n2
∂2
∂ζ2hn εxc(n, ζ)iζ=0,(192)
he spin–s i ness sum ule. Bo h ela ions a e gua an eed he e because
xc
o igina es om a Φ-de i able
ac ion and because ou a las p o ides he uni o m limi s used o ancho
A0
(c . §VIA; de i a ions in he
Kubo–Baym amewo k a e ex book ma e ial [408, 414]).
Ze o o ce and ze o o que (Galilean in a iance).
Unde a uni o m, ime–dependen ansla ion
7→ −a
(
) he in e ac ing many–body Hamil onian changes by a o al ime de i a i e; hus he o al
xc o ce and o que mus anish. In TDDFT his becomes he Wa d iden i ies
Zd n( , )∇ xc( , ) = 0,Zd n( , ) ×∇ xc( , ) = 0,(193)
and, in equency space, cons ain s on
xc
ensu ing ha i s longi udinal p ojec ion p ese es he con inui y
equa ion and i s ans e se pa does no gene a e ne o que. Because
(con)
xc
is he i s a ia ional
de i a i e o an in a ian ac ion e alua ed wi h an Ad-in a ian s a e Φ,
(193)
ollows by di e en ia ion
unde he in eg al (cyclici y o Φ). Thus he cons uc ed
xc
is Galilean in a ian in he sense o he
Wa d iden i ies (193) (compa e he analogous s a emen s in TDCDFT [413, 414]).
Kohn’s heo em / ha monic po en ial heo em.
In a pu ely ha monic ex e nal po en ial
Vex
=
1
2m
Ω
2 2
he cen e o mass unde goes igid oscilla ions a equency Ω, independen ly o in e ac ions
(Kohn’s heo em) [
412
]. The co esponding Wa d iden i y in linea esponse s a es ha he longi udinal
pa o
xc
mus no eno malize he igid– ansla ion pole. Since ou
xc
de i es om a ansla ionally
co a ian ac ion and om e a ded esol en s (no spu ious poles), he Dyson equa ion
(184)
au oma ically
p ese es he igid pole: he le –hand side ans o ms as a scala unde ansla ions, while he igh –hand
side con ains only he Coulomb ke nel
c
(which ca ies he
q−2
singula i y) and a egula
xc
nea
ω
= Ω.
Hence Kohn’s heo em is espec ed, as equi ed by a conse ing app oxima ion [409].
TDCDFT and longi udinal/ ans e se decomposi ion.
I is o en con enien o exp ess cons ain s
in he cu en –densi y o mula ion. Le
LL
xc
(
q, ω
)and
T T
xc
(
q, ω
)be he longi udinal and ans e se cu en
ke nels o TDCDFT. The Wa d iden i y implied by gauge in a iance eads
xc(q, ω) = ω2
q2 LL
xc (q, ω), q · T T
xc (q, ω)=0,(194)
and he
-sum is sa u a ed en i ely by he kine ic and Coulomb e ms, p o ided
xc
decays a leas as
ω−2
. Ou cons uc ion yields
(194)
because bo h scala and ec o ke nels a e gene a ed by he same
esol en sandwiches and Ad-in a ian s a e, di e ing only by inse ions ealizing he cha ge/cu en
couplings (Kubo e ices) [406, 413].
Highe momen s ( hi d momen ) and sho – ime expansion.
The hi d equency momen o
χ
is
ixed by equal– ime commu a o s o he densi y wi h he Hamil onian and imposes addi ional cons ain s
66
on he
ω−2
coe icien
α2
(
q
)in
(189)
(schema ically,
Rdω ω3=χ
equals a polynomial in
q
wi h leading
q4
). Because
α2
(
q
)a ises om he sho – ime expansion o he p oduc o e a ded esol en s in
(133)
,
i s alue is dic a ed by ope a o algeb a and is independen o densi y mixing o empi ical choices; hus
he hi d-momen sum ule is p ese ed au oma ically wi hin ou amewo k (see he Baym–Kadano
discussion o equency momen s [409]).
Summa y.
The dynamical ke nel
xc
ob ained om spec al–law esol en s is causal/analy ic and
conse ing. I obeys he Wa d iden i ies associa ed wi h gauge and Galilean in a iance (con inui y, ze o
o ce/ o que, Kohn’s heo em), and i sa is ies he
-sum and comp essibili y/spin–s i ness sum ules.
The essen ial easons a e s uc u al: (i) Φ-de i abili y o he ac ion (Baym–Kadano ), (ii) analy ici y
and sho – ime expansions o e a ded esol en s (Kubo), and (iii) Ad-in a iance (cyclici y) o he s a e
Φ. No model-speci ic hypo hesis o empi ical i ing is equi ed.
D. Limi heo ems: HEG, SCE (U→∞/low densi y), 1D TLL exponen s; con inui y ac oss iles
Aim.
We es ablish h ee asymp o ic co ec ness esul s o he spec al–law DFT de eloped in his
wo k: (i) he high–densi y (small
s
) jellium/HEG limi ; (ii) he s ic ly co ela ed elec on (SCE) limi ,
equi alen ly he la ge–
U
o low–densi y (
s→∞
) egime; (iii) he one–dimensional Tomonaga–Lu inge
liquid (TLL) exponen s ha con ol long–dis ance co ela ions in quasi–1D me als. In addi ion we p o e
acon inui y ac oss iles s a emen (a las glue) o
(con)
xc
and
Exc
, wi h explici moduli o con inui y
go e ned by cu a u e bounds. All s a emen s ely on he same s uc u al pilla s: (a) esol en unc ional
calculus o
(
L
), (b) Φ–de i able (conse ing) cons uc ion o
xc
, (c) Ad–in a iance (cyclici y) o Φ,
and (d) highe –connec ion anspo wi h cen al/non–cen al cu a u e spli .
No a ion.
Le
s
be he usual densi y pa ame e o he HEG;
U/W
he in e ac ion– o–bandwid h a io
in la ice educ ions; (
, K
) he TLL eloci y and Lu inge pa ame e . Spec al–law cons ain s ead
S k[n] = Φ( k(L[n])) = φk; he cons ained po en ial is (c . Eq. (4.3))
(con)
xc ( ) = X
k
λkΦIΓk
dz
2πi k(z)(z−L)−1δL
δn( )(z−L)−1.
The dynamical ke nel xc(q, ω)is gi en by he e a ded wo– esol en o mula (Sec. 5.3).
(A) HEG (high–densi y) limi : s→
0
.
In he
s→
0limi he elec on gas is weakly co ela ed
and i s co ela ion ene gy and esponse a e cap u ed by he Gell–Mann–B ueckne (GMB) high–densi y
expansion and i s RPA esumma ion [415]. We claim:
Theo em VIII.2
(HEG/GMB limi )
.
Le he p obe se include wo esol en poin s symme ically
s addling he Fe mi le el on he imagina y axis and he a las HEG ile supply he uni o m a ge s
{φHEG
k( s)}. Then as s→0,
E(con)
xc [n] = EGMB
xc [n] + o(1), (con)
xc (q, ω) = RPA
xc (q, ω) + o(1),
uni o mly o qand ωin compac se s ha do no scale o in ini y wi h −1
s.
Ske ch. (i) In he HEG ile he anspo ed in a ian s coincide wi h he uni o m GMB/RPA momen
da a [
415
]. (ii) The esol en sandwich de ining
(con)
xc
educes, a
s→
0, o he quad a ic (Gaussian)
unc ional o densi y luc ua ions (RPA), because highe –o de cumulan s anish in he weak–coupling
limi ; domina ed con e gence on he esol en con ou yields he s a ed limi o
xc
(K ame s–K onig
analy ici y is p ese ed by cons uc ion). (iii) The ACFD unc ional educes o he RPA ing sum wi h
he same ke nel; ene gy equali y hen ollows. The uni o mi y s a emen is a di ec consequence o
bounded esol en no ms o con ou s a ixed dis ance om σ(L)and o compac ness o he p obe se .
Rema k. Mode n QMC pa ame iza ions o he HEG (beyond RPA) can be op ionally embedded in
he ile as a ge s
φHEG
k
; he p oo hen yields con e gence o he co esponding beyond–RPA ke nel and
ene gy (see, e.g., high–p ecision QMC da a in [416]).
(B) SCE (la ge–U/low–densi y) limi : s→∞.
In he opposi e egime elec ons minimize Coulomb
epulsion subjec o ixed densi y, leading o s ic ly co ela ed con igu a ions. Le
W∞
[
n
]deno e he
SCE unc ional ( he mul ima ginal op imal– anspo minimum o he Coulomb cos ), and
W0
∞
[
n
]i s
i s co ec ion.
67
Theo em VIII.3
(SCE/OT limi )
.
Assume he a las SCE ile supplies a ge s
{φSCE
k}
ha encode he
co–mo ion map momen s and he s ong–coupling coe icien s (
W∞, W0
∞
). Then along any densi y bundle
nλ( ) = λ3n(λ ),
lim
λ→0E(con)
xc [nλ] = W∞[n]and E(con)
xc [nλ] = W∞[n] + λ W0
∞[n] + o(λ),
and he pai –dis ibu ion and s uc u e– ac o cons ain s inhe i ed om he ile a e a ained in he limi .
Ske ch. Along
λ→
0(low densi y) he kine ic con ibu ion is subleading and he esol en ke nel
localizes on co–mo ion suppo s; he spec al measu e collapses on o he OT op imize . The spec al–law
cons ain s en o ce he SCE momen s, so he con ou in eg als con e ge o he dis ibu ional limi s
associa ed wi h he co–mo ion map. Exis ence and s abili y o he OT op imize o Coulomb cos unde
densi y escaling a e s anda d [
418
]; he linea co ec ion ollows om he i s a ia ion o he OT alue
unde smoo h de o ma ions o he densi y (shape de i a i e). The e o e
(con)
xc
con e ges (in he sense o
dis ibu ions) o a SCE po en ial and he ene gy o W∞[n]wi h he s a ed co ec ion e m.
(C) 1D Tomonaga–Lu inge liquid (TLL) exponen s.
In one dimension, gapless me als a e
go e ned a low ene gy by a TLL wi h pa ame e s (
, K
); co ela ion unc ions decay wi h exponen s
de e mined by K[419]. Le (con)
xc (q, ω)be he longi udinal ke nel. We claim:
Theo em VIII.4
(TLL consis ency)
.
On he 1D ile, wi h p obes chosen o encode he TLL d essed
cha ge Kand eloci y , he small–(q, ω)limi o he cons ained ke nel sa is ies
χ(con)(q, ω)−−−−→
q,ω→0
K
π
q2
ω2− 2q2,
and all bosoniza ion exponen s o
n
–, 2
kF
–, and 4
kF
–densi y co ela ions de i ed om
χ(con)
ma ch he
TLL p edic ions.
Ske ch. The wo– esol en ep esen a ion o
xc
ep oduces he TLL cu en –cu en esponse (linea
spec um) when he ile ixes (
, K
); he Wa d iden i y ensu es he co ec longi udinal mapping
χ∝
ω−2
Π
LL
(Sec. 8.3). Bosoniza ion o mulae o co ela ion exponen s (e.g.,
αCDW
= 1
−K
,
αSS
= 1
−
1
/K
)
hen ollow by s anda d Fou ie analysis [
420
,
421
]. The key s ep is ha he a las ile p o ides (
, K
)as
exac a ge s (e.g., om Be he/TBA), and ou ke nel is he unique causal/conse ing one consis en wi h
hose and wi h he Wa d iden i ies.
(D) Con inui y ac oss iles (glue).
Le Θbe he desc ip o mani old equipped wi h he cu a u e–
weigh ed me ic
dΛ
o Sec. 6.5 and le
{
T
α}
be a ini e a las wi h ansi ion da a (
Fαβ, Tαβ,
R
αβ
)obeying
he cocycle law. We p o e:
P oposi ion VIII.5
(Con inui y ac oss iles)
.
Along any piecewise
C1
pa h
γ
: [0
,
1]
→
Θ, he glued
a ge s
φglue
(
γ
(
)) a e con inuous and o bounded a ia ion. Consequen ly,
(con)
xc
(
·
;
γ
(
)) is con inuous in
in he weak opology o dis ibu ions and
E(con)
xc
[
n
(
·
;
γ
(
))] is Lipschi z con inuous wi h modulus bounded
by Rb
Λc(γ( ))d .
Ske ch. On an exac leg (Λ
c
=0) he anspo ed in a ian s a e cons an (Theo em 3.1), hence i ially
con inuous. On o e laps wi h small cu a u e, he pa i ion–o –uni y glue (Sec. 6.4, Eq. (6.4)) is a
con inuous con ex combina ion o con inuous maps; he b aided co ec ion Ris algeb aic and no m–
con inuous in he non–cen al cu a u e. Resol en sandwiches a e join ly con inuous in he ope a o
and he in eg and on con ou s a ixed dis ance om he spec um; weak con inui y o
xc
ollows by
domina ed con e gence on he con ou (bounded esol en no ms), and Lipschi z con inui y o
Exc
by
he a p io i bound (7.18) in eg a ed along
γ
. I a ue gap closu e occu s, con inui y is p ese ed and he
sec o jump is quan ized (Sec. 8.2).
Consequences.
The h ee limi s ancho ou cons uc ion a he p incipal asymp o ic egimes (weak
coupling, s ong coupling, and 1D c i icali y), while he glue con inui y ensu es he e is no “kink” a ile
bounda ies beyond he explici ly quan i ied cu a u e con ibu ion. Hence he cons ained XC de ini ions
a e uni o mly co ec ac oss egimes, wi hou any need o empi ical c oss–o e i s.
Summa y.
We p o ed ha (i) in he HEG limi he cons ained ene gy and ke nel educe o GMB/RPA;
(ii) in he SCE limi hey con e ge o he OT/SCE unc ional and po en ial wi h he expec ed i s co ec-
ion; (iii) in he 1D TLL egime he low–ene gy esponse and co ela ion exponen s ma ch bosoniza ion
68
wi h gi en (
, K
); and (i ) con inui y ac oss iles holds wi h moduli con olled by cu a u e. These limi
heo ems con i m ha he a las–based, cons ain –d i en XC cons uc ion eco e s he co ec physics a
he bounda ies o pa ame e space and glues he egimes wi hou ad hoc in e pola ion.
E. Quasi–Hop wis in a iance: scheme independence unde egula iza ion choices
Aim. We show ha all physical objec s ha en e ou spec al–law–cons ained exchange–co ela ion
(XC) cons uc ion—in pa icula he spec al laws
S
[
n
]=Φ
( (L[n]))
and he XC po en ial ob ained
om he esol en unc ional calculus
xc( ) =
m
X
k=1
λkΦIΓk
k(z)R(z;L)δL
δn( )R(z;L)dz
2πi, R(z;L) := (z−L)−1,
a e independen o he egula iza ion scheme used o compose ope a o inse ions and o de ine he
unc ional calculus, p o ided he cons ain s a e anspo ed along he same monoidal equi alence. We
o malize “ egula iza ion choices” as D in el’d wis s o a quasi–Hop algeb a ac ing on he obse ables.
The esul implies ha (i) nume ical choices such as esol en s. hea –ke nel egula o s, dis inc con ou
ep esen a i es o he same spec al p ojec o , o di e en ime-/pa h-o de ings, and (ii) algeb aic choices
such as quasi- iangula s. b aided enso s uc u es, lead o he same XC po en ial and ene gy wi hin a
gi en
K0
-sec o , up o quan ized jumps a gap closu e. Th oughou his subsec ion we keep he no a ion
o §3–§5:
L≡L
[
n
]is he (KS–like) gene a o , Φis an Ad–in a ian s a e on he ope a o algeb a, and
he HH0p ojec ion emo es commu a o con ibu ions.
Se –up.
Le
A
be he (dense)
∗
-algeb a gene a ed by he esol en s o
L
and by bounded ope a o inse -
ions (e.g. densi y pe u ba ions
δL/δn
). Assume an ac ion by a quasi–Hop algeb a (
H,
∆
, ε, S
;
φ, α, β
)
on
A
by
∗
-au omo phisms, (S anda d quasi–Hop no a ion as in [
428
–
431
]; he associa o is
φ∈ H⊗3
,
and
α, β ∈ H
a e he quasi–an ipode s uc u e elemen s.) so ha mul ipoin ke nels (linea esponse,
TDDFT, e c.) a e o med using he monoidal s uc u e encoded by ∆and
φ
. A D in el’d 2–cochain
( wis ) F ∈ H⊗H wi h in e se F−1p oduces a new quasi–Hop s uc u e
∆F(h) = F∆(h)F−1,φF= (1 ⊗F)(id ⊗∆)(F)φ(∆ ⊗id)(F−1)(F−1⊗1),(195)
(and co esponding wis ed quasi–an ipode). The ep esen a ion ca ego ies
Rep
(
H
)and
Rep
(
HF
)a e
monoidally equi alen by he iden i y unc o endowed wi h he enso s uc u e
F
[
428
,
431
]. Physically,
Fencodes a change o egula iza ion/o de ing scheme.
De ini ion VIII.6
(Ad–in a ian /KMS s a e wi h HH
0
p ojec ion)
.
A linea unc ional Φ :
A→C
is
Ad–in a ian i Φ([
X, Y
]) = 0 o all
X, Y ∈A
, i.e. i ac o s h ough he ze o h Hochschild homology
HH0
(
A
) =
A/
[
A,A
](Ha o i–S allings ace [
434
,
436
,
437
]). We use Φas a no malized ace/KMS s a e
as in [434, 435].
Regula iza ion choices as wis s. Typical al e na i es used in p ac ice a e: (i) Dun o d–Taylo esol en
in eg als s. Laplace/hea –ke nel o ze a egula iza ions [
427
,
438
]; (ii) di e en con ou ep esen a i es
o he same Riesz p ojec o ; (iii) dis inc ime-/pa h-o de ings (e.g. Dyson s. Magnus expansion) in
dynamical ke nels; (i ) b aided s. quasi- iangula composi ion o legs in nonlocal ke nels (hexagon/YBE
con ex [
432
,
433
]). Each such choice al e s how enso legs a e combined, exac ly as in
(195)
. We
he e o e model any admissible change o egula iza ion by a wis F.
Lemma VIII.7
(Cyclici y and HH
0
educ ion)
.
Fo any in eg able
and any
A∈A
wi h
A
ace–class
ela i e o R(z;L),
ΦI (z)R(z;L)A R(z;L)dz
2πi= ΦI (z)R(z;L)2Adz
2πi
and he alue depends only on he class o
A
in
HH0
(
A
). In pa icula , any change o egula iza ion ha
modi ies he in eg and by a sum o g aded commu a o s lea es he esul in a ian .
P oo .
By he esol en iden i y
R A R
=
R2A−R
[
R, A
]
R
and linea i y o Φon
HH0
, he commu a o
e m d ops ou . Absolu e con e gence o he con ou in eg al ollows om s anda d bounds on
R
(
z
;
L
)
o zon a con ou Γou side σ(L)[427].
69
P oposi ion VIII.8
(T anspo o cons ain s)
.
Le
{ϕk}m
k=1
be he a ge in a ian s in he spec al
laws, and le
S k
[
n
] = Φ(
k
(
L
[
n
])). Unde a wis
F
he cons ain s a e anspo ed by he monoidal
equi alence, SF
k[n] := Φ k(L[n])=S k[n],i.e. hey a e nume ically iden ical.
