Self-Referential Scattering and the Birth of Fermions: Riccati Square Roots, Spinor Double Cover, and a $\mathbb{Z

Sel -Re e en ial Sca e ing and he Bi h o Fe mions: Ricca i
Squa e Roo s, Spino Double Co e , and a
Z2
Exchange Phase
Haobo Ma
1
Wenlin Zhang
2
1
Independen Resea che
2
Na ional Uni e si y o Singapo e
Abs ac
S anda d quan um eld heo y explains he ela ion be ween spin and s a is ics h ough
he spins a is ics heo em, de i ed om Lo en z co a iance and mic ocausali y on a con-
inuous space ime backg ound. In opological app oaches, he an isymme y o e mionic
wa e unc ions can also be unde s ood ia he non i ial opology o congu a ion spaces
and associa ed line bundles, as in he Finkels einRubins ein cons uc ion o soli ons in
nonlinea eld heo ies. Howe e , hese amewo ks ypically assume ela i is ic quan um
elds as he on ological s a ing poin .
Wi hin a disc e e, causal quan um cellula au oma on (QCA) on ology, he uni e se is
desc ibed as a la ice o local Hilbe spaces upda ed by a global uni a y s ep. In p e ious
wo k, massi e exci a ions we e in e p e ed as localized, sel -sus ained in e e ence s uc u es
whose in e nal dynamics consume pa o a global in o ma ion-upda e budge , gi ing mass
an in e p e a ion as  opological impedance in an unde lying sca e ing ne wo k. Building
on his pic u e, he p esen wo k p oposes a dynamical and geome ic o igin o e mionic
s a is ics in e ms o
sel - e e en ial sca e ing
.
We conside localized exci a ions ealized as eedback loops in an eec i e one-dimensional
sca e ing p oblem ob ained om coa se-g aining he QCA. The bounda y esponse o such
a loop is encoded in an impedance unc ion o eec ion coecien sol ing a nonlinea Ric-
ca i equa ion, a s uc u e well known in wa e p opaga ion and sca e ing heo y. We show
ha he xed-poin condi ion o a s able, sel - e e en ial loop o ces he physical s a e o
he exci a ion o li e on a
squa e- oo b anch
o he unde lying sca e ing da a. This
b anch s uc u e induces a canonical double co e o he congu a ion space o
N
iden ical
exci a ions. We p o e ha he gene a o co esponding o exchanging wo exci a ions li s
o a non i ial loop on his double co e wi h holonomy
(−1)
, so ha he
N
-body wa e unc-
ion ans o ms in he sign ep esen a ion o he pe mu a ion g oup and sa ises e mionic
exchange s a is ics.
In his cons uc ion, spino beha io and he
Spin(3)
double co e o
SO(3)
a e no pos-
ula ed bu eme ge om he necessi y o aking squa e oo s o sel - e e en ial sca e ing
da a. The in e nal Ricca i squa e- oo  a iable plays he ole o a spino ampli ude whose
squa ed modulus ep oduces obse able sca e ing cha ac e is ics. In his sense, spin
1/2

is ein e p e ed as he opological nge p in o in o ma ion sel - e e ence in a QCA-based
uni e se. We ou line an explici ealiza ion in Di ac- ype QCA models and p opose engi-
nee ed sca e ing ne wo ks in pho onic and supe conduc ing pla o ms o es he p edic ed
Z2
exchange phase.
