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

Mass as Topological Impedance: Self-Referential Scattering and Chiral Symmetry Breaking in Dirac-QCA Haobo Ma 1 Wenlin Zhang 2 1 Independent Researcher 2 National University of Singapore November 24, 2025 Abstract In relativistic quantum eld theory, the masses of elementary particles are usually introduced via Yukawa couplings to the Higgs eld and appear as free parameters, with no microscopic information-theoretic or topological origin. Within a quantum cellular automaton (QCA) ontology, however, the universe is modelled as a discrete, strictly causal quantum update rule on a lattice, with an intrinsic speed limit c set by the light-cone of local unitaries. This raises a structural question: why do certain excitations propagate subluminally and behave as massive particles, rather than as massless signals that saturate the update speed? In this work we treat a one-dimensional Dirac-type QCA, formulated as a discretetime quantum walk with internal (coin) degrees of freedom. Building on earlier results that show the Dirac equation emerging as the continuum limit of such models, we prove that the eective Dirac mass is nothing but the amplitude of a locally self-referential scattering process that mixes leftand right-moving components. At the level of the Floquet operator in momentum space, this mixing induces a nontrivial winding of the Bloch vector over the Brillouin zone, placing the model in a chiral-symmetric Floquet topological phase. The non-zero mass then acquires the interpretation of a topological impedance: a topologically protected obstruction that prevents the decoupling of chiral components into purely ballistic, lightlike propagation. We show that this mechanism produces Zitterbewegung as an unavoidable consequence of the local back-scattering. In the Dirac continuum limit the position operator splits into a uniform drift term with group velocity vext and an oscillatory term with frequency ωZB = 2E/ℏ , where E is the quasienergy. We interpret this oscillation as "internal motion" and demonstrate a precise decomposition v2 ext +v2 int =c2, where vint is dened as the standard deviation of the velocity operator. Thus every excitation saturates the microscopic information speed bound c , but a topologically enforced share of this budget is trapped in internal oscillations whenever the mass gap is non-zero. This provides a purely kinematical, topological and informationtheoretic interpretation of mass in Dirac-QCA models, compatible with and complementary to conventional Higgs-type mass generation. 1 Keywords: Quantum cellular automaton; discrete-time quantum walk; Dirac equation; topological mass; winding number; chiral symmetry; Zitterbewegung; Floquet topological phase; information speed conservation  1 Introduction & Historical Context 1.1 Mass and its open conceptual status In relativistic quantum eld theory (QFT), the free Dirac equation (iℏγµ∂µ−mc)ψ= 0 introduces the rest mass m as a parameter in the Lagrangian, while in the Standard Model masses arise from Yukawa couplings to the Higgs eld. Although this mechanism is phenomenologically successful, it leaves two conceptual issues: 1. The numerical values of fermion masses are free parameters, not xed by deeper principles. 2. The mass term m¯ ψψ breaks chiral symmetry explicitly, but this breaking is not directly tied to any topological invariant or discrete information-processing structure. Outside the Standard Model, a number of "topological mass" mechanisms are known, such as ChernSimons terms in (2 + 1) -dimensional gauge theories, or domain-wall and overlap fermions in lattice QCD, where a mass gap is associated with index theorems and spectral ow. However, these mechanisms still presuppose a continuum quantum eld, and they are usually formulated in terms of eective actions rather than underlying discrete information dynamics. 