P oo .
The wis does no change he unde lying endomo phism
L
[
n
]no i s unc ional calculus; i only
al e s how enso legs couple when o ming composi e ke nels. As
k
(
L
)
∈A
is a single–leg obse able, i
is insensi i e o ∆and φ; hence SF
k=S k.
P oposi ion VIII.9
(Riesz p ojec o s and
K0
sec o labels)
.
Le
PI
(
L
)be any Riesz p ojec o associa ed
wi h an isola ed po ion
I⊂σ
(
L
). Unde wis ing,
PI
(
L
)is unchanged and he
K0
-label ca ied by he
ini e– ank bundle i de ines is in a ian .
P oo . PI
(
L
)is a con ou in eg al o he esol en alone. By Lemma VIII.7 and he ac ha no cop oduc
is used,
PI
(
L
)is insensi i e o
F
. Homo opy in a iance o
K0
(and gap p o ec ion) hen implies in a iance
o he sec o label [427, 434].
Theo em VIII.10
(Twis in a iance o he XC po en ial)
.
Le
xc
be he XC po en ial de ined abo e
and le
F
xc
be he po en ial compu ed wi h any wis ed quasi–Hop s uc u e
HF
ha ep esen s a change
o egula iza ion/o de ing scheme. Assume he same cons ain s
{ϕk}
a e en o ced ( anspo ed as in
P oposi ion VIII.8) and he compu a ion is pe o med wi hin a ixed
K0
-sec o (i.e. no gap c ossing).
Then
F
xc( ) = xc( ).(196)
I , due o disc e iza ion, he HH
0
p ojec ion is app oxima e, he di e ence is bounded by he non–cen al
cu a u e,
k F
xc − xck ≤ CkΘnck
δL
δn 
sup
z∈ΓkR(z;L)k2,(197)
o a cons an Cdepending only on he geome y o Γand on he wis cocycle bounds.
P oo .
The wis ed exp ession di e s om he un wis ed one only in how he wo esol en s and he inse -
ion
δL/δn
a e associa ed/ enso ed; algeb aically his eplaces he in eg and
I
:=
k
(
z
)
R
(
z
;
L
)(
δL/δn
)
R
(
z
;
L
)
by
IF
=
I
+
Pj
[
Xj, Yj
]wi h [
Xj, Yj
]gene a ed by he cobounda y e ms in
(195)
. By Lemma VIII.7 hese
commu a o s anish in
HH0
, hence he wo con ou in eg als ha e he same Φ- alue. Summing o e
k
wi h
he same mul iplie s
λk
( he cons ained a ia ional p inciple is unchanged because P oposi ion VIII.8
ixes he esiduals) yields
(196)
. Fo he bound
(197)
, no e ha a disc e iza ion o cu o eplaces Φ
by a nea ly– acial unc ional Φ

wi h
|
Φ

([
X, Y
])
| ≤ Ck
[
X, Y
]
k
. The wis con ibu es commu a o s
p opo ional o he cu a u e enso o he 2–connec ion (c . §3.4), whose non–cen al pa Θ
nc
con ols
non– anishing commu a o s. Es ima ing he con ou in eg al by he esol en bound gi es (197).
Co olla y VIII.11
(Implemen a ion independence)
.
Wi hin a ixed
K0
-sec o and away om gap closu e,
he ollowing choices lead o iden ical xc and iden ical adiaba ic–connec ion ene gies:
1. esol en s. hea –ke nel (Laplace) egula o s (Dun o d–Taylo s. Schwinge ep esen a ions);
2. any wo con ou s Γ,Γ0enclosing he same spec um po ion;
3. ime o de ing (Dyson) s. Magnus-like symme ic o de ings in dynamical ke nels;
4.
b aided s. quasi- iangula composi ions o nonlocal legs compa ible wi h he same Yang–Bax e
da a.
A a gap closu e,
K0
may jump; he unc ionals emain well de ined bu may unde go a quan ized change
consis en wi h sec o elabeling (P oposi ion VIII.9).
Physical meaning. Twis in a iance s a es ha ou XC cons uc ion is a monoidal equi alence in-
a ian . All admissible egula iza ions co espond o di e en choices o associa i i y/o de ing da a
in he unde lying enso ca ego y o ope a o inse ions. Because ou obse ables a e e alua ed in an
Ad–in a ian (Ha o i–S allings/KMS) s a e and depend only on
HH0
classes, hey canno de ec he
cobounda y pieces ha dis inguish wo membe s o he same wis class. Consequen ly, he p edic ed
esponse ke nels (
xc
(
q, ω
)), s a ic g adien coe icien s, and he SCF con e gence c i e ia in §7 a e scheme
independen . Wha emains scheme dependen is only he bookkeeping o in e media e objec s (e.g. how
one dis ibu es con ac ions along enso legs), no he inal physics.

70
Examples. (i) Hea ke nel s. esol en . Using he Laplace ans o m
R
(
z
;
L
) =
R∞
0e− (z−L)d
and
he cyclic p ope y o Φ, one ew i es he esol en in eg al de ining each
Vk
as a hea –ke nel in eg al. The
eb acke ing o ime–o de ed exponen ials is encoded by a wis
F
=
exp{R
Λ(
1, 2
)
d 1d 2}
(a 2–cochain
on he ime–o de ed semig oup). By Theo em VIII.10 bo h gi e he same
Vk
and hence he same
xc
.
(ii) B aided s. quasi– iangula legs. Suppose dynamical co ec ions use an
R
–ma ix o b aid nonlocal
inse ions (§5.3). Changing o a quasi– iangula p esen a ion eplaces
R
by
RF
=
F21RF−1
[
432
]. Since
Ren e s only h ough commu a o –exac di e ences in he Φ– alued aces, he ke nels coincide.
Rela ion o cu a u e and exac –leg in a iance. Recall om §3.4 ha exac legs annihila e he d i :
DΘS
= 0 when he cen al cu a u e anishes and he non–cen al piece is con olled. Twis in a iance
complemen s ha s a emen : changing egula iza ion is ma hema ically a la anspo along a 2–gauge
o bi gene a ed by cobounda ies; hence i has ze o e ec on any
HH0
– alued in a ian . The e o bound
(197) is hus ano he way o ead he cu a u e–d i en e o ba s discussed in §7.4.
Consequences o he a las o iles (§6). Tiles a e popula ed once and o all using in eg able backbones.
Because each ile’s a ge s
{ϕk}
and p ojec o s
PI
(
L
)a e wis in a ian , he a las can be anspo ed
ac oss codes, basis se s, and esponse o malisms wi hou e i ing. This is he p ecise sense in which he
p esen scheme is “ egula iza ion independen .”
Summa y o §8.5.
Twis s
F
cap u e all admissible changes o egula iza ion/o de ing. All spec al
laws and he XC po en ial a e in a ian unde such wis s hanks o HH
0
educ ion (Lemma VIII.7) and
o he monoidal equi alence o ep esen a ion ca ego ies (P oposi ion VIII.8). Sec o labels and p ojec o s
a e p ese ed (P oposi ion VIII.9). The e o e he whole SCF algo i hm o §7, he g adien /dynamic
ke nels o §5, and he heo y gua an ees o §8 a e scheme independen inside a
K0
sec o , wi h quan ized
changes only a gap closu e.
IX. NUMERICAL RESULTS
In oduc ion and p o ocol
Pu pose. This sec ion alida es he spec al–law–cons ained amewo k de eloped in Sec. VI by
con on ing i wi h molecula and condensed–phase benchma ks. Ou guiding p inciples a e: (i) ep o-
ducibili y (e e y quan i y we claim can be compu ed wi h he algo i hms al eady speci ied), (ii) eliable
baselines (compa isons o widely used me hods), and (iii) physics diagnos ics ied o exac cons ain s
(piecewise linea i y, de i a i e discon inui y, conse a ion/sum ules). To a oid o e claiming, we sepa a e
wo kinds o e idence and use each whe e i is mos app op ia e:
•Syn he ic, exac ly ac able es beds (compu ed he e).
These a e small Hamil onians
ha ep oduce he quali a i e physics o in e es and can be sol ed exac ly o wi h con olled
nume ical e o using he in as uc u e al eady p esen in ou code (linea sol es and small ma ix
diagonaliza ions). They le us exe cise he ull pipeline (cons ain s
⇒
mul iplie s
⇒ xc
and ke nels)
and e i y p ope ies such as sec o p o ec ion and CT asymp o ics a negligible cos .
•Li e a u e eanalysis (cu a ed baselines).
Fo ealis ic molecules and eac ions we do no
un ex e nal ab ini io engines he e; ins ead we compile e e ence alues om he li e a u e o
s anda d baselines (GGA, global hyb ids, ange–sepa a ed hyb ids when needed, ACFD–RPA)
and o high–le el wa e unc ion heo y (CCSD(T) nea single– e e ence minima). These se e as
compa ison a ge s o ends and o he diagnos ics we emphasize (e.g., ac ional–cha ge/–spin
pa hologies and CT 1
/R
ails). We ci e hese sou ces explici ly (see Re s. [
439
–
442
,
449
,
461
–
463
]).
Scope o Sec. IX. Sec. IX A (9.1) ea s gas–phase molecules wi h h ee canonical p obes: (i) symme ic
s e ch/dissocia ion o H
2
and N
2
; (ii) isodesmic and he e oly ic eac ions; (iii) dono –accep o dime s
exhibi ing g ound–s a e cha ge ans e (CT). La e subsec ions add ess ex ended sys ems, esponse
spec a, and scaling.
Wha we ac ually compu e (and can ecompu e). All compu a ions we pe o m wi hin his manusc ip
a e based on linea –algeb a p imi i es (ma ix in e sion and diagonaliza ion) o ma ices o dimension
.
10
3
, esol en quad a u es wi h
O
(5
−
8) shi s pe p obe, and a small New on/B oyden upda e o
Lag ange mul iplie s. Speci ically:
C1. Minimal H2and N2models. We use wo exac ly sol able Hamil onians:
71
•
The Hubba d dime wi h wo elec ons (H
2
p oxy):
H
(
R
) =
−
(
R
)
Pσ
(
c†
LσcRσ
+
h.c.
) +
UPs=L,R ns↑ns↓
+∆(
R
)
nL−nR
2
, wi h
(
R
) =
0e−α(R−Re)
and ∆(
R
) i ed o he one–elec on
spli ing. The Hilbe space (6–dimensional single block) is diagonalized exac ly o each R.
•
A ou –o bi al ac i e–space model (N
2
p oxy) ha cap u es he
σg/σu
and
π
mani olds;
pa ame e s a e chosen o ep oduce he quali a i e gap e olu ion along s e ch. The ac i e
block is diagonalized exac ly.
In bo h cases, we cons uc he KS–like gene a o
L
[
n
] om he one–body pa ; esol en ke nels and
Riesz p ojec o s a e o med explici ly; he spec al laws
S
[
n
] = Φ(
(
L
[
n
])) ( i ial/comp essibili y
momen s and p ojec o weigh s) a e e alua ed wi h he Ad–in a ian HH
0
p ojec ion; mul iplie s
a e ob ained pe i e a ion; and
xc
ollows he esol en o mula in Eq.
(6)
. Because all ma ices a e
iny, hese s eps a e i ial o ep oduce.
C2. Two–le el CT dime .
We adop a dono (D) / accep o (A) wo–si e model wi h le el o se

=
D−A
, in e si e coupling
V
(
R
) =
V0e−βR
, and long– ange 1
/R
in e ac ion included a he
linea – esponse le el. The e a ded esol en yields a causal
xc
(
q, ω
) ha sa is ies he
–sum
au oma ically in ou cons uc ion (Sec. V), and we ex ac he e ical CT gap ∆
CT
(
R
)and dipole
momen s. This es allows us o e i y he co ec 1/R asymp o e (Re . [463]) di ec ly.
Wha we do no compu e he e. We do no un la ge–basis ab ini io calcula ions (e.g., ull molecula
H
2
/N
2
cu es wi h Dunning basis se s) in his en i onmen . When such da a a e shown, hey a e aken
om he li e a u e and ci ed acco dingly; we use hem only o compa ison and discussion, no as ou
own p ima y compu a ions.
Baselines and e e ences used o compa ison. Fo each p oblem class we compa e quali a i ely
and, whe e applicable, quan i a i ely o ep esen a i e baselines: PBE (GGA) [
439
], B3LYP/PBE0
(global hyb ids) [
440
,
441
], ACFD–RPA as a co ela ion benchma k [
442
], and CCSD(T) as a high–
le el wa e unc ion e e ence nea single– e e ence minima [
449
]. Fo physics diagnos ics we ely on he
ac ional–cha ge/spin analyses o Cohen–Mo i-Sánchez–Yang [
462
], he de i a i e discon inui y in exac
DFT [
461
], and he CT ailu e mode o s anda d TDDFT [
463
]. We do no duplica e hese e e ences’
esul s; we eplo o abula e om hem only whe e explici ly s a ed, wi h clea a ibu ion.
Me ics and diagnos ics (used h oughou Sec. IX). We will epo :
(i) To al ene gies and cu es.
Fo dissocia ion, we moni o he non–pa alleli y e o (NPE) wi h
espec o an ag eed e e ence cu e, NPE = maxR|E(R)−E e (R)|−minR|E(R)−E e (R)|.
(ii) Reac ion and ba ie heigh s.
Fo isodesmic/he e oly ic examples we epo ∆
E
and ∆
E‡
;
whe e only quali a i e models a e compu ed by us, hese a e used as ends checked agains li e a u e
numbe s.
(iii) Dipoles and CT gaps.
Dipoles
µ
om he densi y ia
p
=
R n
(
)
d
; CT gaps ∆
CT
(
R
)compa ed
o he exac la ge–Rasymp o e I(D)−A(A)−1/R (Re . [463]).
(i ) Exac –cons ain diagnos ics.
Piecewise linea i y de ia ion ∆
PL
[
P
=
e
2πγ
], p esence/absence
o symme y b eaking, and causal/analy ic p ope ies o he esponse (sum ules, Wa d iden i ies).
Compu a ional de ails (common o all ou uns).
•
Cons ain s and p obes. Unless o he wise s a ed we use wo esol en p obes ( i ial and comp ess-
ibili y momen s) and wo p ojec o cons ain s: o al occupied weigh
N
and a on ie –mani old
Riesz p ojec o labelled in K0. Ta ge s {φk}a e ixed once pe es amily (Sec. VI).
•
Quad a u e and linea algeb a. Con ou s Γ
k
a e ci cles o ellipses enclosing he ele an spec al
slices; each in eg al uses
nΓ∈
[6
,
12] nodes; linea sol es euse he same spa se (o dense, o models)
in as uc u e as in DFPT and TDDFT. Because ou models a e iny, we also check esul s agains
di ec diagonaliza ion o L.
•
Mul iplie upda e and s opping. We sol e he KKT sys em by a damped New on s ep on he
mul iplie s wi h B oyden upda es o he Jacobian; s opping when
maxk|S k−φk|<
10
−8
and
kn(i+1) −n(i)k<10−8.
•
E o ba s. Along exac legs we ha e
DΘS
= 0; when non–cen al cu a u e is ini e, we epo
d i bounds o he o m ∆
S ≤C k
Θ
nck
(c . Sec. III D) and p opaga e hem o
xc
ia he esol en
bound kR(z;L)k2on he chosen con ou s.
72
•
In a iance checks. Fo a leas one ep esen a i e case in each class we e–e alua e he same
obse able wi h wo equi alen egula iza ions (e.g., esol en s. hea –ke nel) and e i y quasi–Hop
wis in a iance nume ically wi hin he cu a u e–induced ole ance (Sec. VIIIE).
How compa isons will be p esen ed. Fo each class we p o ide:
(A1)
A igu e om ou ac able model(s) showing he obse able (dissocia ion cu e, dipole p o ile, o
CT gap) oge he wi h exac /model–exac e e ences.
(A2)
A able lis ing li e a u e baseline numbe s (GGA, hyb id, RPA, CCSD(T) when app op ia e) o
ep esen a i e eal molecules o dime s, wi h ci a ions [
439
–
442
,
449
,
461
–
463
]. We ne e imply
hese we e compu ed by us in his en i onmen .
(A3)
A sho diagnos ic panel quan i ying ∆
PL
, CT 1
/R
beha iou , and he p esence/absence o symme y
b eaking, o show whe e he p oposed me hod emedies known DFA ailu es.
Why his design is ai . Semilocal DFAs (PBE) a e s ong a equilib ium s uc u es bu exhibi
con ex
E
(
N
)and ac ional–spin/cha ge e o s [
462
]; hyb ids educe bu do no elimina e hese; TDDFT
wi h s anda d ke nels ails o long– ange CT [
463
]; RPA handles long– ange co ela ion bu lacks he
de i a i e discon inui y and may unde bind depending on he e e ence [
442
]; CCSD(T) is accu a e
nea single– e e ence minima bu un eliable deep in mul i– e e ence egions and cos ly [
449
]. Ou
spec al–law app oach is designed o impose he exac cons ain s esponsible o hese pa hologies
( i ial/comp essibili y, p ojec o –en o ced sec o iza ion, causal esponse), so he es s chosen he e a ge
p ecisely hose egimes.
Roadmap. Sec. IXA now applies his p o ocol o: (i) H
2
and N
2
dissocia ion (wi h ac i e–space
p ojec o s ca ying
K0
labels), (ii) illus a i e isodesmic and he e oly ic eac ions, and (iii) CT dime s
exhibi ing he 1
/R
asymp o e. Each case includes a ac able syn he ic compu a ion p oduced by ou
pipeline and a li e a u e compa ison able wi h clea ci a ion o Re s. [439–442, 449, 461–463].
A. Molecules (gas phase)
Ta ge s and plan. We examine h ee canonical molecula si ua ions whe e exac DFT cons ain s
con ol co ec ness: (i) symme ic s e ch/dissocia ion o H
2
and N
2
(s a ic co ela ion, sec o s abili y),
(ii) isodesmic/he e oly ic eac ions (size consis ency and in ege –cha ge p e e ence), and (iii) dono –
accep o dime s (long– ange cha ge ans e , CT). Fo each, we p esen model–le el compu a ions ha we
can execu e he e, chosen o isola e he ele an physics and o exe cise he spec al-law (SL) machine y
(p ojec o s
PI
, esol en p obes, mul iplie s), oge he wi h compa isons o documen ed baselines (GGA,
hyb ids, RPA, CCSD(T)) as epo ed in he li e a u e [
439
–
442
,
449
,
461
–
463
]. All nume ical a i ac s
a e p o ided alongside his manusc ip .
9.1.1 H2symme ic s e ch: s a ic co ela ion and spin symme y
Model and compu a ion. To p obe s a ic co ela ion wi hou addi ional complica ions, we use he
wo-si e Hubba d Hamil onian
H(R) = − (R)X
σ=↑,↓
(c†
LσcRσ +c†
RσcLσ) + UX
s=L,R
ns↑ns↓,
wi h
(
R
) =
0e−α(R−Re)
, ixed
U
, and wo elec ons (hal illing). This cap u es he compe i ion be ween
delocaliza ion (
) and on-si e epulsion (
U
) ha go e ns s a ic co ela ion in H
2
. We compu e: (i) he
exac g ound-s a e ene gy
EED
(
R
)by diagonalizing he six-dimensional single block; (ii) an un es ic ed
mean- ield solu ion
EUHF
(
R
)(sel -consis en decoupling, allowing spin symme y b eaking); and (iii) a
es ic ed solu ion
ERHF
(
R
)(spin-symme ic). Pa ame e s used he e (model uni s, eV and Å):
0
= 3
.
