Mass as Topological Impedance: Self-Referential Scattering and Chiral Symmetry Breaking in Dirac-QCA

Mass as Topological Impedance: Sel -Re e en ial
Sca e ing and Chi al Symme y B eaking in
Di ac-QCA
Haobo Ma
1
Wenlin Zhang
2
1
Independen Resea che
2
Na ional Uni e si y o Singapo e
No embe 24, 2025
Abs ac
In ela i is ic quan um eld heo y, he masses o elemen a y pa icles a e usu-
ally in oduced ia Yukawa couplings o he Higgs eld and appea as ee pa am-
e e s, wi h no mic oscopic in o ma ion- heo e ic o opological o igin. Wi hin a
quan um cellula au oma on (QCA) on ology, howe e , he uni e se is modelled as
a disc e e, s ic ly causal quan um upda e ule on a la ice, wi h an in insic speed
limi
c
se by he ligh -cone o local uni a ies. This aises a s uc u al ques ion:
why do ce ain exci a ions p opaga e subluminally and beha e as massi e pa icles,
a he han as massless signals ha sa u a e he upda e speed?
In his wo k we ea a one-dimensional Di ac- ype QCA, o mula ed as a disc e e-
ime quan um walk wi h in e nal (coin) deg ees o eedom. Building on ea lie e-
sul s ha show he Di ac equa ion eme ging as he con inuum limi o such models,
we p o e ha he eec i e Di ac mass is no hing bu he ampli ude o a locally
sel - e e en ial sca e ing p ocess ha mixes le - and igh -mo ing componen s. A
he le el o he Floque ope a o in momen um space, his mixing induces a non-
i ial winding o he Bloch ec o o e he B illouin zone, placing he model in
a chi al-symme ic Floque opological phase. The non-ze o mass hen acqui es
he in e p e a ion o a opological impedance: a opologically p o ec ed obs uc ion
ha p e en s he decoupling o chi al componen s in o pu ely ballis ic, ligh like
p opaga ion.
We show ha his mechanism p oduces Zi e bewegung as an una oidable con-
sequence o he local back-sca e ing. In he Di ac con inuum limi he posi ion
ope a o spli s in o a uni o m d i e m wi h g oup eloci y
ex
and an oscilla o y
e m wi h equency
ωZB = 2E/ℏ
, whe e
E
is he quasiene gy. We in e p e his
oscilla ion as "in e nal mo ion" and demons a e a p ecise decomposi ion
2
ex + 2
in =c2,
whe e
in
is dened as he s anda d de ia ion o he eloci y ope a o . Thus e e y
exci a ion sa u a es he mic oscopic in o ma ion speed bound
c
, bu a opologically
en o ced sha e o his budge is apped in in e nal oscilla ions whene e he mass
gap is non-ze o. This p o ides a pu ely kinema ical, opological and in o ma ion-
heo e ic in e p e a ion o mass in Di ac-QCA models, compa ible wi h and com-
plemen a y o con en ional Higgs- ype mass gene a ion.
1
Keywo ds:
Quan um cellula au oma on; disc e e- ime quan um walk; Di ac equa ion;
opological mass; winding numbe ; chi al symme y; Zi e bewegung; Floque opological
phase; in o ma ion speed conse a ion

1 In oduc ion & His o ical Con ex
1.1 Mass and i s open concep ual s a us
In ela i is ic quan um eld heo y (QFT), he ee Di ac equa ion
(iℏγµ∂µ−mc)ψ= 0
in oduces he es mass
m
as a pa ame e in he Lag angian, while in he S anda d
Model masses a ise om Yukawa couplings o he Higgs eld. Al hough his mechanism
is phenomenologically success ul, i lea es wo concep ual issues:
1. The nume ical alues o e mion masses a e ee pa ame e s, no xed by deepe
p inciples.
2. The mass e m
m¯
ψψ
b eaks chi al symme y explici ly, bu his b eaking is no di-
ec ly ied o any opological in a ian o disc e e in o ma ion-p ocessing s uc u e.
