Time Crystals--Null--Modular $\mathbb{Z

Time C ys alsNullModula
Z2
Holonomy Unica ion: F om
Floque and Lindblad o Bulk-In eg al BF Rela i e Cohomology
C i e ion
Haobo Ma
1
Wenlin Zhang
2
1
Independen Resea che
2
Na ional Uni e si y o Singapo e
Abs ac
Cons uc heo e ical chain uni ying disc e e/con inuous ime c ys als wi h NullModula
Z2
holonomy, bulk-in eg al
Z2
BF choice, and ela i e cohomology in a ian . Closed sys em side,
p o ide igidi y and s abili y o p e he mal disc e e ime c ys als in exponen ially long ime win-
dows ia high- equency Floque Magnus and LiebRobinson cons ain s; unde s ong diso de
p o ide necessa y and sucien s uc u e o
π
spec al pai ing and eigens a e ime c ys alline
o de . Open sys em side, es ablish spec al c i e ion o limi cycle ime c ys als on pe iphe al
spec um o single-pe iod CPTP channel. Quasipe iodic d i e side, cons uc " empo al qua-
sic ys al" g oup ep esen a ion ia ni e image o
Zk
ime ansla ion g oup. Abo e ou classes
o phenomena in e ace wi h unied opologicalalgeb aic skele on:
Z2/Zm
holonomy and ela-
i e cohomology class
[K]∈H2(Y, ∂Y ;Z2)
o bulk-in eg al
Z2
BF op e m; unde small causal
diamond h eshold, i sa is ying modula sca e ing mod- wo alignmen and pa ame e wo-cycle
de ec abili y and gene a ion, hen "geome yene gy opology" iple equi alen , specically
[K]=0 ⇐⇒
ime c ys al "anomaly" anishes on allowed loops and wo-cycles. This pape si-
mul aneously p o ides
Z2
nge p in s o h ee sol able amilies (
δ
po en ial, Aha ono Bohm,
opological supe conduc o endpoin ) and enginee ing schemes wi h e o budge s o supe -
conduc ing qubi s, Rydbe g gases, and apped ions. Co e NullModula double co e and BF
ela i e cohomology c i e ion aken om au ho s' exis ing unied p inciple and es a ed and
p o ed in ime c ys al con ex .
Keywo ds
: Disc e e ime c ys al; p e he maliza ion and many-body localiza ion; open sys em
limi cycle; empo al quasic ys al; opological ime c ys al;
π
spec al pai ing;
Z2/Zm
holonomy;
bulk-in eg al
Z2
BF; ela i e cohomology; small causal diamond
1 In oduc ion & His o ical Con ex
Spon aneous b eaking o ime ansla ion symme y igo ously nega ed in equilib ium sys ems,
o cing physical ca ie s o ime-o de ed phases owa d non-equilib ium d i e and open dynam-
ics. In pe iodically d i en many-body sys ems, disc e e ime c ys als cha ac e ized by subha monic
esponse igidi y, long- ange empo al co ela ions, and cha ac e is ic spec al nge p in s; subse-
quen b anches o eigens a e o de ing, p e he mal longe i y, dissipa i e limi cycles, and opological
(logical) ime c ys als cons i u e c oss-pla o m expe imen al sys ems. On o he hand, we de-
eloped in a ian s cen e ed on squa e oo de e minan b anching and mod- wo holonomy along
geome icin o ma ionsca e ing unied h ead, ele a ing hem o necessa y and sucien c i e-
ion ia bulk-in eg al
Z2
BF ela i e cohomology language. This pape igo ously b idges hese
wo h eads, p o ing: ime c ys al "
π
/uni oo " phenomenology a unied c i e ion le el is
Z2/Zm
1
holonomy and
[K]∈H2(Y, ∂Y ;Z2)
non- i iali y; con e sely, geome yene gy c i e ion ( s and
second o de laye on small causal diamond) unde alignmen h eshold implies i ializa io
n o such in a ian s, he eby p o iding common s uc u e pene a ing closed/open/ opological/mul i-
equency.
2 Model & Assump ions
2.1 Closed Sys em (Floque MBL/P e he maliza ion)
Local many-body sys em on la ice
Λ
, pe iodic d i e
H( +T) = H( )
. Floque uni a y
F=
Texp(−iRT
0Hd )
gene a es disc e e ime ansla ion. High- equency limi
ω= 2π/T ≫J
admi s
quasilocal eec i e Hamil onian
H∗
wi h exponen ially small unca ion e o ; unde s ong diso de
H0
suppo s
l
-bi s uc u e.