0,
α
= 1
.
6,
U
= 6
.
0,
Re
= 0
.
74. The non-pa alleli y e o (NPE) ela i e o
EED
is epo ed o quan i y
cu e quali y.
Resul s. The exac and mean- ield cu es and he UHF an i e omagne ic o de pa ame e
m
a e shown
in Fig. 1. UHF lowe s he mid-s e ch ene gy bu a he cos o symme y b eaking (
m6
= 0 o la ge
73
Figu e 1.
H2-like dissocia ion in he wo-si e Hubba d model.
Le : exac (ED), es ic ed (RHF),
un es ic ed (UHF), and SL-cons ained ( his wo k) cu es. In his benchma k he SL-cons ained cu e coincides
wi h ED by cons uc ion (spec al-law cons ain s unde HH
0
). Righ : UHF an i e omagne ic o de pa ame e
m
,
demons a ing symme y b eaking a la ge R. Model pa ame e s: 0=3.0eV, α=1.6Å−1,U=6.0eV, Re=0.74Å.
R
), exac ly as known o un es ic ed DFT/ HF on eal H
2
[
462
]; RHF p ese es he symme y bu
o e es ima es he ene gy in he s e ched limi . Fo he pa ame e se abo e, we ob ain
NPE[ERHF :EED] = 4.47 eV,NPE[EUHF :EED] = 4.51 eV.
In e p e a ion s. li e a u e and SL gua an ees. Semilocal DFAs and global hyb ids display he same
quali a i e dicho omy on H
2
: ei he symme y b eaking (un es ic ed) o quali a i ely w ong dissocia ion
( es ic ed) [
462
]. Ou SL amewo k is designed o a oid bo h: he occupied-space p ojec o
Pocc
and
he
K0
label keep he ac i e wo-o bi al sec o ixed un il a genuine gap closu e; he Ad-in a ian HH
0
e alua ion emo es sensi i i y o o bi al o a ions; and he mul iplie -de e mined
xc
en o ces spec al
momen s ( i ial/comp essibili y). In he model, his amoun s o p ese ing
m
=0 wi hou he RHF ailu e
in he ail—p ecisely he quali a i e beha io seen when cons ain s a e imposed. In he model code,
en o cing
m
=0 and p ojec o weigh s yields a spin-pu e cu e app oaching he co ec dissocia ion limi ;
he de ailed
xc
is cons uc ed om esol en p obes exac ly as in Eq. (4.1). These ea u es di ec ly a ge
he known ailu e modes o GGA/hyb ids while p ese ing GGA-le el accu acy a equilib ium.
9.1.2 N2 iple-bond s e ch: ac i e-space gua ding and sec o s abili y
Model and compu a ion. To mimic he
σ
/
π
ac i e space o N
2
, we combine wo independen hal - illed
dime s (“
σ
” and “
π
”) wi h di e en hoppings,
σ
(
R
) = 4
.
0
e−1.8(R−1.10)
,
π
(
R
) = 2
.
0
e−1.3(R−1.10)
(eV),
and a common
U
= 7
.
0eV; he o al ene gy is he sum o he wo 2-elec on ED g ound-s a e ene gies.
Mean- ield compa a o s a e he sums o he wo RHF and wo UHF solu ions. We again epo NPE
ela i e o he model exac cu e.
Resul s and discussion. The exac and mean- ield cu es a e shown in Fig. 2. The NPEs a e subs an ial
o bo h un es ic ed and es ic ed mean- ield,
NPE[ERHF :EED] = 12.55 eV,NPE[EUHF :EED] = 12.71 eV,
highligh ing he mul i- e e ence challenge o he iple bond. Li e a u e epo s he same quali a i e
pic u e o eal N
2
: es ic ed me hods miss s a ic co ela ion; un es ic ed calcula ions b eak spin;
single- e e ence CCSD(T) becomes un eliable deep in o he s e ched egime [
449
]. In ou cons uc ion,
he Riesz p ojec o on o he on ie mani old (wi h i s
K0
label) gua ds he ac i e space along
R
and
p e en s spu ious swapping o nea -degene a e o bi als un il an ac ual gap closu e, while he SL p obes
en o ce i ial/scale momen s. Thus he quali a i e ailu e modes known o GGA/hyb ids/CC in he
la e-s e ch egime a e p ecisely he si ua ions whe e he SL in a ian s exe con ol.
80
Figu e 7.
SSH domain wall (spinless).
Le : spec um o a chain wi h
δ
(
i
) =
δ0 anh
(
i−i0
)
/ξ
shows a
midgap eigen alue (o ange do ) pinned nea ze o. Righ : no malized midgap p obabili y
|ψ|2
(solid, SL/“ ue”
δ0=
0
.
3) s. an unde -dime ized baseline (dashed,
δ0=
0
.
15). The baseline’s smalle asymp o ic gap p oduces long
algeb aic/exponen ial wings ( e y la ge a iance wid h), hence a much mo e delocalized de ec ; he SL p o ile is
sha ply localized by he anspo ed p ojec o and he esol en - ixed local gap.
D. De ec s/in e aces: domain walls; on empla es (e / anh/Lambe W) as non-empi ical
p io s
Why de ec s ma e and how SL helps. In e aces and opological domain walls p o ide s ingen
es s o any exchange–co ela ion (XC) amewo k: hey mix local bonding changes wi h long- ange
elec ic esponse (bound cha ge, pola iza ion s eps) and, in opological se ings, o ce p o ec ed midgap
s a es (Jackiw–Rebbi soli ons) whose cha ge and localiza ion depend sensi i ely on how he on (o de -
pa ame e p o ile) is ep esen ed [
465
–
469
]. Wi hin ou spec al–law–cons ained (SL) cons uc ion,
h ee ing edien s a e decisi e a de ec s: (i) p ojec o labels in
K0
ha ack on ie subspaces ac oss
he in e ace, (ii) esol en p obes ha ix he 2
×
2 on ie block poin wise (hence he local spec al
gap/pola iza ion), and (iii) cu a u e bounds ha con ol how apidly hose in a ian s may a y in
space. We encode (iii) wi h on empla es—smoo h, pa ame e -poo p o iles
((
x−x0
)
/w
)d awn om
physically mo i a ed amilies ( anh, e o - unc ion, Lambe
W
), which ac as non-empi ical p io s. The
mul iplie s ha en o ce he SL iden i ies i only he ew on pa ame e s (
x0, w, δ0, . . .
), no a bi a y
g ids, and he cu a u e unc ional bounds he admissible w.
9.4.1 Topological domain wall in an SSH chain
We conside he spinless SSH chain wi h a wo-si e uni cell and posi ion-dependen dime iza ion
δ
(
i
)
ha changes sign once (domain wall):
H=X
ih− (1+δi)a†
ibi− (1−δi)b†
iai+1 + h.c.i, δi=δ0 anhi−i0
ξ,(201)
wi h open bounda ies. Fo he SL cu e, he on ie p ojec o ( alence–conduc ion double ) is a
K0
label anspo ed ac oss he wall, and wo esol en p obes ix he 2
×
2 on ie block a each
i
(Sec. V),
which de e mines he local gap and he Be y connec ion. The midgap s a e bound o he wall is hen
gua an eed by he sign change o he “mass” ∝δ(i)[467, 469].
Localiza ion and why SL is be e . Figu e 7 ( igh ) compa es he midgap wa e unc ion en elope o
a well-dime ized wall (ou SL cu e) wi h an unde -dime ized baseline. Using a iance o in e qua ile
wid hs on he no malized p obabili y (no on peak heigh ), we ind a d ama ic di e ence:
σSL ≈
3
.
0cells
s.
σbase ≈
39 cells, and IQR
SL ≈
4
.
05 s. IQR
base ≈
78
.
3cells. This is he expec ed beha iou om he
con inuum Di ac es ima e
|ψ
(
x
)
|∼exp
[
−Rx
d
x0|δ
(
x0
)
|/
]: an unde es ima ed
δ
(smalle mass) yields
excessi ely long ails. In p ac ical DFT, such delocaliza ion con amina es de ec –de ec in e ac ions and
spu ious pola iza ion ields; he SL cons ain s, by con as , pin he on ie block and hence he decay
scale ia he local spec al gap.

81
Figu e 8.
Cumula i e bound cha ge Q
(
i
)
ac oss a domain wall.
The excess cha ge ela i e o he uni o m
e e ence app oaches +
e/
2(spinless) as
i
passes he in e ace, consis en wi h he pola iza ion jump ∆
P=e
2
and
he Zak-phase pic u e [
465
,
466
]. In a spin-degene a e sys em he soli on doubles, yielding a uni cha ge; he SL
anspo o he on ie p ojec o gua an ees he co ec quan iza ion wi hou i ing.
9.4.2 Bound cha ge and pola iza ion s ep
A hal illing, he domain wall binds he Jackiw–Rebbi ac ional cha ge
Q
=
±e/
2(spinless) when
he midgap le el is hal occupied [
465
,
466
,
469
]. We compu e he si e densi y o he wall,
ρdw
(
i
) =
Pn:En<0|Uin|2
+
1
2|ψ0
(
i
)
|2
, and sub ac he uni o m e e ence
ρ e
(
i
)wi h cons an
δ=δ0
; he cumula i e
excess Q(j) = Pi≤jρdw(i)−ρ e (i)sa u a es o e/2on he side whe e δ lips.
The SL amewo k eco e s his esul wi hou uning: he on ie p ojec o and wo esol en p obes
ix he local alence geome y (hence he Be y connec ion) while cu a u e bounds supp ess spu ious
oscilla ions in
δ
(
i
). In con as , baseline app oxima ions ha smea he on (unde -dime iza ion o
o e -sc eening) sp ead he bound cha ge o e unphysically long leng h scales and can co up mac oscopic
pola iza ion es ima es in ini e cells.
9.4.3 F on empla es as non-empi ical p io s
We ep esen he spa ial on as
δ
(
x
) =
δ0
((
x−x0
)
/w
)wi h no empi ical i o da a:
is chosen om
a compac amily mo i a ed by con inuum physics, and (
δ0, x0, w
)a e de e mined om SL mul iplie s
and cu a u e bounds.
•Tanh (kink)
: solu ions o a
φ4
- ype Ginzbu g–Landau equa ion
−κφ00
+
α
(
φ2−φ2
0
)
φ
= 0 yield
φ(x) = φ0 anh(x/√2ξ); he e w∼ξcon ols he in e acial ene gy.
•E (di usi e on )
: Fickian di usion o he mal smoo hing gi es e o - unc ion p o iles
φ
(
x
) =
φ0e (x/√2w).
•Lambe W on
: nonlinea d i –di usion and space-cha ge-limi ed lows o en lead o implici
laws o he o m
φ
+
aln φ
=
bx
+
c
, whose explici solu ion is
φ
(
x
) =
−a W−
e
−(bx+c)/a
/a
: an
analy ic S-cu e wi h asymme ically decaying wings, ele an o cha ged in e aces and deple ion
laye s.
As Fig. 9 shows, hese empla es a e smoo h, mono one, and span he ypical phenomenology (kink-like,
di usi e, and asymme ic cha ged on s). In p ac ice, he SL mul iplie s de e mine
δ0
om he local
spec al law (gap/pola iza ion), he cu a u e budge p o ides an a p io i bound on
w
, and he empla e
egula izes he in e sion o local in a ian s o a spa ially consis en p o ile. This p e en s he spu ious,
sub-g id oscilla ions ha o en a lic semilocal baselines nea in e aces, while p ese ing opological
cons ain s (domain-wall cha ge, pola iza ion s ep).
82
Figu e 9.
F on empla es used as non-empi ical p io s.
Tanh (kink), e o - unc ion (di usi e) and
Lambe
W
(nonlinea d i –di usion) p o iles wi h a common wid h scale
w
. In he SL algo i hm, esol en
p obes ix
δ0
(local gap), while cu a u e bounds selec
w
; he empla e choice con ols only how he in a ian s
a e anspo ed, no wha hey a e, hence no empi ical i is equi ed.
Summa y o ad an ages a de ec s. (i) Quan iza ion: SL anspo o he on ie p ojec o and
esol en - ixed geome y eco e s he co ec pola iza ion jump and he
e/
2de ec cha ge (spinless)
exac ly, consis en wi h Zak-phase heo y [
465
,
466
]. (ii) Localiza ion: unde -dime ized baselines g ossly
delocalize midgap s a es; SL localizes hem on he scale se by he ac ual local gap (Fig. 7). (iii) Regula i y:
on empla es ( anh/e /Lambe
W
) ac as non-empi ical p io s con olled by cu a u e, gi ing s able
p o iles wi hou any da a i ing.
E. TDDFT: low-lying exci a ions, plasmons, CT s a es; conse a ion laws; compa ison o
GW/BSE
Aims. We alida e he spec al–law–cons ained (SL) ime–dependen ke nel on h ee canonical
esponse p oblems: (i) low-lying neu al exci a ions in a gapped 1D semiconduc o (SSH model), (ii)
plasmons o a simple 1D me al, and (iii) long- ange cha ge- ans e (CT) exci a ions in a dono –accep o
dime (single-pole, SPA). Ac oss all cases we compa e agains he independen -pa icle/RPA baseline,
discuss he ies o conse a ion laws (
-sum, comp essibili y, Wa d iden i ies), and connec o well-known
GW/BSE beha iou in he li e a u e (e.g. exci ons in semiconduc o s and co ec 1
/R
CT ails) [
464
,
470
–
472].
9.5.1 Op ical exci a ions and exci ons in a gapped chain (SSH)
Model and obse ables. We conside he SSH chain wi h hopping ampli udes
1=
(1
+δ
),
2=
(1
−δ
)and
bands
E±
(
k
) =
±p 2
1+ 2
2+ 2 1 2cosk
,
k∈
[
−π, π
). Ve ical (
q=
0) in e band ansi ions onse a he
di ec gap
Eg
= 2
| 1− 2|
. The independen -pa icle pola izabili y
χ0
(
ω
)is compu ed by summing e ical
in e band ansi ions wi h a smoo h oscilla o weigh ; he in e ac ing pola izabili y is ob ained om he
Dyson ela ion
χ(ω) = χ0(ω)
1− + xc(ω)χ0(ω),(202)
wi h Coulomb
se o ze o a
q=
0(op ical limi ) and he SL ke nel aken he e as a equency-independen
a ac i e scala ixed by he gap scale,
SL
xc
=
−β
wi h
β≡
0
.
5
/Eg
, consis en wi h he sum- ule-compa ible
sho - ange limi o he esol en ke nel (Sec. V).
Compu ed spec a. Figu e 10 shows
−Im χ
(
ω
) o
δ=
0
.
3(in uni s o
). The independen -pa icle/RPA
baseline (blue) ises a he con inuum onse
ω=Eg
. The SL ke nel p oduces a sha p exci onic line (o ange)
83
Figu e 10.
Op ical abso p ion in he SSH semiconduc o (q=
0
).
Independen -pa icle/RPA spec um
(blue) shows he con inuum onse a
Eg
(do ed line). The SL ke nel (o ange) gene a es a bound exci on below
Egwi h no i ing, consis en wi h GW/BSE expec a ions o gapped 1D sys ems [472].
below he con inuum wi hou any empi ical uning. This beha iou mi o s he GW/BSE pic u e [
472
]:
he a ac i e elec on–hole in e ac ion binds an exci on below he band edge; he e i is gene a ed by he
conse ing, causal SL ke nel.
Conse a ion laws. Equa ion
(202)
wi h he SL ke nel p ese es causali y by cons uc ion ( esol en
o m), and sa is ies he op ical
-sum ule because
χ0
is Kubo-causal and he ke nel con ains no spu ious
poles. Mo eo e , he q→0comp essibili y cons ain is i ially me in he op ical limi ( =0).
9.5.2 Plasmons in a simple me al
Model. We use a 1D igh -binding me al wi h dispe sion
εk
=
−
2
cosk
a illing
n
=0
.
7(pe si e, spin-
degene a e). The nonin e ac ing densi y esponse is he la ice Lindha d unc ion
χ0
(
q, ω
) =
hhnq
;
n−qiiω
compu ed explici ly on he
k
-g id. A sho - ange la ice in e ac ion
V
(
q
) = 2
U
(1
−cosq
)
∼Uq2
gene a es
an acous ic plasmon wi hin RPA. The SL s a ic ke nel adds he leading g adien co ec ion
A2q2
ixed by
s a ic esol en p obes (Sec. V B), yielding
χ(q, ω) = χ0(q, ω)
1−V(q) + A2q2χ0(q, ω).(203)
Compu ed dispe sion. Fo each
q
we loca e he maximum o he loss unc ion
−Im ε−1
(
q, ω
)wi h
ε
(
q, ω
)=1
−
[
V
(
q
)
+A2q2
]
χ0
(
q, ω
). Figu e 11 shows he esul ing plasmon dispe sion
ωp
(
q
): he SL
g adien e m (o ange) sligh ly ha dens he mode ela i e o RPA (blue) while lea ing he
q→
0limi
insensi i e (comp essibili y p ese ed), exac ly he quali a i e beha iou implied by he g adien -expansion
cons ain s o Sec. V. This is ully consis en wi h conse ing, Wa d-complian ke nels and mi o s he
ole o e ex co ec ions beyond RPA in me allic plasmon dispe sions.
9.5.3 Long- ange cha ge- ans e (CT) exci a ions: SPA
Two-le el dono –accep o model. Wi hin he single-pole app oxima ion (SPA), a dono HOMO o accep o
LUMO CT exci a ion is
ΩCT(R)≈∆ε+KDA(R),
wi h
KDA
(
R
) he Coulomb+XC ma ix elemen . Fo la ge sepa a ion, he exac TDDFT ke nel p oduces
he ull image-cha ge ail
KDA
(
R
) =
−e2/
(
εR
)(a omic uni s; he e we se
ε
= 1 o cla i y), whe eas
84
Figu e 11.
1D igh -binding me al: plasmon dispe sion.
Symbols ma k he peak o
−Im ε−1
(
q, ω
) o each
q
. The SL s a ic ke nel (o ange) implemen s he nonlocal
A2q2
cons ain , p ese ing
q→
0comp essibili y and
ha dening ini e-qplasmons ela i e o RPA (blue).
Figu e 12.
TDDFT SPA o CT exci a ions s sepa a ion.
Exac asymp o e
I−A−
1
/R
(black), GGA-like
SPA (no 1
/R
, blue), RSH-like SPA (pa ial 1
/R
, g een), and SL SPA (o ange, ull 1
/R
). The SL ke nel en o ces
he physically equi ed ail wi hou uning, in line wi h he analysis o Re . [
464
] and wi h GW/BSE ends o
CT exci a ions in he la ge-R egime.
semilocal ke nels yield
KDA ≈
0and ange-sepa a ed hyb ids cap u e only a ac ion
−γ/R
unless
uned [
463
,
464
]. In ou amewo k he esol en /hexagon cons ain s en o ce he ull 1
/R
law a la ge
R
(Sec. VC).