Keywo ds:
Quan um cellula au oma a; sel - e e en ial sca e ing; Ricca i equa ion; impedance;
spins a is ics heo em; spino double co e ; Finkels einRubins ein cons ain ; e mionic ex-
change phase
1
1 In oduc ion & His o ical Con ex
1.1 SpinS a is ics Theo em in Con inuous Field Theo y
The spins a is ics heo em s a es ha in h ee-plus-one-dimensional ela i is ic quan um eld
heo y, hal -in ege spin pa icles mus obey Fe miDi ac s a is ics, while in ege spin pa icles
mus obey BoseEins ein s a is ics. Mode n ex book de i a ions ely on he ollowing ing e-
dien s: Lo en z co a iance, acuum s abili y, posi i e-deni e p obabili ies o local obse ables,
and (an i)commu a ion ela ions o ope a o s a spacelike sepa a ions (mic ocausali y). In his
amewo k, he dis inc ion be ween e mions and bosons is one o he inpu condi ions o eld
ope a o commu a ion ela ions, and consis ency wi h spin is subsequen ly gua an eed by s uc-
u al heo ems.
This p oo s uc u e is highly igo ous, ye b ings a widely ecognized puzzle: why is he e
such a p o ound connec ion be ween spin and many-body exchange s a is ics, a he han being
mu ually independen s uc u es? Feynman once lamen ed ha an in ui i e explana ion o
his simple s a emen emains elusi e.
1.2 Topological SpinS a is ics Rela ions and Soli ons
An al e na i e line o hough , ad oca ed by Finkels ein and Rubins ein, examines he homo opy
p ope ies o opological soli on congu a ion spaces in nonlinea eld heo ies. They poin ou
ha when he undamen al g oup o he soli on congu a ion space is non i ial, one can ela e
2π
o a ions o pa icle exchanges ia non i ial line bundles, hus ob aining a spins a is ics
co espondence. In his pe spec i e, he wa e unc ion is no longe a single- alued unc ion on
congu a ion space, bu a he a sec ion o a line bundle; he homo opy class o soli on winding
pa hs de e mines he exchange phase h ough he holonomy o he bundle. Fu he wo k has
applied his idea o YangMills soli ons and Hop opological in a ian s, demons a ing geome ic
ela ionships be ween linking numbe s and s a is ics.
Topological me hods p o ide geome ic in ui ion o he spins a is ics heo em, bu s ill
ake in insic con inuous elds as undamen al objec s, and ypically s a om known soli on
models.
1.3 Quan um Cellula Au oma a and Disc e e On ology
In ecen yea s, quan um cellula au oma a (QCA) ha e eme ged as an a emp o econs uc
quan um eld heo y, and e en he en i e physical uni e se, wi hin a amewo k o disc e e, local
uni a y e olu ion. In he QCA pic u e, space ime consis s o a disc e e la ice
Λ
wi h a local
Hilbe space
Hx
a each si e, and dynamics a e gi en by a global uni a y ope a o
U
ac ing on
ni e neighbo hoods. In he con inuum limi , app op ia e choices o local coin ope a o s can
ep oduce Di ac, Weyl, o Maxwell equa ions, hus eco e ing s anda d quan um eld heo y on
a disc e e on ology.
In a se ies o p e ious wo ks, mass was in e p e ed as a geome ic impedance o in o ma-
ion p opaga ion a es: massless exci a ions co espond o eed o wa d p opaga ion along he
ligh cone, while massi e exci a ions co espond o local s uc u es wi h eedback and looping,
whose in e nal e olu ion speed
in
and ex e nal g oup eloci y
ex
sa is y an in o ma ion a e
conse a ion cons ain . The co esponding mic oscopic pic u e is: a pa icle is a kind o sel -
e e en ial eedback loop in he QCA ne wo k, whose s able exis ence depends on impedance
ma ching be ween inpu and ou pu .
1.4 Goals and Claims o This Wo k
This pape a emp s o ake a u he s ep in he abo e QCA and mass = opological impedance
pic u e, p oposing he ollowing claims:
2
1. Any localized exci a ion ha ealizes a s able es mass in QCA can, a e app op i-
a e coa se-g aining, be iewed as a
sel - e e en ial eedback loop
in an eec i e one-
dimensional sca e ing p oblem.
2. The spa ial e olu ion o he bounda y esponse (impedance o eec ion coecien ) o
his eedback loop is go e ned by a nonlinea Ricca i equa ion; i s s eady-s a e solu ion
is a xed poin o a ce ain Möbius ans o ma ion and has a squa e- oo disc iminan
s uc u e.