1.2 QCA and quantum walks as discrete Dirac dynamics Quantum cellular automata (QCAs) provide an alternative ontology in which dynamics is dened by a causal, translation-invariant unitary on a lattice of nite-dimensional quantum systems. The idea that quantum eld theory might emerge from such automata dates back to Feynman's suggestion that physics could be simulated by quantum computers, and has since been developed in detail by Bisio, D'Ariano, Tosini and others, who construct QCAs whose long-wavelength limit reproduces Weyl, Dirac and Maxwell equations. In one spatial dimension, a particularly simple realization of Dirac-type dynamics is provided by discrete-time quantum walks (DTQWs), in which a two-component "coin" degree of freedom controls conditional shifts to left and right. Strauch showed that for suitable scaling of the coin parameters and lattice spacing the continuum limit of a DTQW reproduces the (1 + 1) -dimensional Dirac equation, and subsequent work generalized this connection to higher dimensions and more general walk protocols. In parallel, a systematic theory of QCAs has been developed, including locality and index theorems classifying one-dimensional automata up to homotopy. Within this setting, DTQWs can be viewed as special cases of one-dimensional QCAs with internal degrees of freedom (coins) and nite propagation radius. 2 1.3 Topological phases and quantum walks Topological band theory, rst developed for static Hamiltonians, has been extended to periodically driven (Floquet) systems, where the time-evolution operator over one period plays the role of a Floquet unitary. The topological classication of such unitaries, especially in the presence of chiral symmetry, naturally applies to DTQWs. Kitagawa and collaborators showed that split-step and multi-step quantum walks realize a rich set of one-dimensional topological phases with integer winding numbers, accompanied by robust edge states at boundaries between domains of dierent invariants. Asbóth and Obuse later formulated a bulk-boundary correspondence and a systematic denition of chiral symmetry for Floquet quantum walks, identifying a Z×Z valued topological invariant controlling zero and π quasienergy edge states. Experiments using photonic and cold-atom platforms have directly observed these topological bound states and measured winding numbers and dynamical topological order parameters. Importantly for our purposes, even the simplest one-coin quantum walk, when formulated with an appropriate coin rotation, exhibits a hidden chiral symmetry and a non-trivial winding number for generic coin angles, with the gap closing and winding changing only at special values of the coin parameter. 1.4 Zitterbewegung in Dirac dynamics and QCAs Zitterbewegung (ZB), the rapid oscillatory motion predicted by the Dirac equation, has long been understood as arising from interference between positive and negative energy components of a wave packet. In the Heisenberg picture the position operator splits into a uniform drift term plus an oscillatory term with frequency 2E/ℏ , and the velocity operator has eigenvalues ±c , so that the velocity uctuates between these extreme values even when average motion is subluminal. In QCA and quantum walk models that approximate Dirac dynamics, the same phenomenon has been analysed both analytically and numerically: the Dirac QCA constructed by Bisio et al. exhibits ZB and Klein tunnelling, with oscillation frequency and amplitude matching those of the continuum Dirac theory in the appropriate limit. 1.5 Information speed conservation and the role of mass In a previous work we proposed an information-theoretic conservation law for local excitations in QCA-type models, namely v2 ext +v2 int =c2, where vext measures the eective group velocity of an excitation in physical space and vint quanties the internal evolution rate in Hilbert space, in terms of the variance of the velocity operator or the FubiniStudy metric. This relation expresses the idea that the "information speed budget" is xed at c , with massless excitations allocating all of it to translational motion, while massive ones divert part of it into internal oscillations. The present paper implements this idea concretely in a Dirac-QCA setting and ties it to topological invariants. 3 1.6 Goals and contributions The central objective is to reinterpret the mass parameter of the Dirac equation, as realized in a Dirac-QCA model, as a topological impedance: a quantized obstruction to decoupling leftand right-moving degrees of freedom into independent ballistic channels. Concretely, we: 1. Specify a one-dimensional Dirac-type QCA as a discrete-time quantum walk with a single coin angle and show, under standard locality and homogeneity assumptions, that its long-wavelength limit reproduces the (1+1) -dimensional Dirac equation with mass m determined by the coin angle. 