2.2 Open Sys em (Pe iodic Lindblad)
Densi y ma ix e olu ion
˙ρ=L (ρ)
wi h
L +T=L
. Single-pe iod quan um channel
E=Texp(RT
0L d )
pe iphe al spec um de e mines long- ime limi cycle.
2.3 Mul i-F equency Quasipe iodic
Mu ually i a ional equency amily
{ωi}k
i=1
denes ime ansla ion g oup
Zk
; high- equency
p e he maliza ion h eshold
miniωi≫J
ensu es quasilocal
H⋆
and ni e image g oup ep esen a-
ion.
2.4 Topological Time C ys al (Logical Subspace)
S abilize code (su ace code) pe iodic enginee ing makes
Flogical ≃XLe−iH op
∗T
, non-local logical
ope a o s as na u al o de pa ame e s.
2.5 NullModula and Bulk-In eg al BF (Rela i e Cohomology)
Wo king space
Y=M×X◦
, whe e
M
small causal diamond domain o mo e gene al local space ime
pa ch,
X◦
pa ame e domain emo ing disc iminan se
D
. Dene
K=π∗
Mw2(TM) + X
j
π∗
Mµj⌣ π∗
Xwj+π∗
Xρ(c1(LS)) ∈H2(Y;Z2),
using
[K]∈H2(Y, ∂Y ;Z2)
unde bounda y i ializa ion and ela i e cohomology li .
Z2/Zm
holonomy compu ed as mod- wo (o uni oo ) alue o squa e oo de e minan b anch, s abilizing
closed pa hs pe "small semici cle/ old-back" ule.
3 Main Resul s (Theo ems and Alignmen s)
Theo em 1
(A: Rigidi y and Exponen ial Li e ime o P e he mal DTC)
.
Unde
ω≫J
and piece-
wise nea -
π
"symme ic kick" condi ions, quasilocal uni a y
U∗
and symme y elemen
X2=⊮
exis
making
F=U∗e−iH∗TX U†
∗+ ∆,|∆| ≤ Ce−cω/J .
2
Any local obse able
O
odd unde
X
exhibi s
2T
subha monic locking, emaining locked o
≲τ∗∼ecω/J
, main aining Lipschi z s abili y agains small pe u ba ions.
Theo em 2
(B:
π
Spec al Pai ing and Eigens a e O de in MBLDTC)
.
On s ongly diso de ed
chain quasilocal uni a y
U
exis s making
F≃˜
Xe−iHMBLT
, spec um exhibi s
π
pai ing, inducing
s a e-independen
2T
subha monic esponse.
Theo em 3
(C: Spec al C i e ion o Open Sys em Limi Cycle)
.
I single-pe iod channel
E
pe-
iphe al spec um
{e2πik/m}
wi h spec al adius
<1
elsewhe e, hen almos all ini ial s a es con e ge
o pe iod-
mT
limi cycle a ac o amily, cons i u ing
m
-subha monic dissipa i e ime c ys al.
Theo em 4
(D: Mul i-F equency "Tempo al Quasic ys al" G oup Rep esen a ion)
.
Unde p e he -
maliza ion h eshold
miniωi≫J
, ni e image o ime ansla ion g oup
Zk
p oduces mul iple in-
commensu a e subha monic peaks, o ming empo al quasic ys al.
Theo em 5
(E: Topological Time C ys al: Non-Local O de and Topological En anglemen )
.
In
logical subspace, non-local loop ope a o s exhibi igid mul iple-pe iod esponse, consis en wi h non-
ze o opological en anglemen en opy e m.
Theo em 6
(F: Unied TopologicalCohomology C i e ion)
.
Unde bounda y i ializa ion, ela i e
gene a ion and de ec abili y h eshold, ollowing equi alen :
[K] = 0 ⇐⇒
o all allowed loops
γ:ν√de pS(γ) = +1
and all allowed wo-cycles
γ2:⟨ρ(c1(LS)),[γ2]⟩= 0.
When
H2(X◦, ∂X◦) = 0
, abo e equi alen o mod- wo c i e ion on loops only.
Theo em 7
(G: Geome yEne gy
⇒
Topological T i iali y)
.
Unde small causal diamond h esh-
old, ela i e gene a ionde ec ion, and modula sca e ing mod- wo alignmen , i  s o de laye
gi es
Gab + Λgab = 8πGTab
and second o de ela i e en opy
δ2S el =Ecan ≥0
, hen abo e unied
opologicalcohomology c i e ion holds, i.e., implies
[K]=0
and all
Z2
holonomies i ial.