3. Fo a sys em o
N
iden ical sel - e e en ial exci a ions, he na u al quan iza ion o he
o al congu a ion space is no longe a single- alued wa e unc ion, bu a he a sec ion o
a
double co e line bundle
induced by he abo e Ricca i s uc u e. Pa icle exchange
pa hs ha e
Z2
holonomy on his double co e , leading o an exchange phase o
(−1)
.
4. The e o e,
massi e sel - e e en ial exci a ions au oma ically ealize Fe miDi ac
s a is ics
in his cons uc ion; bosons co espond o pu e eed o wa d modes wi hou
sel - e e ence o composi es o se e al sel - e e en ial loops.
The s a ing poin o his esea ch is no o a emp o  e-p o e he spins a is ics heo-
em, bu o cons uc a mic oscopic mechanism in disc e e on ology ha es ablishes an explici
connec ion be ween sel - e e en ial sca e ing, Ricca i squa e oo s, and spino double co e s,
hus p o iding a dynamicalgeome ic explana ion o he exis ence o e mions.
2 Model & Assump ions
2.1 Unde lying QCA S uc u e
Le
Λ⊂Zd
be a egula la ice; his pape p ima ily conside s cases
d= 1,3
. Each si e
x∈Λ
is associa ed wi h a ni e-dimensional Hilbe space
Hx∼
=Cq
, and he global Hilbe space is
H=O
x∈ΛHx.
Time e olu ion is gi en by a local uni a y ope a o wi h ni e neighbo hood. Tha is, he e
exis s a ni e ange
R
such ha he single-s ep e olu ion
U
can be w i en as a ni e-dep h
quan um ci cui o local ga es, espec ing causali y:
U†AOU⊂ AO+
, whe e
AO
is he local
ope a o algeb a o egion
O
, and
O+
is a ni e- hickness neighbo hood o
O
.
To connec o he con inuum limi , we conside a class o Di ac- ype QCA, whose single-s ep
e olu ion in momen um ep esen a ion can be w i en as
U(k) = exp (−iHe (k)∆ ),
whe e
He (k)
in he long-wa eleng h limi
ka ≪1
app oaches he Di ac Hamil onian
He (k)≈αk +βm,
wi h
α, β
ma ices sa is ying he Clio d algeb a,
a
he la ice spacing, and
m
he eec i e mass
pa ame e .
2.2 Sel -Re e en ial Sca e ing Uni and Eec i e One-Dimensional Model
Conside in oducing a local s uc u e in a ni e egion
D⊂Λ
such ha he e exis s a eedback
channel wi hin he egion: pa o he inciden ampli ude, a e unde going local sca e ing, is
e-injec ed in o he same egion. Fo modes wi h wa eleng h much la ge han he size o
D
, an
eec i e one-dimensional desc ip ion can be adop ed, comp essing he en i e egion
D
in o an
3
equi alen ansmission line o sca e ing cen e , wi h inciden and ou going signals p opaga ing
along a one-dimensional coo dina e
x
.
In he equency domain ep esen a ion, suppose ha o a xed equency
ω
, he mode
sa ises an eec i e wa e equa ion on he one-dimensional coo dina e. I s p opaga ion in he
hal -space
x > 0
can be cha ac e ized by a posi ion-dependen impedance
Z(x;ω)
. Acco ding o
elec omagne ic wa e and acous ic wa e p opaga ion heo y, unde app op ia e one-dimensional
app oxima ions, he spa ial e olu ion o
Z(x;ω)
sa ises a nonlinea Ricca i equa ion:
dZ
dx=A(x;ω) + B(x;ω)Z+C(x;ω)Z2,
whe e
A, B, C
a e de e mined by medium pa ame e s. Fo laye ed media o disc e e la ice
models,
Z
jumps be ween laye s acco ding o Möbius ans o ma ions:
Zn+1 =anZn+bn
cnZn+dn
,anbn
cndn∈SL(2,C).