2. Show that the timestep unitary in momentum space has chiral symmetry and denes a non-trivial winding of the Bloch vector on the Bloch sphere for any non-zero coin angle, in agreement with earlier classications of Floquet topological phases in quantum walks. This winding number changes only when the gap at quasienergies 0 or π closes, corresponding to m= 0 . 3. Give a scattering-theoretic interpretation of the coin angle as a local self-referential scattering amplitude that repeatedly converts left-moving components into right-moving ones and vice versa. This produces subluminal group velocities and Zitterbewegung, and we interpret the associated energy gap as a topological impedance blocking purely lightlike propagation. 4. Derive in the Dirac continuum limit an exact decomposition v2 ext +v2 int =c2 for arbitrary states, by dening vext as the expectation value of the velocity operator and vint as the square root of its variance, thereby giving a precise information-theoretic meaning to internal motion and linking it to mass. 5. Discuss how the topological impedance picture interacts with conventional mass generation mechanisms, and propose experimental schemes for observing the trade-o between group velocity, internal oscillations and topological invariants in photonic and cold-atom quantum walk platforms.  2 Model & Assumptions 2.1 Lattice, Hilbert space and locality We consider a one-dimensional spatial lattice with spacing a > 0 , whose sites are labelled by integers n∈Z , corresponding to positions x=na . Time is discrete with step ∆t > 0 , so that t=m∆t for m∈Z . The microscopic causal speed is xed by c=a ∆t, which will be identied with the emergent speed of light in the continuum limit. At each lattice site the local Hilbert space is a two-dimensional coin space C2 , interpreted as a chiral or left/right degree of freedom. The global Hilbert space is H=ℓ2(Z)⊗C2. 4 We write states in the position-coin basis as Ψ = X n∈Z (ψL(n)|n⟩⊗|L⟩+ψR(n)|n⟩⊗|R⟩), or in column vector form Ψ(n, t) = ψL(n, t) ψR(n, t). We assume a single-step, translation-invariant QCA evolution operator U:H → H, which is unitary, homogeneous (commutes with spatial translations), causal with - nite propagation radius (each site couples only to a bounded neighbourhood in a single timestep) and parity symmetric. 2.2 Coin operator and conditional shift The timestep unitary is taken to factor into a local coin rotation followed by a conditional shift: U=S C(θ), where θ∈[0, π] is a real parameter. The coin operator is a site-local unitary of the form C(θ) = X n∈Z |n⟩⟨n| ⊗ R(θ), where R(θ)=e−iθσx=cos θ−i sin θ −i sin θcos θ and σx is the usual Pauli matrix. The conditional shift operator moves left and right components in opposite directions: S=X n∈Z (|n−1⟩⟨n|⊗|L⟩⟨L|+|n+ 1⟩⟨n|⊗|R⟩⟨R|). Equivalently, S=X n∈Z |n⟩⟨n| ⊗ (|L⟩⟨L|T++|R⟩⟨R|T−), where T± are lattice translation operators ( T±|n⟩=|n±1⟩ ). The evolution equation over one timestep is Ψ(t+ ∆t) = UΨ(t). In position space this yields coupled update equations: ψL(n, t + ∆t) = cos θ ψL(n+ 1, t)−i sin θ ψR(n+ 1, t), ψR(n, t + ∆t) = −i sin θ ψL(n−1, t) + cos θ ψR(n−1, t). We interpret the parameter θ as controlling the strength of a local "self-scattering" between leftand right-moving components; θ= 0 corresponds to purely ballistic motion at speed c , while θ= 0 introduces back-scattering. 5 2.3 Momentum-space representation and eective Hamiltonian By translation invariance, we can pass to momentum space via Ψ(n, t) = Zπ −π dk 2πeikn e Ψ(k, t), with k∈[−π, π] the dimensionless lattice momentum; the physical momentum is p= ℏk/a . The shift operator is diagonal in k , e S(k) = e−ikσz, and the timestep unitary in momentum space is e U(k) = e S(k)R(θ) = e−ikcos θ−ie−iksin θ ieiksin θeikcos θ. We dene the Floquet quasienergies E(k) and eective Hamiltonian Heff(k) via e U(k) = exp −i ℏHeff(k) ∆t. The eigenvalues of e U(k) are e−iE(k)∆t and e+iE(k)∆t , with cosE(k)∆t= cos θcos k, a well-known dispersion relation for coined quantum walks. We can write Heff(k) = E(k)ˆ n(k)·σ, with ˆ n(k) a unit vector on the Bloch sphere.  