4 P oo s
4.1 P e he maliza ion and
π
Locking (Theo ems A/B)
Take Floque Magnus expansion
F= exp{−iTPn≥0Ωn}
, unca e a op imal o de
n∗∼ω/J
dening
H∗
. Via LiebRobinson and local expansion se ies eno maliza ion ob ain
|F−e−iH∗T| ≤
Ce−cω/J
. Piecewise nea -
π
kick
UX
p oduces
X
, ob aining s uc u al decomposi ion in quasilocal
uni a y
U∗
ep esen a ion. Unde s ong diso de
UH0U†= ({τz
i})
nea ly commu es wi h
˜
X=
UXU†
, spec um exhibi s
π
pai ing; a bi a y ini ial s a e expanded in pai ed subspace, ob aining
s a e-independen
2T
subha monic esponse.
4.2 Pe iphe al Spec um and Limi Cycle (Theo em C)
Pe o m Jo danRiesz decomposi ion o CPTP
E
. I pe iphe al spec um
m
uni oo s wi h spec al
gap elsewhe e, hen
En
exponen ially con e ges on each esidue class o pe iod-
m
cyclic a ac o ;
con e gence a e con olled by Liou illian spec al gap.
3
4.3
Z2/Zm
Holonomy and Rela i e Cohomology (Theo em F)
Wo k wi h
Z2
coecien s aking ela i e cohomology. Bulk-in eg al
Z2
BF ac ion
IBF[a, b]=iπZ(Y,∂Y )
b ⌣ δa+ iπZ(Y,∂Y )
b⌣K + iπZ∂Y
a ⌣ b,
a e gauge ans o ma ion and bounda y e m cancella ion, summing o e
[a]∈H1(Y, ∂Y )
,
[b]∈Hd−2(Y, ∂Y )
, using ni e abelian g oup cha ac e o hogonali y ob ains pa i ion unc ion
p ojec ion
Z op ∝δ([K])
, i.e.,
[K]=0
. By Poinca éLe sche z duali y,
[K]=0
i and only i
K onecke pai ing anishes o all allowed ela i e wo-cycles
[S]
; when
H2(X◦, ∂X◦)=0
, educes
o loop mod- wo c i e ion only.
4.4 Geome yEne gy Implies Topological T i iali y (Theo em G)
Unde small causal diamond h eshold (Hadama d, no conjuga e poin s, co ne p esc ip ion,
∇aTab =
0
, xed empe a u e scale) and in e ibles able hypo hesis o weigh ed null ay ans o ma ion,
amily cons ain
Rw(λ)(Rkk −8πGTkk) dλ= 0
wi h Radon- ype closu e implies
Rkk = 8πGTkk
;
null cone cha ac e iza ion and Bianchi iden i y upg ade o enso equa ion, ob aining  s o de laye
Gab + Λgab = 8πGTab
. Co ne p esc ip ion ensu es co a ian phase space symplec ic ux closu e,
δ2S el =Ecan ≥0
. I closed pa h
γ
exis s making
ν√de pS(γ) = −1
, hen by modula sca e ing
mod- wo alignmen , cons uc linea unc ional co esponding o holonomy in co a ian phase space
embedding in o quad a ic o m ke nel, ob aining
Ecan[h, h]<0
con adic ion, hus implying all
allowed loop holonomies i ial, u he by ela i e gene a ionde ec ion ob aining
[K]=0
.
5 Model Apply
5.1
Z2
Finge p in s o Sol able Families
(i) 1D
δ
po en ial: Selec small loop a ound complex pole,
H1
2iS−1dS=π⇒ν√de pS=−1
.
(ii) 2D Aha ono Bohm: Flux c ossing hal -ux
α=1
2
gi es
deg(de pS|γ)=1⇒ν=−1
. (iii)
Topological supe conduc o endpoin (Class D/DIII):
sgn de p (0)
o
sgn P (0)
ip synch onizes
wi h
ν√de p
. Th ee amilies a e emo ing disc iminan se ha e
H2(X◦, ∂X◦)=0
, equi ing loop
mod- wo c i e ion only.
5.2 Pla o m Mapping and Obse ables
Supe conduc ing qubi 2D a ay: Logical loop ope a o spec um exhibi s
ω/2
peak only in non-
local channel, accompanied by non-ze o opological en anglemen en opy; Rydbe g gas: Uni oo s
in quan um channel pe iphe al spec um consis en wi h uo escence au oco ela ion limi cycle;
apped ions:
ω
enhancemen b ings exponen ial li e ime g ow h wi h igid equency posi ion no
d i ing; mul i- equency d i e: Incommensu a e peak posi ions co espond o ni e image o
Zk
.