The co e equi emen o he sel - e e en ial eedback s uc u e is: a an eec i e bounda y
poin
x= 0
, he inpu impedance
Zin(ω)
ex e nally exhibi ed by he local s uc u e mus equal
he load impedance
Zloop(ω)
seen by he in e nal loop. This sel -consis ency condi ion can be
w i en in disc e e ep esen a ion as
Zin = Φ(Zin),
whe e
Φ
is a composi e mapping consis ing o Möbius ans o ma ions and eedback phases om
mul iple laye s.
Deni ion 2.1
(Sel -Re e en ial Sca e ing Uni )
.
A a gi en equency
ω
, a local s uc u e is
called a sel - e e en ial sca e ing uni i i s ex e nal equi alen inpu impedance
Zin(ω)
is a xed
poin o some complex Möbius ans o ma ion
Φ
:
Zin(ω) = ΦZin(ω),Φ(z) = Az +B
Cz +D, AD −BC = 1.
The s abili y o his xed poin is de e mined by he modulus o
Φ′Zin
.
2.3 Ricca i Fixed Poin and Squa e-Roo Disc iminan
In gene al, he Möbius ans o ma ion xed-poin equa ion
Z=AZ +B
CZ +D
can be educed o a quad a ic equa ion
CZ2+ (D−A)Z−B= 0,
whose solu ion is
Z±=A−D±p(A−D)2+ 4BC
2C,
p o ided
C= 0
. Thus, he equi alen impedance o any sel - e e en ial sca e ing uni na u ally
ca ies a squa e- oo b anch s uc u e, whose disc iminan
∆=(A−D)2+ 4BC
de e mines he physical p ope ies o he wo b anch solu ions h ough i s phase and modulus.
Fo lossless sys ems,
(A, B, C, D)
belong o an app op ia e ep esen a ion o
SU(1,1)
o
SL(2,R)
,
4
∆
lies on a ce ain cu e in he complex plane, and he choice o squa e- oo unc ion
√∆
co esponds o wo ypes o bounda y condi ions.
We iew his squa e- oo s uc u e as he emb yonic o m o a spino : he obse able
impedance
Z
co esponds o he squa e o some ampli ude a iable
ζ
, i.e.,
Z=F(ζ2),
and he mul i aluedness o
ζ
unde closed pa hs in pa ame e space will de e mine he exchange
s a is ics.
2.4 Congu a ion Space o Iden ical Sel -Re e en ial Exci a ions
Conside
N
well-sepa a ed sel - e e en ial sca e ing uni s in h ee-dimensional space, wi h cen e
posi ions
x1,...,xN∈R3
. Igno ing in e nal s uc u al de ails, he geome ic congu a ion space
is
QN=R3N ∆
SN
,
whe e
∆
is he diagonal subse whe e pa icle posi ions coincide, and
SN
is he pe mu a ion
g oup. Fo
d≥3
dimensions, he undamen al g oup o
QN
is
SN
, whose elemen s can be
gene a ed by exchanges o adjacen pa icles.
In s anda d quan um mechanics, he many-body wa e unc ion
Ψ
is iewed as a complex-
alued single- alued unc ion on
QN
. In opological spins a is ics schemes,
Ψ
is iewed as
a sec ion o a line bundle o ec o bundle, wi h die en bundle s uc u es co esponding o
die en exchange s a is ics.
In ou scheme, we will use he Ricca i squa e- oo s uc u e in e nal o each sel - e e en ial
sca e ing uni o cons uc a na u al double co e space
e
QN
o e
QN
, and show ha exchange
pa hs ha e
Z2
holonomy on
e
QN
.
3 Main Resul s (Theo ems and Alignmen s)
This sec ion p esen s he co e esul s o his pape , ocused on h ee le els:
1. The squa e- oo s uc u e o sel - e e en ial sca e ing uni s and he Ricca i equa ion;
2. The spino double co e induced by he squa e- oo s uc u e;
3. The
Z2
phase om pa icle exchange and i s co espondence wi h Fe mi s a is ics.