Ou side he S anda d Model, a numbe o " opological mass" mechanisms a e known,
such as Che nSimons e ms in
(2 + 1)
-dimensional gauge heo ies, o domain-wall and
o e lap e mions in la ice QCD, whe e a mass gap is associa ed wi h index heo ems
and spec al ow. Howe e , hese mechanisms s ill p esuppose a con inuum quan um
eld, and hey a e usually o mula ed in e ms o eec i e ac ions a he han unde lying
disc e e in o ma ion dynamics.
1.2 QCA and quan um walks as disc e e Di ac dynamics
Quan um cellula au oma a (QCAs) p o ide an al e na i e on ology in which dynam-
ics is dened by a causal, ansla ion-in a ian uni a y on a la ice o ni e-dimensional
quan um sys ems. The idea ha quan um eld heo y migh eme ge om such au oma a
da es back o Feynman's sugges ion ha physics could be simula ed by quan um com-
pu e s, and has since been de eloped in de ail by Bisio, D'A iano, Tosini and o he s,
who cons uc QCAs whose long-wa eleng h limi ep oduces Weyl, Di ac and Maxwell
equa ions.
In one spa ial dimension, a pa icula ly simple ealiza ion o Di ac- ype dynamics is
p o ided by disc e e- ime quan um walks (DTQWs), in which a wo-componen "coin"
deg ee o eedom con ols condi ional shi s o le and igh . S auch showed ha o
sui able scaling o he coin pa ame e s and la ice spacing he con inuum limi o a DTQW
ep oduces he
(1 + 1)
-dimensional Di ac equa ion, and subsequen wo k gene alized his
connec ion o highe dimensions and mo e gene al walk p o ocols.
In pa allel, a sys ema ic heo y o QCAs has been de eloped, including locali y and in-
dex heo ems classi ying one-dimensional au oma a up o homo opy. Wi hin his se ing,
DTQWs can be iewed as special cases o one-dimensional QCAs wi h in e nal deg ees
o eedom (coins) and ni e p opaga ion adius.
2
1.3 Topological phases and quan um walks
Topological band heo y,  s de eloped o s a ic Hamil onians, has been ex ended o
pe iodically d i en (Floque ) sys ems, whe e he ime-e olu ion ope a o o e one pe-
iod plays he ole o a Floque uni a y. The opological classica ion o such uni a ies,
especially in he p esence o chi al symme y, na u ally applies o DTQWs. Ki agawa
and collabo a o s showed ha spli -s ep and mul i-s ep quan um walks ealize a ich se
o one-dimensional opological phases wi h in ege winding numbe s, accompanied by
obus edge s a es a bounda ies be ween domains o die en in a ian s.
Asbó h and Obuse la e o mula ed a bulk-bounda y co espondence and a sys ema ic
deni ion o chi al symme y o Floque quan um walks, iden i ying a
Z×Z
alued
opological in a ian con olling ze o and
π
quasiene gy edge s a es. Expe imen s using
pho onic and cold-a om pla o ms ha e di ec ly obse ed hese opological bound s a es
and measu ed winding numbe s and dynamical opological o de pa ame e s.
Impo an ly o ou pu poses, e en he simples one-coin quan um walk, when o -
mula ed wi h an app op ia e coin o a ion, exhibi s a hidden chi al symme y and a
non- i ial winding numbe o gene ic coin angles, wi h he gap closing and winding
changing only a special alues o he coin pa ame e .
1.4 Zi e bewegung in Di ac dynamics and QCAs
Zi e bewegung (ZB), he apid oscilla o y mo ion p edic ed by he Di ac equa ion, has
long been unde s ood as a ising om in e e ence be ween posi i e and nega i e ene gy
componen s o a wa e packe . In he Heisenbe g pic u e he posi ion ope a o spli s
in o a uni o m d i e m plus an oscilla o y e m wi h equency
2E/ℏ
, and he eloci y
ope a o has eigen alues
±c
, so ha he eloci y uc ua es be ween hese ex eme alues
e en when a e age mo ion is subluminal.
In QCA and quan um walk models ha app oxima e Di ac dynamics, he same phe-
nomenon has been analysed bo h analy ically and nume ically: he Di ac QCA con-
s uc ed by Bisio e al. exhibi s ZB and Klein unnelling, wi h oscilla ion equency and
ampli ude ma ching hose o he con inuum Di ac heo y in he app op ia e limi .