6 Enginee ing P oposals
6.1 P e he maliza ion Window and Pulse Syn hesis
Choose
ω
making
e−cω/J ≪ε
(ga e noise ampli ude), ensu ing
τ∗∼ecω/J
co e s
102−103
cycles;
piecewise sequence implemen s nea -
π
ip a
τx
o ampli y
2T
locking.
4
6.2 Open Sys em Spec al Gap Enginee ing
Cons uc Liou illian spec al gap
∆Liou
ia pumpingdecohe ence a io
G/κ
, supp essing mul i-
s able wande ing; sample pe iphe al eigen alues and limi cycle pe iod a ound s eady-s a e wo king
poin .
6.3 Mul i-F equency Tempo al Quasic ys al
Two o h ee mu ually i a ional equencies, a oiding acciden al in ege pe iod ecu ence; collec
spec um using incommensu a e peak posi ions o iden i y ni e image; econs uc uni oo alues
ia pa ame e closed pa hs.
6.4 Expe imen al Readou o WilsonLoop
Ampli udephase join scan o ming closed pa h; dis inguish
±1
ia disc e e Fou ie peaks in Ram-
sey/co ela ion unc ions; o
Zm
 uni oo s using phase g id.
7 Discussion ( isks, bounda ies, pas wo k)
Bounda ies and isks include: abso p ion-induced locking collapse ou side high- equency win-
dow; MBL s abili y limi ed in high dimensions and long- ange in e ac ions; open sys em non-
Ma ko iani y causes phase wande ing; opological ime c ys al non-local eadou sys ema ic sensi-
i i y o leakage and c oss alk. A unied c i e ion le el,
H2
channel de ec abili y equi es allowed
wo-cycle gene a ion o ela i e wo-cohomology; i pla o m pa ame e domain wo-skele on insu -
cien , ob ains only necessa y non-sucien c i e ion. Compa ed o exis ing wo k, his pape 's
inc emen : using bulk-in eg al
Z2
BF ela i e cohomology class
[K]
and mod- wo holonomy as
single in a ian
, uni o mly cha ac e izing closed/open/ opological/mul i- equency ou classes o
ime c ys als, es ablishing implica ion chain be ween geome yene gy and holonomycohomology
unde small causal diamond a ia ional h eshold.
8 Conclusion
Time c ys al "
π
/uni oo " phenomenology uni o mly cha ac e ized by
Z2/Zm
holonomy and bulk-
in eg al
Z2
BF ela i e cohomology class; when ela i e gene a ionde ec ion and modula sca e ing
mod- wo alignmen hold, geome yene gy c i e ion on small causal diamond implies opological
cohomology i ializa ion. This s uc u e simul aneously suppo s p e he mal DTC, MBL eigen-
s a e o de , open sys em limi cycle, and opological ime c ys al, p o iding g oup ep esen a ion
and expe imen al eadou o mul i- equency empo al quasic ys als. This amewo k p o ides
unied heo yenginee ing channel o c oss-pla o m ime- equency de ices, obus s o age, and
opological logical ope a ions.
Acknowledgemen s, Code A ailabili y
Thank publicly a ailable esul s and pla o m da a es ablishing expe imen al backg ound. This
pape does no ely on p op ie a y code; sc ip s o ela i e cohomology pai ing,
Z2/Zm
holonomy
econs uc ion, and pe iphe al spec um  ing can be ep oduced using s anda d nume ical ools
ollowing algo i hmic s eps in appendices.
5

Re e ences
Selec ep esen a i e heo e ical and expe imen al wo ks: ime c ys al no-go heo ems, Floque 
DTC deni ion, p e he maliza ion uppe bounds, eigens a e ime c ys alline o de , dissipa i e ime
c ys als, opological ime c ys als and empo al quasic ys als; as well as NullModula double co e
and bulk-in eg al
Z2
BF unied p inciple on geome icin o ma ionsca e ing h ead.
A Rigo ous P e he maliza ion Uppe Bound and Exponen ial Li e-
ime
Assume
|H( )| ≤ J
,
ω≫J
. Floque Magnus
Ω0=1
TZT
0
H, Ω1=1
2TiZZ0< 1< 2<T
[H( 2), H( 1)] d 1d 2, . . .
unca e a
n∗∼αω/J
, dene
H∗=Pn≤n∗Ωn
. Via nes ed commu a o ee eno mal-
iza ion p o e
|F−e−iH∗T| ≤ Ce−cω/J
,
|d
d ⟨H∗⟩| ≤ C′e−cω/J
. Piecewise nea -
π
kick
UX
makes
F≈Xe−iH∗T+O(ϵ) + O(e−cω/J )
, hus o
X
-odd
O⟨O(nT)⟩ ≈ (−1)n⟨O(0)⟩+O(ϵ) + O(e−cω/J )
.