3.1 Sel -Re e en ial Sca e ing and Ricca i Squa e Roo
Theo em 3.1
(Squa e-Roo Disc iminan o Sel -Re e en ial Sca e ing Uni )
.
Suppose he
equi alen ans e ma ix o a sel - e e en ial sca e ing uni is
M=A B
C D∈SL(2,C),
and i s ex e nal inpu impedance
Zin
is a xed poin o he Möbius ans o ma ion, i.e.,
Zin =
Φ(Zin)
, wi h
Φ(z)=(Az +B)/(Cz +D)
. Assume
C= 0
and he sys em is lossless, so ha he
spec um o
M
lies on he uni ci cle. Then:
1.
Zin
sa ises he quad a ic equa ion
CZ2+ (D−A)Z−B= 0.
5

2. The e exis s a disc iminan
∆=(A−D)2+ 4BC
such ha
Zin =Z±=A−D±√∆
2C,
whe e
√∆
is he wo- alued squa e- oo unc ion on he complex plane.
3. I
M
belongs o
SU(1,1)
o an equi alen ep esen a ion, hen exac ly one o he wo b anch
solu ions
Z±
co esponds o a s able xed poin (
|Φ′(Z)|<1
), and he o he o an uns able
xed poin (
|Φ′(Z)|>1
).
The e o e, he ex e nal esponse o any s able sel - e e en ial sca e ing uni can be equi a-
len ly cha ac e ized by a choice o one o a pai o squa e- oo a iables
±√∆
.
P oo Ske ch.
W i ing he xed-poin condi ion as a quad a ic equa ion immedia ely yields he
disc iminan and wo- alued solu ion; he s abili y condi ion is de e mined by he modulus o
he de i a i e o he Möbius ans o ma ion. The lossless condi ion cons ains he spec um o
M
, he eby cons aining he phase and modulus o
∆
, so ha only one b anch is s able. See
Appendix A o de ails.
3.2 Eme gence o Spino Double Co e
Deni ion 3.2
(Spino In e nal Va iable)
.
Fo each sel - e e en ial sca e ing uni , we in oduce
an in e nal a iable
ζ
such ha
√∆=Λζ2,
whe e
Λ∈C×
is a nonze o cons an ela ed o he specic implemen a ion. Dene a no malized
spino  a iable
χ=ζ
∥ζ∥,
whose o e all phase edundancy is ega ded as a gauge eedom.
Thus, a s able sel - e e en ial sca e ing uni can be desc ibed by ei he he equi alen impedance
Zin
o he spino a iable
χ
, which sa is y
Zin =F(χ2),
whe e
F
is an explici a ional unc ion. No e ha
χ
and
−χ
co espond o he same
Zin
.
Theo em 3.3
(In e nal Spino Double Co e )
.
Unde he abo e assump ions, he in e nal s a e
space o a single sel - e e en ial sca e ing uni can be iewed as a quo ien space
Sin ∼
=C2 {0}/{χ∼ −χ},
and he e exis s a na u al mapping
πin :C2 {0}→Sin , πin (χ)=[χ],
whose ke nel is
{±1}
. Unde app op ia e gauge choices and coa se-g aining, he
SU(2)
o a ion
ep esen a ion on
C2
p ojec s ia
πin
o he
SO(3)
o a ion ep esen a ion on
Sin
, and hus he
in e nal deg ees o eedom a e na u ally o ganized in o a double co e s uc u e o
Spin(3)
.
This s uc u e is comple ely isomo phic o s anda d spino heo y:
χ
plays he ole o a
spin-
1/2
spino , while
Zin
and obse able sca e ing phases co espond o quad a ic in a ian s.
6
3.3 Pa icle Exchange and
Z2
Exchange Phase
Theo em 3.4
(Fe mionic Exchange S a is ics o Sel -Re e en ial Exci a ions)
.