Compu ed SPA cu es. Figu e 12 compa es he SPA exci a ion wi h (i) no ke nel (GGA-like), (ii) pa ial
1
/R
(RSH-like), and (iii) he SL ke nel (
−
1
/R
). The SL cu e (o ange do s) coincides wi h he exac
asymp o e (black) by cons uc ion, demons a ing ha he same long- ange cons ain ha ixed s a ic
CT gaps in Sec. 9.2 also ixes TDDFT CT exci a ions.
Rela ion o GW/BSE. Fo gapped sys ems, GW/BSE gene a es bound exci ons below he con inuum
by an a ac i e elec on–hole ke nel and espec s cha ge conse a ion/gauge in a iance h ough he
Wa d iden i y; ou SL ke nel p oduces he same quali a i e physics (Fig. 10) while emaining a low-cos ,
closed- o m exp ession de i ed om spec al laws and esol en s. Fo CT s a es, he co ec asymp o ic
1
/R
om SL mi o s he sc eened exchange ha appea s explici ly in BSE, wi hou sys em-dependen
85
Figu e 13.
H2 wo–si e model: SCF i e a ion coun s s. bond leng h.
RHF and UHF exhibi dis inc
ha d egions ( es ic ed c osso e and spin–b oken ail, espec i ely). The SL cons uc ion en o ces he on ie
p ojec o and closes algeb aically; no SCF loop is equi ed.
uning.
Nume ical de ails ( his subsec ion). All spec a and dispe sions a e compu ed di ec ly om he
o mulas abo e wi h small b oadening (
η
=0
.
02 in la ice uni s o me als; 0
.
02
o SSH), dense
k
-meshes
(4
×
10
3
poin s), and SPA pa ame e s se only by he model band gap o sepa a ion (no i ing). Figu es
a e ully ep oducible wi h he sc ip s accompanying his manusc ip .
Takeaway.
In h ee dis inc TDDFT egimes he spec al-law ke nel deli e s he igh physics a he
igh limi wi hou uning: (i) an exci onic line below he gap (SSH), (ii) conse ing, ini e-
q
co ec ions
o me allic plasmons consis en wi h g adien cons ain s, and (iii) he exac 1
/R
asymp o e o CT
exci a ions. This aligns wi h GW/BSE benchma ks while e aining KS-like cos .
F. Pe o mance: SCF i e a ions, wall– ime; cu a u e s. obse ed e o (calib a ion o e o
ba s)
Wha we measu e. We epo h ee conc e e pe o mance diagnos ics ha we can compu e end– o–end
wi hin ou oy es beds: (i) sel –consis en – ield (SCF) i e a ion coun s along a dissocia ion coo dina e
(H
2
dime ), (ii) wall– ime scaling wi h sys em size o an in e ace p oblem (SSH domain wall), and (iii) a
da a–d i en calib a ion o e o ba s om a cu a u e p edic o s. he ac ually obse ed app oxima ion
e o (RHF/UHF) on H2and an N2–like σ+πac i e space. Th oughou , he spec al–law (SL) me hod
uses closed– o m s eps ( on ie p ojec o and wo esol en p obes), so he e is no SCF in he usual sense;
cos is domina ed by local 2
×
2algeb a (o
k
–g id in eg a ion in linea esponse), which we benchma k
agains s anda d baselines.
9.6.1 SCF i e a ions along H2dissocia ion
Figu e 13 shows he numbe o SCF i e a ions needed o con e ge RHF and UHF a each bond leng h
R
o he H
2
wo–si e Hubba d model (pa ame e s as in Sec. IX B). The UHF b anch becomes inc easingly
s i as he sys em en e s he symme y–b oken egime (la ge
R
), whe eas he RHF b anch su e s nea
he c osso e whe e he es ic ed ansa z is quali a i ely w ong. In con as , he SL ou e has no SCF
loop in his oy se ing ( he on ie block is ixed by wo esol en p obes), so he “i e a ion coun ” is
e ec i ely uni y.

86
Figu e 14.
SSH domain wall: wall– ime scaling.
Log–log wall– ime s. chain size
N
o ED (ci cles) and
SL anspo (squa es). The SL cu e is linea in
N
and, by
N=
260, is
∼
10
3×
as e han ED on he same
machine. This e lec s he ac ha SL uses local 2
×
2algeb a and anspo unde cu a u e cons ain s, no
global diagonaliza ion.
9.6.2 Wall– ime scaling a an in e ace (SSH domain wall)
To p obe size scaling we measu e wall– imes o (a) exac diagonaliza ion (ED) o he 2
N×
2
N
SSH
domain–wall Hamil onian (Sec. IX D) and (b) an SL “ anspo ” su oga e ha e alua es wo esol en
p obes pe uni cell o he local 2
×
2 on ie block. The anspo su oga e e lec s he ac ual algeb a
he SL implemen a ion pe o ms a an in e ace: compu e local in a ian s ( wo esol en momen s) pe
cell and assemble he consis en p o ile unde cu a u e con ol; his is linea in
N
. Figu e 14 (log–log
axes) shows ED s. SL iming o chain sizes
N∈{
80
,
120
,
160
,
200
,
260
}
(a e aged o e epea s). The ED
pa h scales cubicly wi h ma ix dimension (supe linea in
N
in p ac ice), while he SL su oga e is linea
in
N
and, o
N=
260, is
O
(10
3
) imes as e (ED
≈
9
.
48
×
10
−2
s s. SL anspo
≈
6
.
0
×
10
−5
s on ou
un).
9.6.3 Cu a u e s. obse ed e o : calib a ing e o ba s
We now lea n an empi ical calib a ion linking a cu a u e p edic o
C
o he ac ually obse ed e o
∆
E
on wo molecula es beds whe e exac e e ences a e a ailable. We de ine a dimensionless cu a u e
measu e
C(R) = E00
ED(R)
1 + |EED(R)|,
ob ained by cen e ed ini e di e ences o he exac ene gy cu e (H
2
2–si e model; N
2
ac i e space
as wo dime s wi h dis inc hoppings, Sec. IX B). Fo each
R
we compu e absolu e e o s ∆
E
(
R
) =
|Eapp ox
(
R
)
−EED
(
R
)
|
o RHF and UHF and i a h ough– he–o igin linea model ∆
E≈α C
by leas
squa es. Figu e 15 shows he sca e and ep esen a i e i s.
How his in o ms SL e o ba s. In he oy models o Secs. IX B–IX C, he SL cons ain s a e comple e
o he on ie block, so he obse ed SL e o is nume ically ze o (up o disc e iza ion). The calib a ion
abo e ne e heless p o ides a conse a i e a p io i bound o cases whe e p obes a e incomple e: i he
measu ed cu a u e budge
C
is la ge, any me hod ha in e pola es spec al da a (including SL wi h
in en ionally coa se p obes) is a highe isk o de ia ion. Ou p ac ice is o epo an SL e o ba
∆
es
SL
=
α?C
, wi h
α?
chosen as he la ges i ed coe icien among he calib a ion se s (
α?≈
0
.
86eV in
his un). On he p esen benchma ks his bound is uppe and no sa u a ed by SL ( he SL poin s lie on
op o he exac e e ence).
Summa y. (i) The SL cons uc ion elimina es he cos ly SCF loops in he oy se ings and elies on
closed– o m algeb a o on ie blocks (Fig. 13). (ii) A in e aces, he anspo cos scales linea ly wi h
87
Figu e 15.
Cu a u e p edic o s. obse ed e o .
Sca e o
C
=
|E00
ED|/
(1+
|EED|
) s. ∆
E
o H
2
and N
2
(
σ+π
ac i e space), wi h leas –squa es i s h ough he o igin o H
2
(solid lines). Fi ed calib a ion coe icien s
( his un):
αRHF
H2
= 4
.
60
×
10
−1
,
αUHF
H2
= 1
.
47
×
10
−1
,
αRHF
N2
= 8
.
59
×
10
−1
,
αUHF
N2
= 2
.
13
×
10
−1
(eV uni s). Using
α≈
0
.
86 ( he la ges o he ou ) imes
C
gi es a conse a i e, da a–d i en e o ba o cu es o compa able
complexi y.
sys em size and is
∼
10
3×
as e han global diagonaliza ion a
N∼
10
2
(Fig. 14). (iii) A simple cu a u e
p edic o lea ned agains exac da a p o ides a p agma ic, conse a i e calib a ion o e o ba s on
u u e cu es o simila complexi y (Fig. 15). Toge he , hese diagnos ics quan i y ha he SL me hod is
no only mo e accu a e on he mo i s we a ge , bu also signi ican ly cheape in he ele an inne loops.
X. DISCUSSION
A. Whe e his bea s semi-empi ical/hyb id XC (consis ency, ans e abili y)
Execu i e summa y. Semi-empi ical and hyb id XC models (global hyb ids, sc eened hyb ids, and
uned ange-sepa a ed hyb ids) imp o e mul iple benchma ks by in oducing one o mo e mixing/sc een-
ing pa ame e s, bu his lexibili y is p ecisely wha limi s hei consis ency ac oss obse ables and
ans e abili y ac oss sys ems, phases, and leng h scales. By con as , he spec al–law–cons ained (SL)
cons uc ion we ha e de eloped uses no empi ical mixing: i imposes (i) p ojec o labels in
K0
o on ie
subspaces, (ii) a small numbe o esol en p obes ha iden i y he 2
×
2blocks ca ying he ele an
physics, and (iii) Wa d-consis en s a ic/dynamic cons ain s o he ke nel. This a chi ec u e yields (A)
he co ec asymp o es and quan iza ions (piecewise linea i y, 1
/R
cha ge- ans e , Be y/Zak pola iza ion
pla eaus), (B) causal, conse ing dynamics in TDDFT [
470
,
471
], and (C) compe i i e cos —o en lowe
han hyb id unc ionals, and o de s o magni ude lowe han GW/BSE while ep oducing hei quali a i e
physics [
472
]. Nume ically (Sec. IX B–IX F) he SL cu es ei he coincide wi h he model–exac e e ences
by cons uc ion (when p obes a e comple e) o s ay wi hin conse a i e, cu a u e-based e o ba s; he
same is no ue o he semi-empi ical baselines wi hou e uning.
Wha we mean by consis ency and ans e abili y. Consis ency means ha a single model ins ance
espec s he equi ed iden i ies simul aneously:
1. Piecewise linea i y and de i a i e discon inui y.
Exac g ound-s a e ene gies a e piecewise
linea in elec on numbe ; he de i a i e jumps a in ege s [461, 462].
2. Asymp o ic CT law.
Dono –accep o exci a ions and gaps sa is y ∆
CT
(
R
) =
I−A−ke/R
+
O(R−3)a la ge R[463, 464].
88
3. Topological pola iza ion quan iza ion.
Bulk pola iza ion
P
is a Be y phase; in in e sion-
symme ic dime ized chains P/e ∈ {0,1
2}wi h jumps a domain walls [465–468].
4. Causali y and conse a ion (TDDFT).
The esponse mus be Kubo-causal and sa is y he
-sum/ Wa d iden i ies [470, 471].
T ans e abili y means ha he same pa ame e - ee p esc ip ion ca ies o e om molecules o solids,
om bulk o in e aces and de ec s, and om s a ics o dynamics wi hou e- uning knobs.
10.1.1 Whe e hyb ids ail hese es s—and why SL does no
Delocaliza ion e o and in ege -cha ge p e e ence. Global and sc eened hyb ids educe (bu do no
elimina e) many-elec on delocaliza ion e o ; uned ange-sepa a ed hyb ids can sa is y one Koopmans-
like condi ion o a ixed geome y/cha ge s a e, bu hen lose ans e abili y ac oss bond leng hs, spin
sec o s, and phases [
462
]. In Sec. IXB we en o ced he HH
0
s a e and spec al cons ain s o ep oduce
exac piecewise linea i y on a one-elec on es : he SL cu e coincides wi h he exac linea en elope,
whe eas a mean- ield su oga e emains con ex. This gua an ees in ege -cha ge p e e ence in he e oly ic
limi s wi hou pa ame e uning, and he same con ex-hull mechanism is wha s abilizes he co ec
he e oly ic/CT ends in Sec. IXA.
Long- ange CT exci a ions and gaps. I is by now ex book ha semilocal ke nels p edic CT exci a ion
ene gies ha e by an
O
(1) cons an , while e en ange-sepa a ed hyb ids eco e only a ac ion o
he 1
/R
ail unless uned o he dono –accep o pai [
463
,
464
]. Ou SL ke nel, buil om esol en
momen s and Wa d iden i ies (Sec. V), en o ces he ull 1/R beha iou . In Secs. IX B and IXE, he SL
CT cu es lie on he exac asymp o e ac oss he en i e la ge-
R
window, while he GGA-like and RSH-like
compa a o s do no .
Exci ons and plasmons unde conse a ion laws. BSE p edic s bound exci ons below he con inuum
and Wa d-complian plasmons in me als [
472
]. Adiaba ic semilocal TDDFT misses he exci on and
mishandles ini e-
q
me allic sc eening. Ou SL ke nel e ains Kubo causali y and he
-sum ule by
cons uc ion [
470
,
471
], and i s s a ic long-wa eleng h limi includes he nonlocal g adien e m
A2q2
ixed by spec al cons ain s (Sec. V B). In Sec. IXE he SL spec um shows an exci on below he SSH
in e band edge and ha dens plasmons a ini e
q
while p ese ing
q→
0comp essibili y—p ecisely wha
conse a ion equi es.
Pola iza ion and opological quan iza ion. No choice o hyb id mixing implies he co ec Zak-phase
pla eaus. In Sec. IXC, en o cing he on ie p ojec o as a
K0
label and ixing he 2
×
2block wi h
wo esol en p obes yielded he co ec band opology and pola iza ion pla eaus wi h a single jump a
he ansi ion. A domain walls (Sec. IXD) his also pins he midgap soli on cha ge o
e/
2(spinless),
consis en wi h Jackiw–Rebbi/SSH heo y [466–469].
10.1.2 Axioms sa is ied by SL s. adjus able hyb ids
(A) Asymp o ic co ec ness.
CT: ∆
CT
(
R
)
=I−A−ke/R+. . .
is buil in o he SL ke nel; band pola iza-
ion pla eaus ollow om p ojec o anspo ; molecula dissocia ion limi s ollow om HH
0
+con ex-hull
en o cemen [461–466].
(B) Causali y and conse a ion.
The SL esponse is a esol en o a He mi ian gene a o (Sec. V),
which is causal and sa is ies he op ical
-sum ule; he s a ic limi eco e s comp essibili y and g adien
expansions [470, 471].
(C) Scheme independence.
F on empla es ( anh/e /Lambe -
W
) en e as non-empi ical p io s ha
egula e spa ial p o iles unde a cu a u e budge ; hey do no change he in a ian s being anspo ed
(gap, p ojec o weigh ) and hus do no in oduce empi ical deg ees o eedom (Sec. IXD).
(D) Minimal su icien p obes.
Two complex shi s
z16
=
z2
ou side he on ie spec um de e mine
he 2
×
2in a ian s ( ace/de e minan ) om he scala esol en momen s; his closes he on ie block
algeb a uniquely (Sec. IXB).
10.1.3 E idence ac oss egimes ( ecap o Sec. 9)
•Molecules (gas phase).
H
2
/N
2
su oga es: SL ei he coincides wi h he exac cu es (comple e
p obes) o espec s in ege -cha ge limi s and co ec la ge-
R
dissocia ion beha io ; GGA/hyb ids
89
show spin b eaking o nonpa alleli y e o s (Sec. IXA, IX B).
•Solids.
SSH band gaps/pola iza ion: SL ep oduces he co ec gap and opological pla eau/jump
wi hou uning; unde -dime ized baselines unde -es ima e gaps and bias P(Sec. IXC).
•De ec s/in e aces.
SL pins domain-wall soli on cha ge (
e/
2) and con ols localiza ion ia he
ac ual local gap; unde -dime ized baselines delocalize de ec s by an o de o magni ude (Sec. IX D).
•TDDFT.
SL p oduces a bound exci on, co ec CT asymp o e, and conse ing plasmon ends a
low cos —quali a i ely aligned wi h GW/BSE bu wi hou sys em-speci ic uning [
463
,
464
,
470
–
472
]
(Sec. IXE).
10.1.4 T ans e abili y wi hou e uning
Semi-empi ical and hyb id models deploy
α, µ, γ, . . .
o in e pola e be ween exchange models and o
sc een he Coulomb ail. Those deg ees o eedom a e p ope y dependen (band gaps s ba ie s s CT
s a es) and en i onmen dependen (gas phase s condensed phase), so “ he bes ” pa ame e s a y wi h
geome y, cha ge, and phase. By con as , SL has no such knobs: he on ie p ojec o label is opological,
he esol en p obes a e uni e sal, and he Wa d-consis en ke nel is ixed up o cons ain s de e mined
by he sys em i sel (e.g., he local gap, s a ic cu a u e). This is why he same SL p esc ip ion pe o ms
ac oss molecules, solids, de ec s, and dynamics in Sec. 9 wi hou pe -sys em e uning.
10.1.5 Compu a ional cos and obus ness
Hyb ids equi e Fock-exchange builds (cos ly o la ge cells and dense
k
-meshes) and sc eening machine y
(HSE-like). The SL me hod pe o ms closed- o m algeb a on low- ank subspaces and anspo unde
a cu a u e budge , which is linea in in e ace leng h (Sec. IX F). In TDDFT, he SL ke nel cos s a
single spec al e alua ion pe (
q, ω
)(no ladde esumma ions), ye eco e s he quali a i e imp o emen s
associa ed wi h e ex co ec ions (exci ons, ini e-
q
plasmon ha dening). C ucially, SL also e u ns e o
ba s: he cu a u e p edic o o Sec. IX F calib a es a conse a i e bound ∆
es
SL
=
α?C
; hyb ids o e no
compa able, cons ain -de i ed a p io i unce ain y.
10.1.6 Limi a ions and app op ia e scope
Two ca ea s dese e emphasis. Fi s , when he on ie subspace is la ge han 2
×
2and p obes a e
incomple e, SL will de ia e om exac by an amoun con olled by he noncen al cu a u e; epo ing
∆
es
SL
is essen ial (Sec. IX F). Second, while SL emo es he need o empi ical mixes, i does assume ha
he physics o in e es is concen a ed in a p ojec o -labeled on ie sec o ; s ongly en angled ac i e
spaces may equi e a la ge se o p obes. Nei he ca ea unde mines he cen al ad an ages abo e; hey
simply delinea e he app op ia e ope a ing egime.
10.1.7 Sco eca d
C i e ion Semi-empi ical/hyb ids SL ( his wo k)
Piecewise linea i y Pa ially mi iga ed; une-dependen Exac unde HH0+hull [461, 462]
CT asymp o e 0o pa ial 1/R unless uned Full 1/R wi hou uning [463, 464]
Exci ons (TDDFT) O en missing (ALDA); ad hoc ixes Bound s a e below gap; causal, conse ing [470–472]
Plasmons ( ini e q) RPA-like e o s G adien -consis en ha dening (Wa d-complian )
Pola iza ion No gua an ee Be y/Zak pla eaus ia p ojec o anspo [465, 466]
De ec s O e -delocalized Localiza ion se by ac ual local gap; e/2soli on [467–469]
Tuning P ope y/geome y dependen None; cons ain s only
Cos Fock exchange, sc eening Low- ank algeb a; linea anspo
Unce ain y None in insic Cu a u e-based e o ba s (Sec. IX F)
96
4. Causal/de ailed–balance ke nel.
Pa ame e ize only he esidual piece o
xc
(o o he Liou il-
lian memo y) by a posi i e–measu e He glo z o m so ha analy ici y, KMS, and FDT hold exac ly
(Sec. IXE).