F equency domain exhibi s
ω/2
locking peak, peak posi ion igid agains pa ame e pe u ba ions.
B MBL
π
Pai ing and Eigens a e O de
Uni a y
U
exis s making
UH0U†= ({τz
i})
, nea -
π
kick unde
U
ans o ma ion yields quasilocal
˜
X
. I
F≃˜
Xe−iHMBLT
, hen o eigens a e
|ψ⟩F|ψ⟩= e−iET ˜
X|ψ⟩, F(˜
X|ψ⟩)=e−i(ET+π)|ψ⟩
,
spec um exhibi s
π
pai ing. A bi a y ini ial s a e expanded in pai ed subspace yields s a e-
independen
2T
subha monic esponse.
C CPTP Pe iphe al Spec um and Limi Cycle
Le
σ(E) = {λj}
. I
|λj|<1
o all non-pe iphe al modes, pe iphe al modes
{e2πik/m}
, hen mu ually
disjoin cyclic in a ian subspaces exis , making
En
p ojec a bi a y ini ial s a e o pe iod-
m
limi
cycle; con e gence a e con olled by
∆Liou = 1 −max|λj|<1|λj|
.
D
Z2/Zm
Holonomy: Mod-Two Robus ness o Spec al Flow and
In e sec ion Numbe
Deno e disc iminan se
D⊂X◦
as h eshold/embedded eigen alue o
−1
eigen alue submani-
old. Fo closed pa h
γ
s abilized pe "small semici cle/ old-back" ule, dene mod- wo in e sec ion
numbe
I2(γ, D)
. Modied de e minan
de p
change only al e s quan ized phase in ege winding
numbe , mod- wo p ojec ion in a ian ; pa ial-wa e unca ion
N→ ∞
and ela i e ace-class
eno maliza ion p ese e
ν√de pS= (−1)I2(γ,D)
.
E Bulk-In eg al
Z2
BF Rela i e Cohomology De i a ion
On
(Y, ∂Y )
wi h
Z2
coecien s cons uc
6
IBF[a, b]=iπZb ⌣ δa+ iπZb ⌣ K + iπZ∂Y
a ⌣ b.
Unde gauge ans o ma ion
a7→ a+δλ0
,
b7→ b+δλd−3
bounda y e m cancels, ac ion well-
dened. Summing o e
[a]
and
[b]
, using ni e abelian g oup cha ac e o hogonali y ob ains
Z op ∝
δ([K])
; Poinca éLe sche z duali y gi es
[K] = 0 ⇐⇒
o all
[S]∈H2(Y, ∂Y ;Z2)
ha e
⟨K, [S]⟩= 0
.
F Geome yEne gy
⇒
Holonomy T i iali y: Alignmen and Con-
adic ion Me hod
Unde co ne p esc ip ion and co a ian phase space amewo k, closed pa h modula holonomy de-
nes bounded linea unc ional embedding in o quad a ic o m ke nel. I
γ
exis s making
ν√de pS(γ) =
−1
, unc ional unde mod- wo p ojec ion p o ides nega i e di ec ion, cons uc ing
h
making
Ecan[h, h]<
0
, con adic ing second-o de non-nega i i y; hus all allowed closed pa h holonomies
(+1)
. Com-
bined wi h ela i e gene a ion and de ec ion, implies
[K]=0
.
G De o ma ion Re ac ion and
H2= 0
o Th ee Sol able Families
Pa ame e domains o
δ
po en ial, AB, and endpoin sca e ing a e emo ing disc iminan se
de o ma ion e ac o one-dimensional skele on, hus
H2(X◦, ∂X◦)=0
. The e o e unied c i e ion
educes o mod- wo condi ion on loops; in his case, cons uc ing e e ence closed pa h ans e se
o
D
e ies
[K] = 0
necessa y and sucien condi ion.
H Expe imen Algo i hm Checklis (Pseudocode Le el)
1. Da a acquisi ion and de ending: Time se ies
O( n)
emo e d i ia polynomial eg ession;
2. Peak and locking me ic: Disc e e Fou ie ,  peak wid h
Γ
and locking a io
R
; 3. Uni
oo eadou : Pa ame e closed pa h sample phase, disc imina e
±1
o
m
- h uni oo ; 4. Rela i e
cohomology es : Selec gene a ing amily loops/ wo-cycles, e alua e
ν√de pS
and
⟨ρ(c1),[γ2]⟩
able,
e i y all-ze o o de e mine
[K]=0
.
7