1.5 In o ma ion speed conse a ion and he ole o mass
In a p e ious wo k we p oposed an in o ma ion- heo e ic conse a ion law o local exci-
a ions in QCA- ype models, namely
2
ex + 2
in =c2,
whe e
ex
measu es he eec i e g oup eloci y o an exci a ion in physical space and
in
quan ies he in e nal e olu ion a e in Hilbe space, in e ms o he a iance o he
eloci y ope a o o he FubiniS udy me ic. This ela ion exp esses he idea ha he
"in o ma ion speed budge " is xed a
c
, wi h massless exci a ions alloca ing all o i o
ansla ional mo ion, while massi e ones di e pa o i in o in e nal oscilla ions.
The p esen pape implemen s his idea conc e ely in a Di ac-QCA se ing and ies
i o opological in a ian s.
3
1.6 Goals and con ibu ions
The cen al objec i e is o ein e p e he mass pa ame e o he Di ac equa ion, as
ealized in a Di ac-QCA model, as a opological impedance: a quan ized obs uc ion o
decoupling le - and igh -mo ing deg ees o eedom in o independen ballis ic channels.
Conc e ely, we:
1. Speci y a one-dimensional Di ac- ype QCA as a disc e e- ime quan um walk wi h
a single coin angle and show, unde s anda d locali y and homogenei y assump ions, ha
i s long-wa eleng h limi ep oduces he
(1+1)
-dimensional Di ac equa ion wi h mass
m
de e mined by he coin angle.
2. Show ha he imes ep uni a y in momen um space has chi al symme y and
denes a non- i ial winding o he Bloch ec o on he Bloch sphe e o any non-ze o coin
angle, in ag eemen wi h ea lie classica ions o Floque opological phases in quan um
walks. This winding numbe changes only when he gap a quasiene gies
0
o
π
closes,
co esponding o
m= 0
.
3. Gi e a sca e ing- heo e ic in e p e a ion o he coin angle as a local sel - e e en ial
sca e ing ampli ude ha epea edly con e s le -mo ing componen s in o igh -mo ing
ones and ice e sa. This p oduces subluminal g oup eloci ies and Zi e bewegung,
and we in e p e he associa ed ene gy gap as a opological impedance blocking pu ely
ligh like p opaga ion.
4. De i e in he Di ac con inuum limi an exac decomposi ion
2
ex + 2
in =c2
o a bi a y s a es, by dening
ex
as he expec a ion alue o he eloci y ope a o
and
in
as he squa e oo o i s a iance, he eby gi ing a p ecise in o ma ion- heo e ic
meaning o in e nal mo ion and linking i o mass.
5. Discuss how he opological impedance pic u e in e ac s wi h con en ional mass
gene a ion mechanisms, and p opose expe imen al schemes o obse ing he ade-o
be ween g oup eloci y, in e nal oscilla ions and opological in a ian s in pho onic and
cold-a om quan um walk pla o ms.

2 Model & Assump ions
2.1 La ice, Hilbe space and locali y
We conside a one-dimensional spa ial la ice wi h spacing
a > 0
, whose si es a e labelled
by in ege s
n∈Z
, co esponding o posi ions
x=na
. Time is disc e e wi h s ep
∆ > 0
,
so ha
=m∆
o
m∈Z
. The mic oscopic causal speed is xed by
c=a
∆ ,
which will be iden ied wi h he eme gen speed o ligh in he con inuum limi .
A each la ice si e he local Hilbe space is a wo-dimensional coin space
C2
, in e -
p e ed as a chi al o le / igh deg ee o eedom. The global Hilbe space is
H=ℓ2(Z)⊗C2.
4
We w i e s a es in he posi ion-coin basis as
Ψ = X
n∈Z
(ψL(n)|n⟩⊗|L⟩+ψR(n)|n⟩⊗|R⟩),
o in column ec o o m
Ψ(n, ) = ψL(n, )
ψR(n, ).
We assume a single-s ep, ansla ion-in a ian QCA e olu ion ope a o
U:H → H,
which is uni a y, homogeneous (commu es wi h spa ial ansla ions), causal wi h -
ni e p opaga ion adius (each si e couples only o a bounded neighbou hood in a single
imes ep) and pa i y symme ic.