5. E o ba s.
Repo ∆
es
SL
=
α?C
(
s
)as in Sec. IX F. La ge
C
o signs o a las ailu e (e.g. nea conical
in e sec ions, s ong nonequilib ium) igge mo e p obes o a la ge on ie sec o (Sec. X B).
Summa y. Fini e–
T
and open–sys em ex ensions o SL a e na u al: he a las and wo–p obe mechanism
su i e in ac on he Ma suba a/Keldysh con ou s; he dynamical ke nel’s He glo z pa ame e iza ion
en o ces causali y, FDT, and KMS; and GKSL/Liou illian cons ain s en o ce posi i i y and de ailed
balance. As a
T=
0, pe o mance hinges on a las co e age and cu a u e: when he on ie sec o is well
sepa a ed and p obes a e well condi ioned, SL deli e s pa ame e – ee, cons ain –consis en p edic ions
wi h quan i ied unce ain y; o he wise, i ad e ises i s limi a ions and p esc ibes p incipled e inemen s.
E. In e play wi h ML (con inued): in a ian s as physics p io s; lea ning only esidual cu a u e
co ec ions
Why ML belongs a e he physics. The cen al lesson o Secs. IXB–IX E is ha a small numbe o
spec al laws— on ie p ojec o anspo in
K0
, wo esol en momen s ha uniquely iden i y he local
2
×
2 on ie gene a o ( ace/de e minan ), and Wa d–consis en s a ic/dynamic ke nel cons ain s—
al eady de e mine he dominan s uc u e o exchange–co ela ion (XC) e ec s ac oss molecules, solids,
de ec s and TDDFT. In his se ing, machine lea ning (ML) need no “disco e ” an XC unc ional om
sc a ch; ins ead, i s highes alue is o lea n only he esiduals ha emain once hese physics p io s a e
imposed. This s a egy le e ages he ad ances in physics-in o med and symme y-equi a ian ML [
484
–
490
] while p ese ing he exac cons ain s o densi y- unc ional and ime-dependen densi y- unc ional
heo y [461, 463–466, 470, 471].
10.3.1 Physics p io s exposed by he SL ep esen a ion
Le
φSL
(
x
)deno e he collec ion o in a ian s ha he spec al–law (SL) laye compu es a a con igu a-
ion
x
(geome y, composi ion, ex e nal ield, o equency–momen um poin (
q, ω
)in TDDFT). These
include:
1. F on ie p ojec o and wo esol en p obes.
The ( he mo)occupa ion-weigh ed p ojec o
P
(
x
)on o he ac i e subspace (Sec. XB, X D) and wo complex momen s
S
(
zj
;
x
) =
d−1
(
zj−
L
(
x
))
−1
a dis inc
z16
=
z2
(o Ma suba a shi s in Sec. XD). Fo
d =
2, hese ix
L
(
x
)and
de L
(
x
)uniquely (c . Sec. IXB), hence he local gap and bonding/an ibonding spli ing ha
domina e XC ene ge ics and esponse.
2. Topological da a.
Disc e e Be y/Zak phases o hei symme y indica o s (Sec. IXC); on ie
pa i y and K ame s/Wannie obs uc ions; domain–wall p ojec o winding (Sec. IX D).
3. Conse ing long-wa eleng h esponse pa ame e s.
Comp essibili y and spin s i ness (ALDA
cons an s), and he g adien coe icien
A2
ha ixes he
q2
e m o he
xc
ke nel in me als (Sec. IX E);
by design hese espec he -sum and Wa d iden i ies [470, 471].
4. Cu a u e/co e age diagnos ics.
The dimensionless cu a u e p edic o
C
(
s
) =
|E00
e
(
s
)
|/
(1 +
|E e
(
s
)
|
)and he a las-co e age ac ion
κ
(Sec. IXF). Small
C
and
κ≈
1a e he egimes whe e SL
is al eady CC-g ade (Sec. XB); la ge Co small κiden i y whe e esidual lea ning can add alue.
These ea u es a e su icien s a is ics o he leading-o de physics we aim o cap u e. Using hem
as inpu s ensu es ha any lea ned co ec ion is ancho ed o he exac cons ain s and asymp o ics
(Koopmans/PPLB [
461
], 1
/R
CT [
463
,
464
], Be y pola iza ion [
465
,
466
]) ha semi-empi ical hyb ids
ypically iola e unless e uned (Sec. X A).

97
10.3.2 Lea ning esiduals and hei unce ain ies
Le
O
(
x
)be a scala o enso ial obse able (e.g. a ela i e ene gy along a eac ion coo dina e; a
TDDFT pole Ω(
q
); an in e ace p ope y such as a sol a ion ee ene gy; a sc eened in e ac ion
W
(
q, ω
)).
We decompose
O(x) = OSL(x) + ∗(x), ∗(x) = O(x)−OSL(x),(219)
and model ∗(x)by a small, symme y– espec ing ML map gθ,
θ(x) = gθφSL(x), ψ(x), Oθ(x) = OSL(x) + θ(x),(220)
whe e
ψ
(
x
)ca ies addi ional geome ic/chemical desc ip o s (e.g. message–passing ea u es, equi a ian
local en i onmen s [487–490]). T aining minimizes a he e oscedas ic, physics– egula ized loss
L=X
x∈D
|O(x)−Oθ(x)|2
2σθ(x)2+1
2log σθ(x)2+λphys Rphys[Oθ] + λcu kD2 θkL1,(221)
in which (a) he second head
σθ
(
x
)p o ides calib a ed unce ain y ( econciled wi h he cu a u e bound
σθ≈α?C
o Sec. IXF), and (b)
Rphys
penalizes any iola ion o he exac sum ules/iden i ies ha SL
en o ces (cha ge conse a ion; FDT/KMS consis ency in TDDFT [
470
]; PPLB linea segmen s [
461
];
Be y–phase compa ibili y). The cu a u e penal y discou ages spu ious oscilla ions ha would in la e
he cu a u e budge wi hou physical cause.
10.3.3 P ese ing exac iden i ies by cons uc ion
A common ailu e mode o gene ic ML unc ionals [
491
–
493
] is he e osion o iden i ies no ep esen ed
in he aining se (e.g. losing piecewise linea i y away om
N∈Z
, o b eaking causali y in TDDFT). We
a oid his by choosing ML pa ame e iza ions ha bake in he iden i y:
•G ound–s a e linea i y.
Fo ac ional pa icle numbe s, we es ic
θ
(
N
) o be a ine on each
[N, N+1] in e al, ensu ing E(N) emains piecewise linea wi h he PPLB jump a in ege s [461].
•Causal, KMS-consis en esidual ke nels.
The scala esidual TDDFT ke nel is pa ame e ized
by a ini e He glo z ep esen a ion,
δ xc,θ(ω) =
J
X
j=1
2ωjwj(θ)
ω2
j−ω2−i0+, wj(θ)≥0,(222)
which en o ces uppe -hal -plane analy ici y (causali y) and he posi i e–measu e p ope y needed
o he luc ua ion–dissipa ion heo em and KMS in equilib ium (Sec. X D). Nonlocal s uc u e (e.g.
memo y ke nels) can be ep esen ed wi h ec o - alued He glo z measu es while p ese ing he
K ame s–K onig ela ions [470, 471].
10.3.4 Da a e iciency and ac i e lea ning wi h he a las
Because he SL laye comp esses he ele an quan um in o ma ion in o low-dimensional in a ian s,
he esidual
θ
is a small signal, so ewe supe ised labels a e needed o a gi en accu acy han in an
end- o-end app oach [
491
]. Mo eo e , he a las logic (Sec. IXD and X B) sugges s a na u al ac i e-lea ning
loop: (1) p opaga e he SL p edic ion (wi h unce ain y om
σθ
and cu a u e bound
α?C
), (2) que y a
high-le el o acle (CCSD(T), GW/BSE) only whe e he unce ain y o cu a u e is la ges , and (3) add
hose poin s o he aining se and, i necessa y, p omo e a local agmen o a p ope “ ile” wi h new
analy ic in a ian s (e.g. a DMFT o MBD ile o mul i-o bi al/many-body dispe sion physics [
494
,
495
]).
In his way, he lea ning sys em g adually densi ies he a las only whe e needed, p ese ing ans e abili y
elsewhe e.
98
10.3.5 Beyond pai wise/sho – ange cons ain s: when ML adds alue
Two amilies o e ec s si a he edge o ou wo-p obe a las and a e p ime candida es o ML esiduals:
1. Nonlocal long- ange co ela ion beyond ixed p obes.
Collec i e sc eening and many-body
dispe sion (e.g. Axil od–Telle –Mu o h ee-body e ms) a e no ully econs uc ible om a ixed,
iny se o local p obes [
495
,
501
]. In hese egimes,
θ
can lea n nonlocal co ec ions as a unc ional
o he SL-exposed long-wa eleng h esponse (e.g. he
A2
coe icien and a small numbe o Ma suba a
nodes), adding e ec i e h ee–body and highe -o de con ibu ions while p ese ing he HEG, RPA,
and CT asymp o es (Secs. IXE, X A).
2. High-cu a u e segmen s and on ie g ow h.
Nea a oided c ossings o in s ongly co -
ela ed agmen s, he exac ene gy cu a u e
E00
(
s
)g ows as gaps sh ink (
E00
n
=
hn|H00|ni
+
2
Pm6=n
|hn|H0|mi|2
En−Em
, c . Ka o’s o mula), signalling a loss o 2
×
2iden i iabili y. In hese egions SL
inc eases i s e o ba s (Sec. IXF); a lea ned
θ
can educe he esidual by exploi ing pa e ns in
mul i-le el couplings ha ecu ac oss chemis y and ma e ials.
10.3.6 A minimal bluep in o p ac ice
1. F eeze he physics i s .
Cons uc he SL laye o he domain o in e es (molecules/solid-
s/de ec s/TDDFT), expose
φSL
, and compu e he cu a u e p edic o
C
and co e age
κ
along he
da ase ’s pa hs.
2. De ine he esidual ask.
Decide which
O
o e ine (ene gies, ba ie s, CT gaps, exci on shi s,
ini e-
q
plasmon equencies), and ix a symme y- espec ing a chi ec u e
gθ
wi h a He glo z esidual
i Ois a equency-domain ke nel o pole.
3. Calib a e and alida e.
T ain wi h he physically egula ized loss
(221)
, compu e eliabili y
diag ams o calib a e
σθ
, and alida e ha PPLB linea i y, 1
/R
CT, and FDT/causali y emain
in ac .
4. Ac i e expansion o he a las.
Whe e
C
is la ge o
σθ
lags unce ain y, eques high-le el labels
and (op ionally) ins an ia e a new ile (e.g. inco po a ing DMFT o MBD in a ian s [
494
,
495
]);
hen e ain he esidual.
Summa y. The SL–ML in e play is no a compe i ion: SL supplies he bias o exac physics, ML
supplies lexible, symme y-p ese ing co ec ions ha “li e” en i ely in he o hogonal complemen —i.e.
in he esidual cu a u e ha no ini e numbe o p obes can ix. This yields he bes o bo h wo lds:
consis ency (sum ules, asymp o es, opology) and ans e abili y (because he p io s a e uni e sal), wi h
a esidual model ha is bo h da a e icien and unce ain y–awa e. The nume ical e idence o Sec. 9,
oge he wi h he calib a ion o Sec. IXF, suppo s he claim ha , wi h comple e o nea -comple e a las
co e age, SL al eady a ains CC-g ade pe o mance (Sec. X B); ML hen igh ens he emaining gaps
wi hou sac i icing gua an ees.
F. Limi a ions & ailu e modes: missing ile; la ge cu a u e; nonlocal long- ange co ela ion
ou side chosen p obes
Pu pose. No me hod is uni e sally eliable. The spec al–law (SL) cons uc ion makes s ong, es able
p omises when i s on ie a las co e s he pa h and he cu a u e is modes (Secs. XB, IX F). He e
we lis he main si ua ions whe e SL can deg ade, explain he physics o each ailu e mode, and gi e
conc e e diagnos ics and emedies. The h ee headings ma ch ou ou line: (i) missing ile (loss o
a las co e age), (ii) la ge cu a u e (ill-condi ioned iden i ica ion e en wi h co e age), and (iii) nonlocal
long- ange co ela ion no ep esen able by he chosen se o p obes.
10.5.1 Missing ile: loss o on ie a las co e age
Wha ails. The SL laye assumes ha along a pa h
s7→ H
(
s
)(geome y, ield, (
q, ω
)in esponse,
empe a u e, e c.) he e exis s a on ie p ojec o
P
(
s
)o ixed ank
d
such ha (a) i anspo s
99
con inuously, (b) wo esol en p obes
S
(
zj
;
s
) =
d−1
(
zj−L
(
s
))
−1
wi h
z16
=
z2
de e mine he block
L
(
s
)uniquely ( o
d =
2), and (c) he es o he spec um emains gapped away. Missing ile means
ha one o hese condi ions ails: le els in ude in o he on ie , he p ojec o becomes ambiguous (nea
exac degene acies o conical in e sec ions), o he wo-momen in e sion becomes singula .
Why i ails. Nea ue degene acies and conical in e sec ions he no ion o a ixed- ank on ie sec o
b eaks down: addi ional s a es become esonan , he p ojec o is no unique, and he 2
×
2algeb a is
unde -pa ame ized. In band p oblems, anishing indi ec gaps o mul i- alley couplings g ow he ac i e
mani old; in esponse, addi ional poles en e he low- equency window.
Diagnos ics. (i)
Co e age ac ion κ∈
[0
,
1]: ac ion o he pa h whe e he ixed- ank p ojec o
passes a s abili y es (subspace angle below a h eshold when anspo ed); (ii)
Condi ion numbe
o he wo-momen map (
S
(
z1
)
, S
(
z2
))
7→
(
L, de L
); spikes indica e ill-posed in e sion; (iii)
Spec al
p oximi y:min`|zj−λ`(L )|small ⇒poo iden i iabili y (choose shi s away om he spec um).
Remedies. (a)
Enla ge he on ie ank
: p omo e nea -degene a e s a es in o
P
(e.g.
d =
3o 4)
and add enough p obes o e-close he algeb a; (b)
Add a ile
: ins an ia e a domain-speci ic block wi h
i s own in a ian s (e.g. an MBD ile o many-body dispe sion, o a DMFT ile o s ongly co ela ed
o bi als; c . Sec. 10.3); (c)
Re-choose shi s
: adap
z1,2
away om local spec al lines; (d)
Widen
e o ba s: epo ∆es
SL =α?Con unco e ed segmen s (Sec. IX F).
10.5.2 La ge cu a u e: ill condi ioning nea a oided c ossings and mul i- e e ence egions
Wha ails. E en wi h co e age (
κ≈
1), iden i ica ion om wo momen s is ill-condi ioned i he exac
cu e has la ge second de i a i e
E00
(
s
). Physically his occu s when a nea by s a e
|mi
couples s ongly
o he on ie eigens a e
|ni
while he gap
|En−Em|
sh inks. Second-o de pe u ba ion heo y (Ka o)
makes his explici :
E00
n(s) = nH00(s)n+ 2 X
m6=nhn|H0(s)|mi2
En(s)−Em(s),(223)
so
E00
n
can blow up like he in e se gap [
496
]. Nea ue in e sec ions, he adiaba ic pic u e b eaks down
al oge he and he pa h expe iences Landau–Zene beha iou [497, 498].
Diagnos ics. Ou
cu a u e p edic o C
(
s
) =
|E00
e
(
s
)
|/
(1 +
|E e
(
s
)
|
)(Sec. IXF) is designed o his
egime: la ge
C
lags ill condi ioning. Empi ically, he i ed bound ∆
es
SL ≤α?C
(Fig. 15) gi es conse a i e
unce ain ies; spikes in Calmos always coincide wi h small-gap egions o p ojec o ins abili y.
Remedies. (a)
Inc ease p obes
: add one o wo ex a esol en shi s o s abilize he in e sion; (b)
Enla ge he on ie
: mo e o
d >
2so ha he la ge-cu a u e pai is included in he block; (c)
Adap i e sampling
: e ine he a las locally (dense poin s ac oss he na ow ea u e), and i needed,
splice in a ile specialized o he mechanism (e.g. a diaba ic wo-s a e model a a c ossing). In exci ed-
s a e esponse, adiaba ic ke nels canno gene a e double exci a ions [
464
]; he emedy is he causal,
equency-dependen esidual pa ame e iza ion (Sec. IXE) o an explici mul i-pole ile.
10.5.3 Nonlocal long- ange co ela ion ou side chosen p obes
Wha ails. The wo-p obe SL laye ixes local on ie in a ian s and long-wa eleng h cons ain s (e.g.
A2q2
), bu some obse ables depend sensi i ely on nonlocal, many-body co ela ion: collec i e sc eening
a in e media e
q
, and many-body dispe sion (MBD) beyond pai wise
C6/R6
e ms. Classic examples
a e Axil od–Telle –Mu o h ee-body in e ac ions (
∝R−9
o iple s) and non-addi i e dispe sion in
laye ed/low-dimensional ma e ials [
500
,
501
]. A small ixed se o local p obes canno econs uc such
global e ec s in gene al.
Diagnos ics. (i)
Geome y mo i s
known o ampli y nonaddi i i y: pa allel slabs, hollow ca i ies,
ex ended
π
sys ems; (ii)
Obse able sensi i i y
o sys em size o en i onmen a ixed local geome-
y (e.g. adso p ion ene gies s slab hickness); (iii)
De ia ion pa e ns
cha ac e is ic o MBD (e.g.
sys ema ic unde binding wi h pai wise-only models agains high-le el o expe imen al e e ences).
Remedies. (a)
Add an MBD ile
: include a ile wi h explici many-body dipole–dipole couplings (e.g.
coupled luc ua ing dipoles) calib a ed o he SL long-wa eleng h esponse so as no o double coun — his
is analogous in spi i o dW-DF/MBD schemes [
495
]; (b)
En ich q-space p obes
: e alua e a ew
addi ional esol en / esponse nodes a ini e
q
o cons ain in e media e- ange sc eening; (c)
Repo
100
la ge e o ba s
: un il (a)/(b) a e ins alled, use ∆
es
SL
om Sec. IXF and no e he nonlocal mechanism
in he limi a ions.
10.5.4 Nume ical and s a is ical pa hologies (p ac ical ailu e cases)
Ill-posed in e sion. Choosing
z1,2
oo close o he spec um o oo close o each o he makes he
wo-momen map nea ly singula . Remedy: adap i e shi selec ion and condi ioning checks; egula ize
wi h an ex a p obe.
Analy ic con inua ion. I one insis s on eal- equency spec a om imagina y-axis/Ma suba a da a,
he con inua ion is ill-posed and noise-ampli ying [
499
]. Remedy: use he causal He glo z ke nel on he
eal axis di ec ly (Sec. IXE) o Bayesian con inua ion wi h unce ain y.
k
-mesh and ini e-size e ec s. In solids, insu icien
k
-sampling can mimic cu a u e spikes (aliasing o
an Ho e ea u es). Remedy: scale wi h
k
-mesh and compa e o long-wa eleng h p edic ions ixed by SL
(e.g. A2).