Conside
N
iden-
ical sel - e e en ial sca e ing uni s in h ee-dimensional space, wi h congu a ion space
QN
as
dened in Sec ion 2.4. Le
e
QN
be he double co e induced by he in e nal spino a iables:
e
QN=n(x1,...,xN;χ1, . . . , χN)o∼,
whe e he equi alence ela ion iden ies
χj∼ −χj
o each pa icle, while equi ing in a iance
o he ex e nal
Zin
. Then:
1.
e
QN
is a double co e space o
QN
, whose co e ing ans o ma ion g oup is gene a ed by
simul aneously changing he sign o all
χj
, o ming a
Z2
.
2. Fo any pai o pa icles
i, j
, hei exchange ope a ion co esponds o a closed pa h
γij
on
QN
. On
e
QN
,
γij
li s o wo pa hs
eγ±
ij
, whose endpoin s die by a global sign change:
eγ+
ij (1) = −eγ−
ij (1).
3. I we ega d he many-body s a e as a sec ion o a line bundle on
e
QN
and equi e ha his
sec ion changes sign unde he co e ing ans o ma ion, hen pa allel anspo along
γij
gi es he wa e unc ion a phase o
(−1)
:
Ψ(γij(1)) = −Ψ(γij(0)).
The e o e, in his quan iza ion scheme, he geome ic ealiza ion o pa icle exchange neces-
sa ily co esponds o e mionic an isymme ic s a is ics.
P oo Ske ch.
This is a conc e e ealiza ion o he Finkels einRubins ein scheme on sel - e e en ial
spino in e nal deg ees o eedom. The key is: iewing pa icle winding pa hs as closed cu es
on
e
QN
, hei li in he co e ing space has non i ial
Z2
holonomy. Choosing a line bundle whe e
he co e ing ans o ma ion co esponds o wa e unc ion sign change yields Fe mi s a is ics. See
Appendix B o de ails.
4 P oo s
This sec ion p o ides p oo ou lines o he abo e heo ems, wi h echnical de ails expanded in
he appendices.
4.1 P oo o Theo em 3.1
F om he xed-poin condi ion
Z=AZ +B
CZ +D
we ob ain
CZ2+ (D−A)Z−B= 0.
This is a quad a ic equa ion in
Z
; as long as
C= 0
we can w i e he explici solu ion
Z±=A−D±√∆
2C,∆=(A−D)2+ 4BC.
7
Assuming he sys em is lossless, i.e.,
M∈SU(1,1)
o a simila g oup, means ha he
eigen alues o
M
lie on he uni ci cle, and
M
is closely ela ed o he in insic phase o he
co esponding sca e ing ma ix
S
. The Möbius ans o ma ion
Φ(z) = Az +B
Cz +D
has de i a i e
Φ′(z) = 1
(Cz +D)2.
Subs i u ing
Z±
, we can e alua e
|Φ′(Z±)|
. Unde he lossless condi ion, he moduli o
(CZ±+D)
a e ecip ocals, so exac ly one o he wo b anch solu ions sa ises
|Φ′(Z)|<1
,
co esponding o a s able xed poin and a s able sel - e e en ial sca e ing s uc u e; he o he
b anch is uns able, co esponding o a nonphysical solu ion o exci ed s a e. A de ailed analysis
compa ing wi h he a iable phase me hod and Le inson heo em is gi en in Appendix A.