2.2 Coin ope a o and condi ional shi
The imes ep uni a y is aken o ac o in o a local coin o a ion ollowed by a condi ional
shi :
U=S C(θ),
whe e
θ∈[0, π]
is a eal pa ame e .
The coin ope a o is a si e-local uni a y o he o m
C(θ) = X
n∈Z
|n⟩⟨n| ⊗ R(θ),
whe e
R(θ)=e−iθσx=cos θ−i sin θ
−i sin θcos θ
and
σx
is he usual Pauli ma ix.
The condi ional shi ope a o mo es le and igh componen s in opposi e di ec ions:
S=X
n∈Z
(|n−1⟩⟨n|⊗|L⟩⟨L|+|n+ 1⟩⟨n|⊗|R⟩⟨R|).
Equi alen ly,
S=X
n∈Z
|n⟩⟨n| ⊗ (|L⟩⟨L|T++|R⟩⟨R|T−),
whe e
T±
a e la ice ansla ion ope a o s (
T±|n⟩=|n±1⟩
).
The e olu ion equa ion o e one imes ep is
Ψ( + ∆ ) = UΨ( ).
In posi ion space his yields coupled upda e equa ions:
ψL(n, + ∆ ) = cos θ ψL(n+ 1, )−i sin θ ψR(n+ 1, ),
ψR(n, + ∆ ) = −i sin θ ψL(n−1, ) + cos θ ψR(n−1, ).
We in e p e he pa ame e
θ
as con olling he s eng h o a local "sel -sca e ing" be-
ween le - and igh -mo ing componen s;
θ= 0
co esponds o pu ely ballis ic mo ion
a speed
c
, while
θ= 0
in oduces back-sca e ing.
5

2.3 Momen um-space ep esen a ion and eec i e Hamil onian
By ansla ion in a iance, we can pass o momen um space ia
Ψ(n, ) = Zπ
−π
dk
2πeikn e
Ψ(k, ),
wi h
k∈[−π, π]
he dimensionless la ice momen um; he physical momen um is
p=
ℏk/a
.
The shi ope a o is diagonal in
k
,
e
S(k) = e−ikσz,
and he imes ep uni a y in momen um space is
e
U(k) = e
S(k)R(θ) = e−ikcos θ−ie−iksin θ
ieiksin θeikcos θ.
We dene he Floque quasiene gies
E(k)
and eec i e Hamil onian
He (k)
ia
e
U(k) = exp −i
ℏHe (k) ∆ .
The eigen alues o
e
U(k)
a e
e−iE(k)∆
and
e+iE(k)∆
, wi h
cosE(k)∆ = cos θcos k,
a well-known dispe sion ela ion o coined quan um walks.
We can w i e
He (k) = E(k)ˆ
n(k)·σ,
wi h
ˆ
n(k)
a uni ec o on he Bloch sphe e.

3 Main Resul s (Theo ems and Alignmen s)
In his sec ion we summa ize he main heo ems; de ailed p oo s a e gi en in he subse-
quen sec ion and in he appendices.
Theo em 3.1
(Di ac con inuum limi )
.
Le he la ice spacing and imes ep obey
a=
c∆
, and scale he coin angle as
θ=mc2
ℏ∆ +O(∆ 3),
wi h xed pa ame e s
c > 0
and
m≥0
. Conside ini ial s a es whose ampli udes a y
slowly on he la ice scale, in he sense ha
ψL/R(n±1,0) −ψL/R(n, 0) = O(a)
uni o mly in
n
. Then in he limi
∆ →0
he disc e e e olu ion con e ges, up o e o s
o o de
O(∆ 2)
, o he con inuum Di ac equa ion in
(1 + 1)
dimensions,
iℏ∂ Ψ(x, ) = −iℏc σz∂x+mc2σxΨ(x, ),
wi h
x=na
. The con e gence holds in ope a o no m on any ni e ime in e al and o
wa e packe s suppo ed in a compac momen um egion.
6
Theo em 3.2
(Chi al symme y and winding numbe )
.
Dene a chi al symme y ope -
a o
Γ = σx.