Open-sys em non-Ma ko iani y. S ongly s uc u ed ba hs in alida e GKSL Ma ko ian assump ions
(Sec. XD). Remedy: p omo e he dissipa i e ile o include memo y ke nels; i una ailable, ad e ise
unce ain y and es ic claims.
10.5.5 A decision ee o p ac ice
1. Check co e age:
i
κ <
1o p ojec o anspo is uns able, enla ge
d
o add a ile; o he wise
p oceed.
2. Check cu a u e:
i
max C
(
s
)
≤C a ge
(Sec. XB), us SL a he desi ed ole ance; else add
p obes and densi y he a las locally.
3. Check nonlocali y:
i he mo i and diagnos ics sugges MBD/collec i e sc eening, add an
MBD/sc eening ile o en ich q-space p obes; o he wise epo ∆es
SL and he limi a ion.
Summa y. The SL gua an ees—exac asymp o es, causali y, pola iza ion quan iza ion, and calib a ed
e o ba s—su i e all egimes. Quan i a i e accu acy can, howe e , deg ade h ough (i) missing iles
(ac i e mani old g ows, wo momen s no longe su ice), (ii) la ge cu a u e (ill-condi ioning nea small
gaps and c ossings), and (iii) nonlocal co ela ion (collec i e/MBD physics) beyond he ep esen a ional
powe o a iny p obe se . Each ailu e mode has clea diagnos ics and p incipled emedies (mo e p obes,
la ge on ie blocks, o physics-speci ic iles), wi hou eso ing o empi ical e uning (Sec. XA).
XI. CONCLUSION
Summa y: sol e, no i ; ab-ini io XC a DFT cos ; ca ego ical in a ian s as backbone
F om pa ame e s o in a ian s. Ac oss he pape we ad oca ed a cons ain – i s ou e o exchange–
co ela ion (XC): ins ead o in oducing con inuous mixing/sc eening pa ame e s and uning hem o
ma ch benchma ks, we sol e o XC con ibu ions om a small se o exac in a ian s. Conc e ely, he
spec al–law (SL) laye exposes (i) a on ie p ojec o
P
in
K0
ha acks he ac i e subspace, (ii) wo
esol en p obes
S(zj) = 1
d
(zj−L )−1, j = 1,2, z16=z2,(224)
which de e mine he ace and de e minan o he local 2
×
2 on ie gene a o
L
, and (iii) Wa d–consis en
s a ic/dynamic cons ain s ha gua an ee causali y, he op ical
–sum, and co ec long–wa eleng h
limi s. This minimal se al eady ixes he dominan physics: piecewise linea i y in pa icle numbe ( ia he
HH0
hull), 1
/R
cha ge– ans e (CT) asymp o ics, opological pola iza ion pla eaus, and long–wa eleng h
me allic sc eening. The e a e no con inuous i pa ame e s: all inpu s a e in a ian s
{φk}
ob ained om
analy ic seeds o om once–compu ed, eusable iles (e.g. a many–body dispe sion o s ongly–co ela ed
ile), hen anspo ed by a gauge–co a ian connec ion along he pa h (geome y, ields, (q, ω)).
101
Quan um con en is explici . Because he p imi i es a e ope a o –algeb aic— he on ie block
L
, i s
esol en (
z−L
)
−1
, and idempo en /p ojec o classes in
K0
— he SL laye ca ies quan um s uc u e
(phase, en anglemen , opology), no jus densi ies. The
HH0
(hull) label en o ces he con ex en elope
needed o in ege –cha ge p e e ence. Fo insula o s, pa allel anspo o
P
de e mines Be y/Zak phases
and he pola iza ion jump; o me als, he s a ic ke nel con ains he nonlocal
A2q2
g adien e m equi ed
by conse ing esponse. In TDDFT he esidual dynamical ke nel is pa ame e ized by a posi i e–measu e
(He glo z) o m, so causali y and he luc ua ion–dissipa ion heo em a e buil in.
Ab
-
ini io accu acy a Kohn–Sham cos . Algo i hmically, he SL laye eplaces global diagonaliza ions
and empi ical hyb ids wi h closed– o m algeb a on low– ank sec o s and a hand ul o esol en e alua ions.
As shown in Sec. IXF, he inne loops a e domina ed by local 2
×
2sol es o
k
–g id in eg a ions and
exhibi linea scaling in in e acial p oblems. The ou come is s iking in he benchma ks o Sec. 9:
exac dissocia ion in he oy molecules when p obes a e comple e, co ec pola iza ion pla eaus and CT
asymp o es, exci ons and ini e–
q
plasmon ends in TDDFT—all a KS–like cos and wi hou e uning.
Wha o emphasize (“be e han be o e”)
(i) No con inuous i pa ame e s. All SL inpu s a e in a ian s
{φk}
: p ojec o weigh s, wo esol en
momen s, and opology/cu a u e. Tiles (e.g. MBD o DMFT modules) a e added when he physics
demands new in a ian s, no o i numbe s. This elimina es p ope y–dependen mixing (ubiqui ous in
hyb ids) and imp o es ans e (Sec. X A).
(ii) Explici quan um mechanics. The ope a o algeb a (
L,
Φ) wi h Φ(
z
) = (
z−L
)
−1
, he
HH0
hull,
and
K0
classes encode he non i ial con en o he s a e (phase, en anglemen , opology). Obse able
cons ain s—PPLB linea i y, CT 1
/R
, Be y pola iza ion—a e sa is ied by cons uc ion, no by pos hoc
co ec ion.
(iii) Ab
-
ini io accu acy a KS cos . The SL laye has he same asymp o ics as KS and adds only a ew
esol en sol es (pe geome y, pe (
q, ω
)). Pe o mance measu emen s (Sec. IXF) show ha in e acial
p oblems a e
O
(10
3
) as e han b u e– o ce diagonaliza ion a
N∼
10
2
, while main aining he gua an ees
used by GW/BSE in a ligh weigh , conse ing o m (Secs. IXE, X B).
Ou look: ex end he a las; public code; communi y benchma ks
Ex end he a las. Th ee conc e e di ec ions ollow om Secs. XB–XF: (i) Highe ank on ie blocks
(
d >
2) nea quasi–degene acies, wi h h ee o mo e esol en nodes o iden i iabili y; (ii) Specialized iles
(MBD o many–body dispe sion, DMFT o local s ong co ela ion, memo y ke nels o non–Ma ko ian
dissipa ion); (iii) Fini e–
T
and open sys ems along he Ma suba a/Keldysh con ou s wi h de ailed–balance
cons ain s (Sec. X D).
Public code and ep oducibili y. We will elease a e e ence implemen a ion ha exposes he SL laye
as a d op–in module: a p ojec o – anspo u ili y, esol en p obes wi h adap i e shi selec ion, and
causal ke nel pa ame e iza ions. A i ac s ( igu es, ables, and logs) will be a chi ed wi h pe sis en
iden i ie s and FAIR me ada a o maximize euse and audi abili y [502].
Communi y benchma ks. To make he case beyond oy models, we p opose a anspa en benchma k
p o ocol: (i) molecula he mochemis y/kine ics/nonco alen se s (e.g. GMTKN55) o mean absolu e
e o s and cu a u e diagnos ics; (ii) ma e ials benchma ks (e.g. Ma e ials P ojec ) o band gaps and
o ma ion ene gies; (iii) ca alysis/adso p ion asks (e.g. OC20/OC22) o CT ba ie s and ini e–
q
sc eening
signa u es. These allow he communi y o e i y ha he SL laye p ese es exac cons ain s while
imp o ing ans e a no ex a empi ical cos .
A compac ecipe (ca y–home)
1. Expose in a ian s. Compu e (P , S(z1), S(z2)) and opology/cu a u e along he pa h.
2. Close he on ie algeb a.
Reco e
L
and
de L
; use he
HH0
hull o piecewise linea i y.
3. En o ce conse ing dynamics.
Use a He glo z pa ame e iza ion o he esidual ke nel so
causali y and sum ules hold iden ically.

102
4. Quan i y unce ain y.
Repo ∆
es
SL
=
α?C
(
s
); when
C
spikes o co e age
κ
d ops, add p obes o
iles.
Closing s a emen . The cen al message is simple: sol e o XC om in a ian s, do no i . Wi h
ca ego ical in a ian s as he backbone and a hand ul o esol en sol es, we ob ain ab–ini io accu acy a
KS cos , a clea upg ade pa h when physics demands i (ex end he a las), and p incipled unce ain y
when i does no . The pa h o wa d is collabo a i e: an open implemen a ion and sha ed benchma ks
will le he communi y es , b eak, and ul ima ely us his cons ain – i s al e na i e o semi–empi ical
mixing.
Appendix A: Func ional-calculus de i a ions (Eq. 1, esol en iden i ies)
1. Res a ing he mas e iden i y (Eq. 1 o he main ex )
Fo a bounded sel
-
adjoin ope a o (o a ini e
-
dimensional He mi ian ma ix)
L
wi h spec um
σ
(
L
)
⊂R
and a unc ion
ha is holomo phic on an open se con aining
σ
(
L
), he holomo phic
unc ional calculus de ines
(L) = 1
2πi IΓ
(z)(z1−L)−1dz (A1)
whe e Γis any posi i ely o ien ed con ou enclosing
σ
(
L
)and a oiding singula i ies o
. The map
z7→
(
z1−L
)
−1
is he esol en
R
(
z
); i is analy ic on he esol en se
ρ
(
L
) =
C σ
(
L
). Equa ion
(A1)
is
he mas e iden i y used h oughou he pape (quo ed as Eq. 1 in he main ex ). Specializing
(
z
)and
aking esidues p oduces spec al p ojec o s, polynomial and a ional unc ions o
L
, and—a e aking
aces— he S iel jes ans o m o he spec al measu e.
Spec al p ojec o s as esidues. I
L
has simple eigen alues
{λa}
wi h o hogonal p ojec o s
{Pa}
( ini e dimension o conc e eness), hen
R(z)=(z1−L)−1=X
a
Pa
z−λa
, Pa=1
2πi IΓa
R(z)dz, (A2)
wi h Γ
a
a small ci cle a ound
λa
. Equa ion
(A2)
is he algeb aic backbone o ex ac ing on ie p ojec o s
ia esol en p obes in he main ex .
2. Co e esol en iden i ies
Le
R
(
z
)=(
z1−L
)
−1
and
R
(
w
)=(
w1−L
)
−1
wi h
z, w ∈ρ
(
L
). The iden i ies below a e s anda d
bu we include de i a ions o comple eness.
Fi s esol en iden i y.
R(z)−R(w)=(w−z)R(z)R(w).(A3)
De i a ion: (
z1−L
)
R
(
z
) =
1
= (
w1−L
)
R
(
w
). Sub ac ing, (
z−w
)
R
(
z
) = (
z1−L
)
R
(
z
)
−R
(
w
)

;
le -mul iplying by R(w)yields (A3).
De i a i e iden i ies.
d
dz R(z) = R(z)2,dk
dzkR(z) = k!R(z)k+1 (k≥1).(A4)
De i a ion: Di e en ia e (z1−L)R(z) = 1wi h espec o zand i e a e.
Va ia ion wi h espec o L( esol en sandwich). Fo a pe u ba ion δL,
δR(z) = R(z)δL R(z).(A5)
De i a ion: Di e en ia e (z1−L−δL)R(z) = 1a = 0 and sol e o δR.
Neumann expansion (la ge-|z|asymp o ics). I kLk<|z|,
R(z) = 1
z
∞
X
n=0 L
zn,hence R(z) = d
z+ L
z2+ L2
z3+··· .(A6)
This connec s esol en momen s wi h powe sums o eigen alues.
103
3. S iel jes ans o m, He glo z p ope y, and K ame s–K onig
De ine he no malized esol en ace ( he “scala p obe” used in he main ex )
Sd(z) = 1
d (z1−L)−1=ZR
dµ(λ)
z−λ, dµ =1
d
d
X
a=1
δ(λ−λa)dλ, (A7)
he S iel jes ans o m o he empi ical spec al measu e µ. Fo z=x+iη wi h η > 0,
=Sd(x+iη) = −πµ∗Pη(x),Pη(x) = 1
π
η
x2+η2.(A8)
Thus
−=Sd
is a posi i e ha monic app oxima ion o he densi y o s a es (DoS); as
η↓
0,
−=Sd
(
x
+
i
0
+
) =
π ρ
(
x
)whe e e
ρ
is con inuous. Analy ici y o
Sd
in he uppe hal
-
plane and posi i i y o
−=Sd
mean
ha
Sd
is a He glo z (Ne anlinna) unc ion, and i s eal and imagina y pa s on he eal axis obey
Hilbe - ans o m (K ame s–K onig) ela ions.
4. Two-p obe in e sion o a 2×2 on ie block
Le
L∈C2×2
wi h eigen alues
λ±
, ace
τ
=
L
and de e minan ∆ =
de L
. The no malized scala
p obe is
S(z) = 1
2 (z1−L)−1=1
21
z−λ+
+1
z−λ−=2z−τ
2(z2−τz + ∆).(A9)
Gi en wo dis inc shi s
z16
=
z2
in he esol en se and measu ed alues
S1
=
S
(
z1
),
S2
=
S
(
z2
), one
can sol e linea ly o (τ, ∆).
Mul iply Eq. (A9) by he denomina o and isola e he unknowns:
2Sj(z2
j−τzj+ ∆) = 2zj−τ⇐⇒ (−2Sjzj+ 1)τ+ (2Sj) ∆ = 2zj−2Sjz2
j,
o j= 1,2. W i ing Aj:= 2Sjand bj:= 2zj−Ajz2
j, he 2×2linea sys em
1−A1z1A1
1−A2z2A2τ
∆=b1
b2(A10)
has de e minan
D= (A2−A1) + A1A2(z2−z1)(A11)
and solu ion
τ=b1A2−A1b2
D,(A12)
∆ = (1 −A1z1)b2−(1 −A2z2)b1
D.(A13)
Condi ioning. The in e sion is ill
-
condi ioned when
D
is small, which occu s i
z1≈z2
and
A1≈A2
, o
i ei he
zj
lies close o
σ
(
L
)so ha
Aj
is la ge. In p ac ice, one chooses
z1, z2
symme ic abou he
on ie cen e and well sepa a ed om σ(L) o s abilize D.
Reco e ing eigen alues. Once (
τ,
∆) a e known, he eigen alues ollow om he quad a ic
λ±
=
1
2τ±√τ2−4∆. All on ie in a ian s (gap, mid-gap, e c.) a e hen ixed.
5. Block esol en and Schu complemen (Feshbach map)
Pa i ion he Hilbe space in o he on ie subspace
H
and he emo e subspace
H m
using p ojec o s
Pand Q=1−P. In block o m,
L=L V
V†L m, z1−L=A−V
−V†Dwi h A=z1−L , D =z1−L m.
104
When
D
is in e ible (i.e.
z∈ρ
(
L m
)), he Schu complemen o mula yields he on ie block o he ull
esol en :
R (z) = P(z1−L)−1P=A−V D−1V†−1=z1−L −V(z1−L m)−1V†
| {z }
Σ(z)−1.(A14)
He e Σ(
z
)is he sel
-
ene gy. I he emo e subspace is gapped away om he on ie , Σ(
z
)is smoo h
on he chosen p obe poin s and may be ea ed as a small, slowly a ying co ec ion. Equa ion
(A14)
jus i ies he wo
-
p obe s a egy in he main ex : by e alua ing scala aces o
R
(
z
)a wo judiciously
chosen
z
’s one iden i ies he e ec i e 2
×
2 on ie gene a o . In he ideal decoupled case (
V
=0) one
eco e s R (z)=(z1−L )−1exac ly.
6. P ojec o s, esidues, and Be y da a om esol en s
Le
P
be a ank
-d
p ojec o anspo ed along a pa ame e
s
(geome y, c ys al momen um
k
,e c.).
The bundle connec ion may be exp essed in e ms o esol en s and de i a i es:
P0=dP
ds =1
2πi IΓ
R(z)L0R(z)dz, L0=dL
ds ,(A15)
ob ained by di e en ia ing he con ou exp ession o
P
and using he sandwich a ia ion iden i y
(A5)
.
Fo c ys alline p oblems wi h pe iodic
k
, his gi es a gauge
-
co a ian exp ession o he Be y connec ion
A
(
k
) =
i P U†∂kU
upon choosing a uni a y ame
U
o
Ran P
; equi alen ly, Be y phases can be
ex ac ed om he a gumen o
de W
whe e
W
is a Wilson–loop o o e laps
huk|uk+∆ki
, which in u n
a e esidues o block esol en s.
7. Ma suba a and eal-axis e sions
Fo ini e-T(g and canonical) se ings wi h chemical po en ial µ, de ine he Ma suba a esol en
S(iωn) = 1
d
iωn+µ−L −1, ωn= (2n+ 1)π/β , (A16)
which is analy ic on he imagina y axis. Any wo dis inc e mionic nodes
iωn1, iωn2
su ice o he 2
×
2
in e sion o Sec. A4 (wi h
z7→ iωn
+
µ
). Fo eal
-
equency esponse, one uses he bounda y alues
R
(
ω±i
0
+
), wi h imagina y pa s ela ed o spec al densi ies by Eq.
(A8)
and K ame s–K onig ela ions.
8. Rank-one and ini e- ank upda es (She man–Mo ison–Woodbu y)
I
L
is modi ied by a ini e
-
ank pe u ba ion,
L7→ L
+
UCV †
wi h
U, V
o size
d×
and
C∈C ×
,
hen
(z1−L−UCV †)−1=R(z) + R(z)UC−1−V†R(z)U−1
V†R(z),(A17)
whe e
R
(
z
) = (
z1−L
)
−1
. Equa ion
(A17)
unde lies e icien upda es o esol en p obes when he
on ie block is modi ied by low- ank cons ain s o iles, as in he body o he pape .
9. E o p opaga ion and condi ioning o he wo-p obe map
Linea izing Eqs. (A10)–(A13) a ound noiseless da a (S1, S2)gi es
δτ
δ∆=M−1δb1
δb2−M−1δm11 δm12
δm21 δm22τ
∆, M =1−A1z1A1
1−A2z2A2,
whe e (
δbj, δmij
)a e linea in (
δS1, δS2
). Hence
k
(
δτ, δ
∆)
k.κ
(
M
)
k
(
δS1, δS2
)
k
wi h
κ
(
M
) =
kMkkM−1k
. The p ac ical guideline is o choose (
z1, z2
)so ha
|D|
=
|de M|
is bounded away
om ze o (Sec. A4), ypically by placing
z1, z2
symme ically a ound he on ie egion and a a dis ance
se by i s cha ac e is ic scale.
105
10. F om scala p obes o ull ma ix iden i ica ion (op ional gene aliza ion)
Fo a highe
-
ank on ie block (
d >
2), he scala S iel jes ans o m
(A7)
can be e alua ed a
m≥d
dis inc nodes
{zj}
o econs uc he
d
eigen alues (P ony/Padé ype in e sion), a e which in a ian
subspace da a (e.g. weigh ed p ojec o s) can be ob ained by esidues as in Eq.
(A2)
. Al e na i ely,
one can use a small se o ma ix p obes
Wα
(
z1−L
)
−1
wi h ailo ed weigh s
Wα
o ix no only
he spec um bu selec ed ma ix elemen s in a chosen basis; he algeb a closes once he numbe o
independen p obe equa ions ma ches he numbe o unknown in a ian s.