4.2 P oo o Theo em 3.3
The disc iminan
∆
is an in a ian o he ace and de e minan o
M
; in he lossless case,
∆
ypically lies on a complex plane cu e passing h ough he o igin. Fo each equency
ω
and
momen um
k
, we can w i e
∆(ω, k)=Λ2(ω, k)ζ4(ω, k),
whe e
Λ= 0
is a gauge choice. Thus
√∆=Λζ2,
and
Zin
can be ew i en as a a ional unc ion
F(ζ2)
. Viewing
ζ
as coo dina es o a wo-
dimensional complex ec o , we in oduce no maliza ion
χ=ζ
∥ζ∥∈C2 {0},
and iden i y
χ∼ −χ
o ob ain he quo ien s uc u e o he in e nal s a e space. S anda d g oup
heo y esul s show ha he na u al ep esen a ion o
SU(2)
on
C2
p ojec s ia he quo ien
map o he ep esen a ion o
SO(3)
on
S2
, and
SU(2)
is he double co e o
SO(3)
. Thus, he
in e nal s a e has a double co e s uc u e comple ely equi alen o a spin-
1/2
spino .
4.3 P oo o Theo em 3.4
The opological p ope ies o
QN
ha e ma u e conclusions: in h ee-dimensional space, he
undamen al g oup o
QN
is he pe mu a ion g oup
SN
, whose gene a o s can be iewed as
adjacen pa icle exchange pa hs. The Finkels einRubins ein p oo shows: i he e exis s a
double co e
e
QN→QN
wi h co e ing ans o ma ion g oup
Z2
, and
2π
spa ial o a ions and
pa icle exchange pa hs a e ela ed o non i ial closed loops in
e
QN
, hen one can cons uc a
line bundle such ha he many-body s a e, as a sec ion o his line bundle, changes sign unde
he co e ing ans o ma ion, hus ealizing Fe mi s a is ics.
In ou cons uc ion, each pa icle ca ies an in e nal spino a iable
χj
, wi h
χj∼ −χj
co esponding o he same physical impedance. Combining he in e nal a iables o all pa icles
na u ally yields
e
QN
. E olu ion along he exchange pa h
γij
no only winds in posi ion space, bu
also winds a ound he squa e- oo b anch cu in in e nal pa ame e space, causing he o e all
phase o
(χi, χj)
o unde go a
2π
winding. This co esponds o non i ial
Z2
holonomy on
e
QN
.
Choosing a line bundle whe e he co e ing ans o ma ion co esponds o wa e unc ion sign
change yields
Ψ(. . . , xi,xj, . . . ) = −Ψ(. . . , xj,xi, . . . ).
The e o e, sel - e e en ial sca e ing exci a ions na u ally ealize e mionic exchange s a is-
ics.
8
5 Model Apply
This sec ion shows how o implemen he abo e sel - e e en ial sca e ing s uc u e in conc e e
QCA models, and ela e i o Di ac mass and opological impedance.
5.1 Sel -Re e en ial De ec s in One-Dimensional Di acQCA
Conside a one-dimensional Di ac- ype QCA, whose single-s ep e olu ion can be w i en as a
quan um walk:
U=S+⊗|↑⟩⟨↑|+S−⊗|↓⟩⟨↓|◦(I⊗C),
whe e
S±
shi he s a e le o igh by one s ep, and
C
is a
2×2
coin ma ix, e.g.,
C(θ) = cos θsin θ
−sin θcos θ.
In momen um ep esen a ion, he eigen alues o
U(k)
a e
e∓iω(k)
, sa is ying he dispe sion
ela ion
cos ω(k) = cos θcos(ka),
which in he long-wa eleng h limi yields an eec i e Di ac equa ion, wi h mass
m
ela ed o
θ
.
Agains his backg ound, modi ying he coin ma ix o in oducing a local loop a a single si e
o ni e sub-chain can ealize an eec i e sca e ing cen e . Fo modes wi h wa eleng h much
la ge han he de ec egion size, hei sca e ing is cha ac e ized by a
2×2
single-channel
sca e ing ma ix
S(k) =  (k) ′(k)
(k) ′(k).
By explici ly modeling he in e nal loop o he de ec egion as addi ional bounda y and
eedback channels, i s equi alen impedance can be w i en as a unc ion
Z(x;k)
sa is ying a
Ricca i equa ion, whose alue a he de ec egion pe iphe y gi es
(k)
. Rela ed echniques a e
closely ela ed o he a iable phase me hod.