Then o all momen a
k
he Floque uni a y sa ises
Γe
U(k) Γ = e
U†(k),
so he QCA is in he chi al-symme ic Floque class AIII. The co esponding one-dimensional
opological in a ian (winding numbe ) is
W=1
2πiZπ
−π
dkT Γe
U−1(k)∂ke
U(k).
Fo he p esen model,
W=(0, θ = 0
o
θ=π,
1,0< θ < π.
Thus any non-ze o mass pa ame e
m
, as dened in Theo em 1, co esponds o a opo-
logically non- i ial Floque phase ha canno be adiaba ically de o med o he massless
case wi hou closing he quasiene gy gap a
E= 0
o
E=π/∆
. This is consis en wi h
ea lie analyses o opological phases in one-coin and spli -s ep quan um walks.
Theo em 3.3
(Mass as sel - e e en ial sca e ing)
.
In posi ion space, he single-s ep e o-
lu ion a si e
n
can be w i en as he ac ion o a local sca e ing ma ix on an incoming
wo-componen eld comp ised o le - and igh -mo ing ampli udes. Explici ly, in mo-
men um space, which is equi alen o a wo-channel uni a y sca e ing wi h eec ion and
ansmission ampli udes
(θ) = −i sin θ, (θ) = cos θ.
Fo a wa e packe sha ply peaked a ound momen um
k0
, he g oup eloci y sa ises
ex (k0, θ) = ∂E(k)
∂p k=k0
=1
ℏ
∂E(k)
∂k a,
wi h dispe sion
cos(E∆ ) = cos θcos k
. In he Di ac con inuum limi desc ibed in Theo-
em 1, his yields
ex (p)≈c2p
pp2c2+m2c4
and an ene gy gap
E(p)≈pp2c2+m2c4.
Thus he same pa ame e
θ
simul aneously con ols (i) he size o he mass gap in he
Di ac limi and (ii) he s eng h o local back-sca e ing be ween le - and igh -mo ing
componen s. Repea ed applica ion o his local sca e ing p oduces an eec i e ine ia:
he s onge he mixing (la ge
θ
), he smalle he asymp o ic g oup eloci y a gi en
momen um. We in e p e his as mass a ising om a sel - e e en ial sca e ing p ocess
ha ies he exci a ion o i s own his o y.
7
Theo em 3.4
(Zi e bewegung and in o ma ion speed decomposi ion)
.
Le
H=cpσz+
mc2σx
deno e he Di ac Hamil onian in he con inuum limi , and dene he eloci y
ope a o
ˆ =i
ℏ[H, X] = cσz.
Then o any no malized s a e
Ψ
,
⟨ˆ 2⟩=c2,
and we can dene
ex =⟨ˆ ⟩, in =p⟨ˆ 2⟩−⟨ˆ ⟩2.
Fo all s a es,
2
ex + 2
in =c2.
Mo eo e , in he Heisenbe g pic u e he posi ion ope a o decomposes as
X( ) = X(0) + ex + Ξ( ),
whe e
Ξ( )
is an oscilla o y e m wi h equency
ωZB = 2E/ℏ
and ampli ude p opo ional
o
ℏc/(2E)
. This is he usual Zi e bewegung e m, which we he eby in e p e as he
mani es a ion o he in e nal in o ma ion speed
in
. Massi e exci a ions necessa ily ha e
0<| ex |< c
and co esponding non-ze o
in
; massless ones sa is y
| ex |=c
and
in = 0
.
In he disc e e QCA, a simila decomposi ion holds a he le el o he eec i e Hamil-
onian and eloci y ope a o in he long-wa eleng h egime, and Zi e bewegung appea s
as a as oscilla ion o he expec a ion alue o he posi ion ope a o supe imposed on he
g oup- eloci y d i .

4 P oo s
In his sec ion we ske ch he main a gumen s; de ailed s ep-by-s ep de i a ions a e de-
e ed o he appendices.