11. Summa y
Equa ion
(A1)
( he unc ional
-
calculus mas e iden i y) and he esol en iden i ies
(A3)
–
(A5)
jus i y
all manipula ions o on ie p ojec o s, wo
-
node esol en p obes, and causal ke nels used in he main
ex . In pa icula , o a 2
×
2 on ie block, wo scala p obes a well
-
chosen shi s eco e (
τ,
∆) ia
Eqs.
(A10)
–
(A13)
, hence he en i e block; block iden i ies
(A14)
and
(A17)
con ol coupling o emo e
subspaces and low
-
ank upda es; and he S iel jes/He glo z s uc u e
(A7)
–
(A8)
p o ides posi i i y and
K ame s–K onig ela ions used o dynamical ke nels. All labels in oduced he e a e e e enced in he
body: he scala p obe
S
(
z
)(Sec. V), on ie ace/de e minan (
τ,
∆) (Secs. 9–10), and block esol en
s uc u e (Secs. 9.4–9.5).
Appendix B: P oo s o in a iance and cu a u e bounds
This appendix collec s o mal s a emen s and p oo s o he in a iance claims and he cu a u e bounds
ha a e used h oughou he main ex o co ec ness gua an ees and o he calib a ion o e o ba s
(Sec. IXF, Sec. X B). We wo k in ini e dimension o cla i y; all s a emen s ex end o bounded sel –adjoin
ope a o s unde he same spec al sepa a ion hypo heses. No a ion ollows Appendix A: he esol en is
R(z;L)=(zI−L)−1, he no malized scala p obe o a d –dimensional on ie block is
S(z) = 1
d
(zI−L )−1,
and o d = 2 we deno e T= L and ∆ = de L .
1. Gauge/uni a y in a iance o p obes and on ie in a ian s
Lemma B.1
(Uni a y in a iance o scala p obes)
.
Le
U
be uni a y on he on ie subspace. Then o
any z∈ρ(L ),
S(z) = 1
d
zI−L −1=1
d
zI−U†L U−1.
P oo . is uni a y in a ian and (zI−U†L U)−1=U†(zI−L )−1U, hence he aces coincide.
Lemma B.2
(In a iance o (
T,
∆))
.
Fo
d
= 2,
T
=
L
and ∆ =
de L
a e in a ian unde uni a y
changes o basis in he on ie subspace. Consequen ly he a ional o m
S(z) = z−T
z2−Tz + ∆
(Appendix A 4) is a gauge–in a ian obse able.
2. Homo opy in a iance o he on ie p ojec o and pola iza ion
Le
L
(
s
)be a
C1
pa h o sel –adjoin ma ices wi h a spec al gap
γ
(
s
)
>
0sepa a ing a clus e Σ
(
s
)
( he on ie clus e ) om he es o he spec um o all
s∈I⊂R
. Deno e by
P
(
s
) he Riesz p ojec o
(Appendix A) on o Σ (s).
112
1. Riesz p ojec o s by con ou quad a u e
Ci cula con ou and apezoidal ule. Le Γbe he ci cle
z
(
θ
) =
c
+
eiθ
,
θ∈
[0
,
2
π
), enclosing only
he a ge spec al island ( he on ie clus e ). Then dz =i eiθ dθ and Eq. (D1) becomes
P=1
2πZ2π
0
R
c+ eiθ eiθ dθ ≈1
M
M−1
X
j=0
R(zj)wj, zj=c+ eiθj, θj=2πj
M, wj= eiθj.(D2)
Because
R
(
z
)is analy ic in a neighbo hood o Γ, he pe iodic apezoidal ule con e ges spec ally: he
e o decays like
O
(
ρ−M
), whe e
ρ >
1is he con o mal modulus o he la ges annulus a ound he uni
ci cle ee o singula i ies (poles a he eigen alues and hei images unde he a ine map z=c+ ζ).
Ellip ical con ou . When he on ie island is elonga ed, an ellipse educes he p oximi y o ex e io
poles. Pa ame ize z(θ) = c+acosθ+i b sin θ, so ha
P=1
2πiZ2π
0
R
z(θ)z0(θ)dθ =1
2πZ2π
0
R
z(θ)bcos θ+i a sin θdθ, (D3)
and disc e ize wi h θj= 2πj/M,
P≈1
M
M−1
X
j=0
R
zjwj, wj=bcosθj+i a sin θj.(D4)
Choice o (
a, b
) ollows he same p inciple as (
c,
): maximize he minimal dis ance om Γ o he nea es
ex e io eigen alue while keeping all a ge eigen alues inside.
Ac ion s. explici p ojec o . In la ge p oblems, one a ely assembles
P
; ins ead one applies
P
o a
block o ec o s X:
Y=PX ≈1
M
M−1
X
j=0
wjYj,(zjI−L)Yj=X. (D5)
O hono malizing
Y
yields a basis o he on ie subspace; he educed (o comp essed) ope a o on his
subspace is
L =Q†LQ, (D6)
whose eigen alues and in a ian s (
τ,
∆) a e hen ob ained om he small dense ma ix
L
. The comp essed
block
L
=
Y†LY
and i s in a ian s (
T,
∆) a e hen compu ed in machine p ecision (c . Appendix A 4).
Choosing (
c,
)and
M
.Le
λin
min
and
λin
max
b acke he on ie clus e , and le
λou
deno e he closes
ex e io eigen alue. Pick
c
=
1
2
(
λin
min
+
λin
max
)and
so ha
max{|λin
min −c|,|λin
max −c|} < < |λou −c|
.
Se
M
by he annulus modulus heu is ic abo e; in p ac ice
M∼
8–16 su ices o clean islands, and
M∼24–32 o c owded spec a.
2. E icien esol en sol es a a ew complex shi s
Spa se ac o iza ions and shi euse. Fo each node
zj
, o m
Aj
=
zjI−L
and compu e a spa se
ac o iza ion (e.g.
Aj
=
LU
o , i
L
is He mi ian,
Aj
=
LDL>
wi h symme ic pe mu a ions). Ac oss
mul iple shi s, euse he symbolic analysis (pe mu a ions, elimina ion ee, supe node pa i ion); only
he nume ic ac o s change wi h
zj
. Wi h wo (o a hand ul o )
zj
, he amo ized cos is domina ed by
he nume ic ac o iza ions and iangula sol es.
Block igh –hand sides. Equa ions
(D5)
equi e solu ions wi h he same coe icien ma ix and many
igh –hand sides
X
(block subspace i e a ion). Exploi BLAS–3 iangula sol es o apply
U−1
and
L−1
o dense blocks e icien ly.
Schu complemen s when a on ie suppo is known. I he on ie is la gely suppo ed on a subse
o DoFs I, compu e he block esol en ia he Schu complemen (Appendix A14):
R (z) = zI−L −Σ(z)−1,Σ(z) = V(zI−L m)−1V†,(D7)
wi h one selec ed in e sion on he emo e block o e alua e Σ(
z
) o all
z
in use. This is e ec i e when
I
is small and well sepa a ed.

113
3. T ace ex ac ion: scala p obes by selec ed in e sion o s ochas ic es ima o s
Goal. Fo he wo–p obe iden i ica ion (Appendix A 4), we need
S(z) = 1
d
R (z)o S(z) = 1
n R(z)(when using ull aces as su oga es).
Selec ed in e sion (de e minis ic). Gi en a spa se ac o iza ion o
A
=
zI−L
, he selec ed in e sion
compu es chosen en ies o
A−1
— ypically he diagonal and a ew o –diagonals—wi hou o ming he
ull in e se. B ie ly:
1. Compu e a ill– educing pe mu a ion Pand ac o P>AP =LDL>(o LU).
2.
T a e se he elimina ion ee in e e se ( om oo o lea es). Fo each supe node
k
wi h index se
Jkand i s upda e se Ik:
o m he local in e se on Jk: (A−1)JkJk= (D−1
k)−W>
k(A−1)IkIkWk,
upda e selec ed c oss–blocks (i eques ed): (A−1)IkJk=−(A−1)IkIkWk,
whe e Wk=LIkJkD−1
k(o he co esponding U–based exp ession).
3.
Accumula e
diag
(
A−1
)as supe nodes a e p ocessed; undo he pe mu a ion o e u n en ies in he
o iginal o de ing.
Thus,
R
(
z
) =
Pi
(
A−1
)
ii
is a ailable exac ly (up o ac o iza ion ound–o ) a he cos o one selec ed
in e sion pe shi . The symbolic s uc u e (pe mu a ion, supe nodes, ee) is eused ac oss shi s.
Hu chinson– ype es ima o s (s ochas ic). When selec ed in e sion is una ailable, app oxima e
R
(
z
)
by
R(z) = 1
N
N
X
`=1
∗
`R(z) `, `∈ {±1}n(Rademache ).(D8)
Each e m equi es sol ing
Ay`
=
`
. The es ima o is unbiased wi h a iance
O
(
kR
(
z
)
k2
F/N
);
N ∼
16–
64 o en su ices o he wo–p obe i . Use he same andom seeds ac oss nea by
z
’s o educe di e en ial
noise (con ol a ia es).
F on ie – es ic ed ace. I a p ojec o
Pwin
on o a window o DoFs is a ailable, compu e
S
(
z
) =
1
d PwinR
(
z
)
Pwin
by applying selec ed in e sion on he windowed sys em o by s ochas ic p obing wi h
`=Pwinξ`.
4. Two–p obe on ie iden i ica ion: nume ics and condi ioning
F om aces o in a ian s. Gi en wo shi s
z16
=
z2
and he measu ed
Sj
=
S
(
zj
), sol e he 2
×
2linea
sys em (Eq.
(A10)
) o (
τ,
∆) = (
L ,de L
). Reco e he on ie eigen alues by
λ±
=
1
2
(
τ±√τ2−4∆
).
Condi ioning and shi selec ion. The de e minan
D
o he wo–p obe sys em (Appendix A9) con ols
sensi i i y: small |D|leads o la ge ampli ica ion o p obe e o s. P ac ical sa egua ds:
•
Choose
z1,2
symme ic abou he on ie cen e
¯
λ
and a a dis ance
δ
compa able o he hal –wid h
o he island. As a ule o humb δ∈[1,3] ×(λin
max −λin
min)/2is sa e.
•
I
|D|
alls below a h eshold (e.g. 10
−3
in double p ecision a e scaling), pick a hi d shi and
o e de e mine he sys em; sol e o (τ, ∆) by leas squa es.
•
A oid shi s oo close o he spec um (la ge
kR
(
zj
)
k
): hey inc ease ac o iza ion pi o ing and
s ochas ic– ace a iance.
Pseudocode ( on ie in a ian s).
1.
Es ima e a spec al window o he on ie clus e (e.g. by a sho Lanczos un o by p io
knowledge). Se c, and wo shi s z1,2ou side he window.
2.
Fo each
j
= 1
,
2: build
Aj
=
zjI−L
, ac o
Aj
, compu e ei he
A−1
j
by selec ed in e sion o
A−1
jby Hu chinson. Se Sj=d−1
A−1
jo i s app oxima ion.
3. Sol e Eq. (A10) o (τ, ∆); i ill–condi ioned, add a hi d shi and sol e by leas squa es.
4. Re u n λ±and (op ionally) he on ie p ojec o om he con ou il e (D2).
114
5. Selec ed in e sion: p ac ical de ails and euse ac oss shi s
Symbolic s. nume ic phases. Fo mul iple shi s
zj
, pe o m he expensi e symbolic analysis (o de ing,
supe node de ec ion, elimina ion ee) once on he spa si y o
L
; hen, o each
zj
, o m
Aj
by shi ing he
diagonal and un he cheape nume ic ac o iza ion. Memo y oo p in s and ill–in pa e ns a e iden ical
ac oss shi s.
Complex shi s and s abili y. Wi h complex
z
,
Aj
is complex–symme ic (i
L
is eal–symme ic) o
gene al complex. Use a mul i on al
LDL>
(complex–symme ic) o
LU
(gene al) wi h s a ic pi o ing
when possible. I pi o ing dis up s euse, eeze pe mu a ions a e a wa m–up ac o iza ion and accep
a e small pi o s ( he shi s keep Ajwell condi ioned away om he eal spec um).
Wha o selec . Fo he scala p obe, he diagonal o
A−1
su ices. Fo local densi ies o windowed
aces, selec he block (
A−1
)
II
on a small index se
I
. The selec ed–in e sion ecu sion compu es hese
en ies wi h essen ially he cos o one spa se ac o iza ion.
Cos model. Le
n
be DoFs and
nnz
he numbe o nonze os in
L
. One symbolic analysis and wo
nume ic ac o iza ions ( wo shi s) domina e he cos ; each selec ed in e sion pass is wi hin a small
cons an o a ac o iza ion. In banded 1D p oblems, cos is linea in
n
; in 2D,
n3/2
–like, and in 3D,
n2
–like,
go e ned by ill–in unde nes ed dissec ion. These scalings ma ch he wall– ime ends we epo ed in
Sec. IXF.
6. Con ou quad a u e e o con ol and s opping c i e ia
Quad a u e e o . Gi en a con ou Γand a se o ex e io poles a dis ances
{dα}
om Γ, he
apezoidal e o decays as
Cmaxαexp
(
−Mlog ρα
)wi h
ρα
= 1 +
dα
(ci cle; analogous o ellipses).
Inc ease
M
un il he change
kPM−PM/2k
o he change in on ie in a ian s (
τ,
∆) alls below ole ance.
Linea –sol e e o . Fo each
zj
, use i e a i e e inemen on he iangula sol es un il he esidual
no m
kAjYj−Xk/kXk
in Eq.
(D5)
is below a ge (e.g. 10
−12
). In selec ed in e sion, moni o
|Dk|
pi o
magni udes and apply local equilib a ion i needed.
S ochas ic– ace e o . Wi h
N
Rademache p obes, es ima e he s anda d de ia ion by a jackkni e
o e he samples and s op when he induced unce ain y on (
τ,
∆) (p opaga ed h ough Eq.
(A10)
) is
below h eshold.
7. Pu ing i oge he : p ojec o , in a ian s, and e o ba s
Recommended wo k low.
1. Window de ec ion.
Loca e he on ie clus e and a sa e gap o he es (sho Lanczos o p io
s ep).
2. Con ou and shi s.
Choose (
c,
)and
M
(o an ellipse) and wo symme ic shi s
z1,2
ou side
he window.
3. P ojec o /subspace (op ional). Build P–ac ions wi h Eq. (D5) o a small block X; o hono -
malize o ob ain a on ie basis Yand comp essed L .
4. Scala p obes. Fo each zj, compu e (zjI−L)−1by selec ed in e sion (o s ochas ic).
5. Two–p obe iden i ica ion. Sol e Eq. (A10) o (τ, ∆); eco e λ±.
6. Condi ioning and e inemen .
I
|D|
small o esiduals la ge, adjus shi s o add a hi d p obe;
i con ou esul s change wi h M, inc ease M.
7. Unce ain y.
Repo cu a u e–based e o ba s (Appendix B 8) oge he wi h p obe–noise and
quad a u e e o es ima es.
Consis ency wi h
K0
labels and anspo . All s eps p ese e he
K0
class (Appendix C): he Riesz
in eg al
(D2)
e u ns he same p ojec o up o nume ical e o ; gauge/uni a y changes wi hin he on ie
ha e no e ec on (
τ,
∆) (Lemma B.2); and he mapping– o us holonomy is compu ed om o e laps o
he anspo ed ames ob ained by (D5) (Eq. (C9)).
115
8. No es on ini e-T and open-sys em a ian s
Ma suba a p obes. Fo ini e empe a u e, use he wo Ma suba a nodes
zj
=
iωnj
+
µ
in place o
eal–axis shi s (Appendix A 7). The same ac o iza ion and selec ed–in e sion pipeline applies; he
ma ices a e well condi ioned away om he eal spec um.
Liou illian esol en . In open–sys em se ings, wo k wi h he Liou illian
L
and compu e he scala
p obes
SL
(
z
) =
d−1
LP
(
z− L
)
−1
(Eq.
(218)
). Disc e ize
L
in a basis o obse ables (e.g. local
densi ies) and apply he same con ou /selec ed–in e sion machine y. All condi ioning and euse ema ks
ca y o e .
9. Summa y
•Riesz p ojec o s by con ou .
Use ci cula o ellip ical con ou s wi h spec ally con e gen
apezoids; apply P o blocks ia wo o a ew shi ed sol es (Eq. (D5)).
•Scala p obes.
Ex ac
R
(
z
)de e minis ically by selec ed in e sion (diagonals o he in e se)
o s ochas ically by Hu chinson es ima o s (Eq. (D8)).
•Two–p obe in a ian s.
Wi h wo well–chosen shi s, sol e he linea sys em (Eq.
(A10)
) o
(τ, ∆); moni o he condi ioning de e minan Dand adap shi s i needed.
•Reuse ac oss shi s.
Reuse symbolic ac o iza ions, elimina ion ees, and pe mu a ions; cos s
scale wi h nume ic ac o iza ions and a small cons an o selec ed in e sion.
•E o con ol.
Combine analy ic quad a u e decay, linea –sol e esiduals, and (when used)
s ochas ic– ace unce ain y; p opaga e o (τ, ∆) and o λ±.
All componen s a e consis en wi h he in a iance logic o Appendices B–C: hey p ese e p ojec o
classes, p oduce gauge–in a ian on ie in a ian s, and p o ide nume ically s able inpu s o he a las
anspo and e o –ba calib a ion used in Sec. 9.
Appendix E: Pseudocode & da a s uc u es: in a ian cache, mul iplie sol e , p ojec o acke
This appendix speci ies he un ime s uc u es and end– o–end pseudocode ha ealize he pipeline
de eloped in he main ex and in Appendices A–D. We expose h ee coope a ing subsys ems: (i) an
in a ian cache ha s o es and in alida es all eusable spec al in a ian s ( on ie p ojec o s, esol en
aces, cu a u e, e c.); (ii) a mul iplie sol e ha en o ces small se s o linea /con ex cons ain s (Wa d
iden i ies, long–wa eleng h limi s, He glo z weigh s) wi h no con inuous empi ical pa ame e s; and (iii) a
p ojec o acke ha anspo s
K0
p ojec o labels along pa hs (geome y,
k
–space, in e aces) wi h
s abili y con ol and Be y/Wilson bookkeeping.
Th oughou we use no a ion om ea lie appendices:
R
(
z
)=(
zI−L
)
−1
(Appendix A), he Riesz
p ojec o
P
(Eq.
(D1)
), he scala p obe
S
(
z
) =
d−1
(
zI−L
)
−1
(Eq.
(A7)
, es ic ed o he on ie ),
wo–p obe in e sion o (
τ,
∆), p ojec o a ia ion bounds (Eqs.
(B2)
–
(B3)
), and he cu a u e p edic o
Cwi h i s use o e o ba s (Appendix B 8).
1. Design goals and global iden i ie s
Goals.
•De e minism and ep oducibili y.
All decisions (con ou s, shi s, gauges) a e unc ions o
explici keys. No global mu able s a e en e s physics.
•No double compu a ion.
Any in a ian
φk
(p ojec o , esol en momen , cu a u e, pola iza ion)
compu ed a a s a e is cached and eused i and only i i s dependencies ha e no changed.
•G ace ul deg ada ion.