5.2 Mass, Bound S a es, and Sel -Re e en ial Feedback
Fo de ec s wi h sel - e e en ial eedback, he e exis ce ain equencies
ω
and momen a
k
such
ha he pole condi ion
1− (ω)eiθloop = 0
holds, whe e
θloop
is he addi ional loop phase. These poles co espond o bound o quasi-bound
s a es, which in he con inuum limi o QCA mani es as localized pa icles whose equency
de ia es om he massless mode dispe sion ela ion, dening an eec i e mass
m
.
On he o he hand, he eec ion coecien can be w i en as
(ω) = eiδ(ω),
whe e
δ(ω)
is he sca e ing phase. Th ough he Le inson heo em and a iable phase me hod,
δ(ω)
can be ela ed o he numbe o bound s a es and Ricca i phase unc ions. The pole
condi ion can be ew i en as a sel -consis en equa ion o impedance, whose solu ion has a
wo- alued squa e- oo s uc u e, hus in oducing a na u al in e nal spino a iable o he
bound s a e.
9
B.3 FR Da a o Sel -Re e en ial Sca e ing Exci a ions
In his pape 's cons uc ion, he Ricca i disc iminan squa e oo in e nal o each sel - e e en ial
sca e ing uni p o ides a na u al wo- alued deg ee o eedom
χ∼ −χ
. Combining he posi-
ions and in e nal spino s o all pa icles yields
e
QN=n(x1,...,xN;χ1, . . . , χN)o∼,
whose p ojec ion on o
QN
o ge s all
χj
. The li o exchange pa hs on
e
QN
has he ollowing
p ope y:

Pe o ming spa ial exchange alone wi hou changing in e nal spino s co esponds o a
closed pa h;

Howe e , due o he exis ence o he Ricca i squa e- oo b anch cu , adiaba ic e olu ion
along his pa h causes he o e all phase o
(χi, χj)
o wind a ound he o igin once, hus
co esponding o a non i ial elemen on
e
QN
.
The e o e, he sel - e e en ial sca e ing s uc u e au oma ically p o ides he double co e
and holonomy da a equi ed by he FR scheme. Choosing a quan iza ion ule whe e he co e ing
ans o ma ion co esponds o wa e unc ion sign change yields Fe mi s a is ics; choosing he
co e ing ans o ma ion as i ial ac ion yields Bose s a is ics. This pape emphasizes: unde he
na u al dynamics o massi e sel - e e en ial exci a ions, s abili y and uni a i y equi e choosing
he o me , hus locking massi e and Fe mi s a is ics oge he in his cons uc ion.
C Rema ks on 2+1 Dimensions and Anyonic Gene aliza ions
Al hough his pape p ima ily ocuses on h ee-plus-one dimensions, in wo-plus-one dimen-
sions, sel - e e en ial sca e ing and Ricca i squa e- oo s uc u e may p oduce iche s a is ical
beha io .
In wo-plus-one dimensions, he undamen al g oup o he
N
-pa icle congu a ion space
is he b aid g oup
BN
, whose ep esen a ions can yield anyonic s a is ics, allowing con inuous
in e pola ion o phase o ma ix ep esen a ions du ing pa icle exchange be ween bosons and
e mions. In sel - e e en ial sca e ing ne wo ks, he combina ion o he mul i aluedness o
in e nal spino a iables
χ
and b aid g oup ep esen a ions p omises o gi e a conc e e ealiza ion
o a class o sel - e e en ial anyons: hei s a is ical phase is no longe limi ed o
±π
, bu is
ela ed o he winding numbe o he Ricca i disc iminan in pa ame e space.
Sys ema ic analysis o his case equi es ex ending his pape 's QCAsca e ing cons uc ion
o wo-dimensional la ices and pe o ming unied modeling o he opology o mul i-pa icle
b aid pa hs and he geome ic s uc u e o in e nal sel - e e en ial eedback, le o u u e wo k.
16