4.1 P oo o Theo em 1 (Di ac con inuum limi )
We se
a=c∆
and
θ= (mc2/ℏ)∆ +O(∆ 3)
and wo k o  s o de in
∆
. Rew i ing
he posi ion-space upda e equa ions as
ψL(n, + ∆ ) = cos θ ψL(n+ 1, )−i sin θ ψR(n+ 1, ),
ψR(n, + ∆ ) = −i sin θ ψL(n−1, ) + cos θ ψR(n−1, ),
we Taylo -expand all e ms in
∆
. Using
cos θ= 1 + O(∆ 2),sin θ=mc2
ℏ∆ +O(∆ 3),
and
ψL/R(n±1, ) = ψL/R(x±a, ) = ψL/R(x, )±a∂xψL/R(x, ) + O(a2),
8
wi h
x=na
, we ob ain o  s o de
ψL(x, )+∆ ∂ ψL(x, ) = ψL(x, ) + c∆ ∂xψL(x, )−imc2
ℏ∆ ψR(x, ) + O(∆ 2),
ψR(x, )+∆ ∂ ψR(x, ) = ψR(x, )−c∆ ∂xψR(x, )−imc2
ℏ∆ ψL(x, ) + O(∆ 2).
Sub ac ing
ψL/R(x, )
om bo h sides and di iding by
∆
yields
∂ ψL−c∂xψL=−imc2
ℏψR+O(∆ ),
∂ ψR+c∂xψR=−imc2
ℏψL+O(∆ ).
Mul iplying by
iℏ
and collec ing e ms in spino o m
Ψ = (ψL, ψR)T
, we nd
iℏ∂ Ψ = −iℏc σz∂x+mc2σxΨ + O(∆ ),
which is he desi ed Di ac equa ion up o
O(∆ )
co ec ions. A mo e ca e ul analysis using
Fou ie me hods and no m es ima es shows ha he e o emains
O(∆ )
in ope a o
no m on any xed ime in e al o wa e packe s wi h bounded momen um suppo , in
line wi h igo ous con inuum-limi esul s o quan um walks.
4.2 P oo o Theo em 2 (chi al symme y and winding numbe )
We  s e i y chi al symme y. Using
Γ = σx
and he explici o m o
e
U(k)
,
e
U(k) = e−ikcos θ−ie−iksin θ
ieiksin θeikcos θ,
we compu e
Γe
U(k)Γ = σxe
U(k)σx=eikcos θie−iksin θ
ieiksin θe−ikcos θ=e
U†(k),
whe e he las equali y ollows om uni a i y and complex conjuga ion. Thus he chi al
symme y condi ion holds.
To compu e he winding numbe , we w i e
e
U(k)=e−iE(k)ˆ
n(k)·σ
wi h
cos(E∆ ) = cos θcos k,
and a Bloch ec o
ˆ
n(k)
lying on he Bloch sphe e. A con enien pa ame iza ion, used
in Lam's analysis o he Hadama d quan um walk, is
ˆ
n(k) = 1
sin(E∆ )

sin θsin k
sin θcos k
cos θsin k
,
alid away om gap closing poin s whe e
sin(E∆ )= 0
.
The chi al symme y implies ha he ele an winding is ha o he p ojec ion o
ˆ
n(k)
on o he plane o hogonal o
Γ
, which he e is he
(yz)
-plane. As
k
uns om
−π
9
4. In he Di ac limi he eloci y ope a o has eigen alues
±c
, implying ha any
exci a ion sa ises
2
ex + 2
in =c2
, whe e
ex
is he a e age g oup eloci y and
in
quan ies in e nal uc ua ions associa ed wi h Zi e bewegung. Massi e exci a ions
hus di e pa o he xed in o ma ion speed budge in o in e nal mo ion.
5. In e aces whe e he eec i e mass changes sign suppo opologically p o ec ed
bound s a es, na u ally in e p e ed as de ec s in he pa e n o opological impedance.
Expe imen al pla o ms based on pho onic and cold-a om quan um walks can es
hese p edic ions and di ec ly p obe he in e play be ween mass, opology and in-
o ma ion ow.
Taken oge he , hese esul s sugges ha wi hin a QCA on ology, mass need no be an
a bi a y pa ame e inse ed in o he con inuum eld heo y. Ins ead, i can eme ge as
a opologically p o ec ed p ope y o disc e e in o ma ion dynamics, cons aining how
exci a ions alloca e hei ni e in o ma ion speed budge be ween ex e nal p opaga ion
and in e nal sel - e e en ial mo ion.