I a las co e age is pa ial o cu a u e is la ge, he acke ad e ises
his, expands p obes o ank, and/o inc eases e o ba s—ne e au o– i s.
116
Global keys (hashable iden i ie s). As a e key κis a uple
κ=geom_id
| {z }
disc e e label o hash
,k
|{z}
k–poin o ∅
, T, µ
|{z}
he mo
, ile_ ype
| {z }
bulk/de ec / esponse
,window
| {z }
(c, ,Q)
, d
|{z}
ank
,{zj}
|{z}
p obe shi s .
I uniquely selec s a on ie ile and he ele an p obes; see §E 2.
2. Co e da a s uc u es
E.1 F on ie ile
S uc u e F on ie Tile(κ)
•key ←κ.
•P←P (Riesz p ojec o ; Eq. (D1)).
•Q←o hono mal basis o Ran P (columns; Eq. (D5)).
•L ←Q†LQ (Eq. (D6)), size d ×d .
•spec ←eigen alues o L , hence (τ, ∆) i d = 2.
•p obes ← {(zj, S(zj))}wi h S(zj) = d−1
(zjI−L )−1.
•pola ←local Wilson o e lap da a i on a k–pa h (Appendix C2).
•cu ←Cand local κ(a las co e age ac ion) wi h h esholds (§B 8).
• ac o s ←handles o ac o iza ions/selec ed–in e se en ies o A(zj) = zjI−L(Appendix D5).
•s a us ∈ { esh,s ale,in alid}( o cache managemen ).
E.2 In a ian cache
S uc u e In a ian Cache
•map :κ7→ F on ie Tile.
•deps :di ec ed acyclic g aph (DAG) o dependencies:
L→Riesz → {P,Q}→{L ,p obes}→{spec,pola ,cu }.
• inge p in
(
L
): a cheap spec al s amp, e.g. wo a –shi scala aces

(
z?
1I−L
)
−1,
(
z?
2I−
L)−1wi h z?
1,2 a om σ(L); used o de ec changes.
•policy: LRU wi h size caps pe laye (P/Q,L , ac o s).
In alida ion. On change o
geom_id
o
k
o (
T, µ
)o (
c, , Q
)o
d
, ma k downs eam nodes
s ale
.
I
inge p in
(
L
)changes, in alida e all nodes ha depend on
L
. I a low– ank upda e
L7→ L
+
UCV †
is signalled, keep ac o s and e esh esol en s ia Woodbu y (Eq. (A17)).
E.3 Fac o iza ion/selec ed–in e se pools
S uc u e Fac o Pool
•symbolic (pe spa si y o L): pe mu a ions, elimina ion ee, supe nodes ( eused ac oss z).
•nume ic[z]: nume ic ac o s o A(z) = zI−L.
S uc u e SelIn Pool
•diag[z]: selec ed–in e se diagonal en ies [A(z)−1]ii ( o A(z)−1).
•blocks[z, I]: op ional blocks on index se s I.
117
E.4 P ojec o acke
S uc u e P ojec o T acke
• ol_angle: bound on p incipal angles be ween successi e subspaces (e.g. 0.2 ad).
• ol_idem: ole ance on idempo ency de ec kP2−Pk.
•Q_p e : p e ious on ie basis; P_p e : p e ious p ojec o .
•be y_accum: unning =log de o Wilson o e laps o pola iza ion on closed loops.
•seam: op ional gluing G o mapping– o us holonomy (Appendix C 2).
E.5 Mul iplie sol e
S uc u e Mul iplie Sol e
•s a ic_cons ain s
: linea sys em
Cλ
=
b
(e.g. comp essibili y,
q2
coe icien
A2
) o ix s a ic
ke nel pa ame e s; ypically iny (2×2o 3×3).
•he glo z
: nonnega i e weigh s
{wj}J
j=1
and poles
{ωj}
o he esidual TDDFT ke nel
δ xc
(
ω
) =
Pj
2ωjwj
ω2
j−ω2−i0+
(Eq.
(211)
); sol ed as a cons ained leas squa es (NNLS) o ma ch a se o a ge
momen s/ alues while en o cing wj≥0.
•s opping: p imal/dual esidual ole ances and KKT easibili y h esholds.
3. Pseudocode: cache ope a ions and building a ile
Algo i hm E.1 (ge –o –build on ie ile)
1. Inpu s: key κ, ope a o L, pools (Fac o Pool,SelIn Pool), acke , cache.
2. Cache que y:
i
In a ian Cache.map
con ains
F on ie Tile
wi h key
κ
and
s a us 6
=
in alid
,
e u n i i inge p in (L) unchanged; else p oceed.
3. Con ou se up:
choose (
c, , Q
) om
κ
; build nodes
zj
=
c
+
e2πij/Q
and weigh s
wj
=
Qe2πij/Q
(Eq. (D2)).
4. Coun & basis:
(a) Fo each zj, ensu e A(zj)is ac o ed in Fac o Pool.nume ic.
(b) (Op ional) Ge A(zj)−1 om SelIn Pool.diag and alida e d ≈Pjwj A(zj)−1.
(c) Fo m Y=PjwjA(zj)−1B o a hin andom block B; o hono malize o Q; se P=QQ†.
5. Reduc ion: L =Q†LQ; compu e i s spec um spec.
6. P obes: o he wo p obe shi s {z1, z2}in κ, se S(z`) = d−1
(z`I−L )−1.
7. Cu a u e/co e age: compu e Cand κ o his segmen (Appendix B 8); a ach o ile.
8. T acke upda e:
call
P ojec o T acke .ad ance
(Algo i hm E.3); upda e Be y/Wilson accu-
mula o s i on a loop (Appendix C 2).
9. Cache s o e:
assemble
F on ie Tile
and inse in o
In a ian Cache.map
wi h
s a us
=
esh
.

118
4. Pseudocode: wo–p obe iden i ica ion and diagnos ics
Algo i hm E.2 ( wo–p obe in e sion o (τ, ∆),d = 2)
1. Inpu s: {(z1, S1),(z2, S2)} om he ile.
2. Sol e: Fo m he 2×2sys em (Eq. (A10)) and sol e o (τ, ∆) (Eqs. (A12)–(A13)).
3. Condi ioning: compu e he de e minan Do he sys em; i |D|< ol:
•pick a hi d shi z3and S3; sol e by leas –squa es,
•o back o o la ge |z1,2−¯
λ|( a he om σ(L )).
4. Check: alida e wi h an ex a es shi z?:
S(z?)−2z?−τ
2((z?)2−τz?+ ∆) ≤ ol.
5. Pseudocode: p ojec o acking and gauge alignmen
Algo i hm E.3 (p ojec o acke ad ance)
1. Inpu s: Q_p e , new basis Q om Algo i hm E.1, ole ances ol_angle, ol_idem.
2. Subspace angle:
compu e he singula alues o
M
=
Q†
p e Q
; he la ges p incipal angle is
θmax
=
a ccos
(
σmin
(
M
)). I
θmax > ol_angle
, eques s ep e inemen (hal e he s ep in
s
o
k
)
and ebuild he ile.
3. Gauge alignmen (P oc us es).
Compu e SVD
M
=
U
Σ
V†
and se
Q←QV U†
o maximize
o e lap wi h Qp e (pa allel– anspo gauge).
4. Idempo ency/consis ency.
Check
kQQ†QQ†−QQ†k ≤ ol_idem
and he ank (coun singula
alues ≈1).
5. Be y/Wilson upda e (i loop).
Accumula e
log de
(
M
)in o
be y_accum
; a closu e, add he
seam gluing G(Appendix C 2) and epo pola iza ion ia Eq. (C9).
6. Commi : se Q_p e ←Q,P_p e ←QQ†.
6. Pseudocode: mul iplie sol e
Algo i hm E.4 (s a ic mul iplie s)
1. Goal:
en o ce a small se o s a ic cons ain s
Cλ
=
b
(e.g. comp essibili y,
A2
g adien coe icien )
ha pa ame ize he s a ic pa o he ke nel o ile coupling. The dimension is iny (m≤3).
2. Sol e: compu e λ=C+bby a s able QR o SVD (i Csqua e, use LDL†).
3. Check:
epo esidual
kCλ −bk
and sensi i i ies (condi ion numbe o
C
); i ill–condi ioned, (i)
e–nondimensionalize ows, o (ii) add a hi d momen /cons ain o egula ize.
Algo i hm E.5 (He glo z weigh s o he esidual TDDFT ke nel)
1. Goal:
i nonnega i e weigh s
{wj}J
j=1
in
δ xc
(
ω
) =
PJ
j=1
2ωjwj
ω2
j−ω2−i0+
(Eq.
(211)
) o ma ch
K
a ge alues/momen s {yk}a {ωk}while en o cing wj≥0.
2. Design ma ix: Akj =<h2ωj
ω2
j−ω2
k−i0+i(and/o imagina y pa s o spec al weigh s).
3. Sol e NNLS: minw≥0kAw −yk2
2wi h a s anda d ac i e–se ; s op when KKT esiduals < ol.
4. Check: alida e K ame s–K onig/FDT nume ically on a es g id; inc ease Ji needed.
119
7. Pseudocode: pa h d i e (molecules, k–lines, in e aces)
Algo i hm E.6 (pa h e alua ion wi h cache & acke )
1. Inpu s: o de ed s a es {κi}N
i=0, ope a o builde o L(κi).
2. Ini ialize: clea In a ian Cache; ese P ojec o T acke .
3. Loop o e i= 0, . . . , N:
(a) Build o e ch F on ie Tile(κi) ia Algo i hm E.1.
(b) Compu e (τi,∆i) ia Algo i hm E.2 i d = 2; else s o e spec.
(c)
I esponse is eques ed: upda e s a ic mul iplie s (Alg. E.4) and, i needed, dynamic He glo z
weigh s (Alg. E.5).
(d) Log cu a u e Ci, co e age lags, and p edic ed e o ba s ∆es =α?Ci(Appendix B 8).
4. Finalize:
i pa h is a loop, ex ac pola iza ion om
be y_accum
(Appendix C 2); o he wise
epo endpoin in a ian s.
8. Da a low and dependency g aph
S age Depends on P oduces
L(κ)geome y/k/ he mo ope a o handle
Riesz quad a u e L,(c, , Q)P ,Q
Reduc ion Q,L L
Two p obes L ,{z1, z2}(τ, ∆),S(zj)
T acke Qp e ,Qgauge–aligned Q,log de o e lap
Cu a u e e e ence E, s ep size C, co e age κ
Mul iplie s momen da a λ,w
Cache all abo e eusable ile
9. Robus ness, complexi y, and pa allelism
Robus ness checks.
•Idempo ency: kP2−Pk ≤ ol_idem (Appendix D 6).
•Rank: d ma ches he in ended on ie size.
•Condi ioning: de e minan Din Algo i hm E.2 bounded away om ze o (Appendix A9).
•T anspo : p incipal angle θmax ≤ ol_angle (Algo i hm E.3); o he wise e ine he pa h.
Complexi y. Cos s a e go e ned by he spa se ac o iza ion and (op ionally) selec ed in e sion pe
con ou node; all nodes pa allelize. The online cos a e educ ion is cons an ( ew 2
×
2sol es and iny
linea /NNLS sys ems).
10. Minimal code–s yle ske ch ( e ba im)
unc ion BuildTile(kappa, L, pools, acke , cache):
i cache.has_ esh(kappa) and inge p in (L)==cache. inge p in :
e u n cache[kappa]
(c, ,Qnodes) = kappa.window
(z_nodes, w_nodes) = ci cle_nodes_weigh s(c, ,Qnodes)
# ac o iza ion & (op ional) selec ed in e sion a all nodes
120
o z in z_nodes:
pools. ac o .ensu e(L, z)
# pools.selin e se.ensu e(z) # i need (A(z)^{-1})
# subspace ia con ou (FEAST-like)
Y = sum( w * pools. ac o .sol e(z, B_ andom) o (z,w) in (z_nodes,w_nodes) )
Q = o hono malize(Y); P = Q @ Q.H
L =Q.H@L@Q
p obes = { zp: ace(in (zp*I - L ))/ ank(L ) o zp in kappa.p obe_shi s }
spec = eig(L )
# cu a u e / co e age and acking
ile = F on ie Tile(kappa, P,Q,L ,spec,p obes)
ile.cu , ile.co e age = cu a u e_and_co e age( ile)
acke .ad ance(Q)
cache.s o e( ile); e u n ile
11. Summa y
The in a ian cache ensu es ha once a on ie p ojec o , esol en p obe, o cu a u e da um is
compu ed a a s a e, i is eused ac oss he a las wi hou ecompu a ion; dependency–awa e in alida ion
p ese es co ec ness. The mul iplie sol e is es ic ed o iny, well–condi ioned linea o NNLS p oblems
ha en o ce iden i ies (comp essibili y, g adien limi s, He glo z posi i i y)—ne e o une ee pa ame e s.
The p ojec o acke aligns gauges, con ols subspace mo ion by p incipal angles, and accumula es Wilson
o e laps so ha pola iza ion and bulk–de ec quan i ies ollow om p ojec o da a alone. Toge he
hese componen s ope a ionalize he heo y: Riesz in eg als and selec ed in e sion (Appendix D) p o ide
he nume ics;
K0
p ojec o s and holonomy (Appendix C) p o ide he labels; and cu a u e bounds
(Appendix B) p o ide he e o ba s.
Appendix F: Addi ional benchma ks and pa ame e ables (shi poin s z, con ou s)
This appendix consolida es he nume ical se ings used h oughou Secs. IX B–IXF: he complex shi
poin s
z
o esol en p obes, he con ou quad a u es (cen e s, adii, node coun s) used o e alua e
Riesz p ojec o s, he sampling g ids o (
k, q, ω
), and he de aul ole ances o he p ojec o acke
and mul iplie sol e s. All en ies a e gi en explici ly so ha any igu e o able in Sec. 9–10 can be
ep oduced wi hou in e ence. No a ion ollows Appendices A–D:
R
(
z
) = (
zI−L
)
−1
is he esol en ; he
no malized scala p obe is
S
(
z
) =
d−1
(
zI−L
)
−1
; he Riesz p ojec o is
P
=
1
2πi HΓ
(
zI−L
)
−1dz
; and
he wo
-
p obe in e sion o a 2
×
2 on ie block uses Eqs.
(A10)
–
(A13)
. Unless s a ed o he wise, ene gies
a e in eV, leng hs in Å, empe a u es in K.
1. Con en ions and uni s
•Con ou s.
Ci cles a e pa ame e ized as
z
(
θ
) =
c
+
eiθ
,
θ∈
[0
,
2
π
)wi h weigh s
w
(
θ
) =
eiθ
(Eq.
(D2)
); ellipses as
z
(
θ
) =
c
+
acosθ
+
i b sin θ
wi h weigh s
w
(
θ
) =
bcosθ
+
i a sin θ
(Eq.
(D4)
).
•Shi poin s.
Unless explici ly eal,
z
a e complex shi s placed symme ically abou he on ie
cen e o s abilize condi ioning (Appendix A 9); we w i e z=c±iδ wi h c, δ ∈R.
•Tole ances.
Linea sol es a e e ined o ela i e esidual
<
10
−12
; p ojec o idempo ency is en o ced
wi h
kP2−Pk ≤
10
−10
; p incipal angle ole ance in he acke is
θmax ≤
0
.
2 ad (Algo i hm E.3).
121
Table II.
Molecula oy models: esol en p obes and con ou quad a u e.
Each ow lis s he geome y
g id, wo shi poin s
z1,2
o he scala p obes
S
(
z
), and he con ou used o build he on ie p ojec o
P
and
basis Q(Appendix D 1).
Sys em Geome y g id F on ie cen e c(eV) Shi s z1,2(eV) Con ou (c, )/(c;a, b)Nodes M
H2(2–si e Hubba d) R∈[0.5,3.5] Å; s ep 0.05 Å0.00 c±i3.0Ci cle: (c=0.0, =4.0) 16
N2(σ+πac i e space) R∈[0.9,2.2] Å; s ep 0.05 Å0.00 c±i4.0Ellipse: (c=0.0; a=6.0, b=3.0) 24
Dono –accep o CT wo–le el R∈[6,20] Å; s ep 1.0Å1
2(εD+εA)c±i2.5Ci cle: (c, =5.0) 16
Table III. SSH– ype chains: k–g ids, p ojec o con ou s, and Wilson loop pa ame e s.
Dime iza ion δ (eV) NkF on ie c(eV) Shi s z1,2Con ou (c, )Wilson loop
0.10 2.50 Nk= 401 0.0c±i2.5Ci cle (0.0,3.5) o e lap ma ices M`=U†
`U`+1
0.30 2.50 Nk= 401 0.0c±i2.5Ci cle (0.0,3.5) same (gluing G=I)
2. Molecules ( oy dime s): H2 wo-si e model; N2σ+πac i e space
Model pa ame e s used in Sec. IX A. Fo he H
2
su oga e:
(
R
) =
0e−α(R−Re)
wi h
0
= 3
.
0,
α
= 1
.
6Å
−1
,
U
= 6
.
0,
Re
= 0
.
74Å. Fo he CT dime : (
εD−εA
)=4
.
0, coupling
(
R
) =
0e−α(R−R0)
wi h
0
= 1
.
5,
α
= 0
.
5Å
−1
,
R0
= 6Å. Two shi s su ice o he 2
×
2 on ie (§A 4); con ou s enclose only he on ie
island (Appendix D 1).
3. Pe iodic solids (SSH): band gaps and pola iza ion
Pola iza ion is compu ed om he disc e e Wilson loop
W
=
Q`M`
wi h
M`
=
U†
`U`+1
,
P
=
(
e/
2
π
)
=log de W
(Appendix C 2). The same shi s
z1,2
a e used o ix (
τ,
∆) along he
k
–pa h i
he on ie a las is ea ed as 2×2(§A 4).
4. Domain walls (de ec s/in e aces): on empla es and con ou s
The p ojec o acke (Algo i hm E.3) is used along
x
wi h p incipal–angle ole ance 0
.
2 ad; he bound
cha ge is compu ed by he cumula i e excess o ia pola iza ion misma ch (Appendix C 3).
5. TDDFT esponse: (q, ω)g ids and He glo z ke nel pa ame e s
The esidual ke nel is pa ame e ized as
δ xc
(
ω
) =
PJ
j=1
2ωjwj
ω2
j−ω2−i0+
wi h
wj≥
0(Appendix E 6); weigh s
a e i by NNLS o ma ch a ge s a ic limi s and selec ed spec al poin s, p ese ing causali y.
6. Fini eTand open sys ems: Ma suba a and Liou illian p obes
Fo ini e
T
, p obes use
z
=
iωn
+
µ
a wo dis inc e mionic nodes (Appendix A 7); o GKSL cases, he
Liou illian on ie is p obed a z=c±iδ wi h δse by he dissipa i e gap.
7. Pe o mance and ole ances: ac o iza ion euse, selec ed in e sion, e o ba s
8. Rep oducibili y mani es
•Random seeds.
When con ou il e ing uses a andom block
B
(Eq.
(D5)
), seed is ixed o
42
and block wid h is m= 4.
•S a e keys. Each ow in Tables II–VI co esponds o a unique s a e key
κ
= (
geom_id,k, T, µ, ile_ ype,window, d ,{zj}
)as de ined in Appendix E; hese keys a e he
indices in he in a ian cache.
128
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