9 Acknowledgemen s, Code A ailabili y
The au ho acknowledges he exis ing body o wo k on quan um walks, quan um cellula
au oma a and opological phases ha unde pins his s udy, in pa icula he con ibu ions
o S auch, Childs, Ki agawa, Asbó h, Bisio, D'A iano, Tosini, Fa elly and many o he s.
No nume ical simula ions beyond s anda d analy ical calcula ions we e equi ed o
he de i a ions p esen ed he e. Simple quan um-walk simula o s sucien o ep oduce
he dispe sion ela ions, Zi e bewegung and domain-wall bound s a es discussed in his
pape can be implemen ed s aigh o wa dly in s anda d scien ic compu ing en i on-
men s; no dedica ed code eposi o y is p o ided.

A Appendix A: De ailed De i a ion o he Di ac Con-
inuum Limi
A.1 A.1 Scaling and smoo hness assump ions
We adop he scaling
a=c∆ , θ =mc2
ℏ∆ ,
and assume ha he la ice wa e unc ion
Ψ(n, )
a
= 0
is ob ained by sampling a
smoo h con inuum spino
Ψ(x, 0)
a posi ions
x=na
. Explici ly,
Ψ(n, 0) = Ψ(x=na, 0),
wi h
Ψ
wice con inuously die en iable and decaying sucien ly as a inni y.
The goal is o cons uc a con inuum spino
Ψ(x, )
sa is ying he Di ac equa ion such
ha
Ψ(n, ) = Ψ(x=na, ) + O(∆ 2)
o
in a bounded in e al. The e o es ima e can be made p ecise in Sobole no ms,
ollowing me hods used in igo ous con inuum-limi analyses o quan um walks.
16

A.2 A.2 Expansion o he disc e e upda e
We w i e he disc e e upda e equa ions as
Ψ(n, + ∆ ) = UΨ(·, )(n),
whe e
U
ac s on la ice spino s acco ding o
UΨ(n) = cos θ ψL(n+ 1) −i sin θ ψR(n+ 1)
−i sin θ ψL(n−1) + cos θ ψR(n−1).
We now in e p e
Ψ(n, )
as sampling o a con inuum eld
Ψ(x, )
, and Taylo -expand
a ound
x=na
. Using
Ψ(x±a, ) = Ψ(x, )±a∂xΨ(x, ) + a2
2∂2
xΨ(x, ) + O(a3),
and
cos θ= 1 −θ2
2+O(θ4)=1− O(∆ 2),
sin θ=θ+O(θ3) = mc2
ℏ∆ +O(∆ 3),
we ob ain
ψL(x, + ∆ ) = ψL(x, ) + a∂xψL(x, )−imc2
ℏ∆ ψR(x, ) + O(∆ 2),
ψR(x, + ∆ ) = ψR(x, )−a∂xψR(x, )−imc2
ℏ∆ ψL(x, ) + O(∆ 2).
He e we used
a=c∆
and neglec ed e ms o o de
∆ 2
o highe .
Rew i ing his as a  s -o de ime disc e iza ion o a con inuum equa ion,
Ψ(x, + ∆ ) = Ψ(x, )+∆ ∂ Ψ(x, ) + O(∆ 2),
we iden i y
∂ Ψ = −cσz∂xΨ−imc2
ℏσxΨ + O(∆ ),
and hence
iℏ∂ Ψ = −iℏcσz∂x+mc2σxΨ + O(∆ ).
S anda d s abili y and consis ency a gumen s o one-s ep schemes hen show ha he
disc e e e olu ion con e ges o he Di ac e olu ion wi h e o
O(∆ )
o e ni e imes.
B Appendix B: Zi e bewegung in Di ac and QCA Dy-
namics
He e we gi e he s anda d de i a ion o Zi e bewegung o he Di ac Hamil onian and
ou line how i appea s in he QCA.
17
B.1 B.1 Zi e bewegung in he Di ac heo y
Conside he ee Di ac Hamil onian in
(1 + 1)
dimensions,
H=cpσz+mc2σx,
ac ing on spino s
Ψ(x)
. In he Heisenbe g pic u e,
dX( )
d =i
ℏ[H, X( )],dσz( )
d =i
ℏ[H, σz( )].
Using
[p, X] = −iℏ
, one nds
dX( )
d =cσz( ) = ˆ ( ),
and
dσz( )
d =2mc2
ℏσy( ),dσy( )
d =−2
ℏcpσx( ) + mc2σz( ).
Sol ing hese coupled equa ions yields
ˆ ( ) = c2pH−1+ e2iH /ℏˆ (0) −c2pH−1,
and in eg a ing o e ime,
X( ) = X(0) + c2pH−1 +iℏc
2He2iH /ℏ−1ˆ (0) −c2pH−1.
The  s e m ep esen s uni o m mo ion wi h g oup eloci y
ex =c2p/E
, while he
second is an oscilla o y e m wi h equency
ωZB = 2E/ℏ
and ampli ude o o de
ℏc/(2E)
.
Fo wa e packe s consis ing pu ely o posi i e ene gy eigens a es,
ˆ (0)
coincides wi h
c2pH−1
, and he oscilla o y e m anishes; Zi e bewegung a ises only when bo h posi i e
and nega i e ene gy componen s a e p esen .
B.2 B.2 Zi e bewegung in he Di ac-QCA
Fo he QCA, he Heisenbe g equa ions mus be o mula ed wi h espec o he eec i e
Hamil onian
He (k)
o , mo e di ec ly, ia he disc e e- ime Heisenbe g e olu ion
Xm+1 =U†XmU,
wi h
m
labelling imes eps. Wo king in momen um space, one nds ha o wa e packe s
peaked a ound small momen a and small masses, he disc e e e olu ion o
⟨Xm⟩
ep o-
duces he Di ac beha io o good accu acy, including an oscilla o y con ibu ion wi h
equency close o
2E/ℏ
. De ailed calcula ions and nume ical simula ions o he Di ac
au oma on ha e been ca ied ou in p e ious wo k, con ming ha Zi e bewegung is an
in insic ea u e o he QCA dynamics.
This suppo s he in e p e a ion ha Zi e bewegung, bo h in he con inuum and in
he QCA, is he mani es a ion o he in e nal componen
in
o he xed in o ma ion
speed budge , while he g oup eloci y
ex
accoun s o he ne anspo .
C Appendix C: Compu a ion o he Winding Numbe
We b iey de ail he compu a ion o he winding numbe o he one-coin quan um walk
conside ed he e.
18
C.1 C.1 Chi al decomposi ion
Wi h chi al symme y
Γ = σx
and Floque uni a y
e
U(k)
, we can swi ch o a basis in
which
Γ
is diagonal:
Γ = 1 0
0−1,
and
e
U(k)
can be w i en in block o m
e
U(k) = A(k)B(k)
C(k)D(k).
Chi al symme y implies
Γe
U(k)Γ = e
U†(k),
which leads o cons ain s
A=D†
,
B=−B†
,
C=−C†
. In pa icula , he o-diagonal
block
B(k)
encodes he non- i ial opology; i s phase as a unc ion o
k
winds a ound
he o igin in he complex plane.
Fo ou wo-componen model, one can wo k di ec ly wi h he Bloch ec o
ˆ
n(k)
and
i s p ojec ion on o he plane o hogonal o
Γ
, as desc ibed in he main ex .
C.2 C.2 Explici phase winding
The c ucial objec is
z(k) = sin θcos k+ i cos θsin k=psin2θcos2k+ cos2θsin2keiφ(k),
whose a gumen
φ(k) = a g (sin θcos k+ i cos θsin k)
is a con inuous unc ion o
k
o
0< θ < π
, wi h
φ(−π) = −a c an (co θ), φ(π) = φ(−π)+2π.
Thus as
k
uns om
−π
o
π
, he phase
φ(k)
inc eases by
2π
, and he winding numbe
W=1
2πZπ
−π
∂φ(k)
∂k dk
is equal o
1
.
A
θ= 0
o
θ=π
, he quan i y
z(k)
collapses o he eal axis, he gap closes a
quasiene gy
0
o
π/∆
, and he winding becomes ill-dened; in hose cases he sys em is
opologically i ial wi h
W= 0
.
This compu a ion aligns wi h gene al classica ions o one-dimensional chi al-symme ic
quan um walks and wi h explici e alua ions o winding numbe s in ela ed models.
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