3 Main Results (Theorems and Alignments) In this section we summarize the main theorems; detailed proofs are given in the subsequent section and in the appendices. Theorem 3.1 (Dirac continuum limit) . Let the lattice spacing and timestep obey a= c∆t , and scale the coin angle as θ=mc2 ℏ∆t+O(∆t3), with xed parameters c > 0 and m≥0 . Consider initial states whose amplitudes vary slowly on the lattice scale, in the sense that ψL/R(n±1,0) −ψL/R(n, 0) = O(a) uniformly in n . Then in the limit ∆t→0 the discrete evolution converges, up to errors of order O(∆t2) , to the continuum Dirac equation in (1 + 1) dimensions, iℏ∂tΨ(x, t) = −iℏc σz∂x+mc2σxΨ(x, t), with x=na . The convergence holds in operator norm on any nite time interval and for wave packets supported in a compact momentum region. 6 Theorem 3.2 (Chiral symmetry and winding number) . Dene a chiral symmetry operator Γ = σx. Then for all momenta k the Floquet unitary satises Γe U(k) Γ = e U†(k), so the QCA is in the chiral-symmetric Floquet class AIII. The corresponding one-dimensional topological invariant (winding number) is W=1 2πiZπ −π dkTr Γe U−1(k)∂ke U(k). For the present model, W=(0, θ = 0 or θ=π, 1,0< θ < π. Thus any non-zero mass parameter m , as dened in Theorem 1, corresponds to a topologically non-trivial Floquet phase that cannot be adiabatically deformed to the massless case without closing the quasienergy gap at E= 0 or E=π/∆t . This is consistent with earlier analyses of topological phases in one-coin and split-step quantum walks. Theorem 3.3 (Mass as self-referential scattering) . In position space, the single-step evolution at site n can be written as the action of a local scattering matrix on an incoming two-component eld comprised of leftand right-moving amplitudes. Explicitly, in momentum space, which is equivalent to a two-channel unitary scattering with reection and transmission amplitudes r(θ) = −i sin θ, t(θ) = cos θ. For a wave packet sharply peaked around momentum k0 , the group velocity satises vext(k0, θ) = ∂E(k) ∂p k=k0 =1 ℏ ∂E(k) ∂k a, with dispersion cos(E∆t) = cos θcos k . In the Dirac continuum limit described in Theorem 1, this yields vext(p)≈c2p pp2c2+m2c4 and an energy gap E(p)≈pp2c2+m2c4. Thus the same parameter θ simultaneously controls (i) the size of the mass gap in the Dirac limit and (ii) the strength of local back-scattering between leftand right-moving components. Repeated application of this local scattering produces an eective inertia: the stronger the mixing (larger θ ), the smaller the asymptotic group velocity at given momentum. We interpret this as mass arising from a self-referential scattering process that ties the excitation to its own history. 7 Theorem 3.4 (Zitterbewegung and information speed decomposition) . Let H=cpσz+ mc2σx denote the Dirac Hamiltonian in the continuum limit, and dene the velocity operator ˆv=i ℏ[H, X] = cσz. Then for any normalized state Ψ , ⟨ˆv2⟩=c2, and we can dene vext =⟨ˆv⟩, vint =p⟨ˆv2⟩−⟨ˆv⟩2. For all states, v2 ext +v2 int =c2. Moreover, in the Heisenberg picture the position operator decomposes as X(t) = X(0) + vextt+ Ξ(t), where Ξ(t) is an oscillatory term with frequency ωZB = 2E/ℏ and amplitude proportional to ℏc/(2E) . This is the usual Zitterbewegung term, which we thereby interpret as the manifestation of the internal information speed vint . Massive excitations necessarily have 0<|vext|< c and corresponding non-zero vint ; massless ones satisfy |vext|=c and vint = 0 . In the discrete QCA, a similar decomposition holds at the level of the eective Hamiltonian and velocity operator in the long-wavelength regime, and Zitterbewegung appears as a fast oscillation of the expectation value of the position operator superimposed on the group-velocity drift.  4 Proofs In this section we sketch the main arguments; detailed step-by-step derivations are deferred to the appendices. 4.1 Proof of Theorem 1 (Dirac continuum limit) We set a=c∆t and θ= (mc2/ℏ)∆t+O(∆t3) and work to rst order in ∆t . Rewriting the position-space update equations as ψL(n, t + ∆t) = cos θ ψL(n+ 1, t)−i sin θ ψR(n+ 1, t), ψR(n, t + ∆t) = −i sin θ ψL(n−1, t) + cos θ ψR(n−1, t), we Taylor-expand all terms in ∆t . Using cos θ= 1 + O(∆t2),sin θ=mc2 ℏ∆t+O(∆t3), and ψL/R(n±1, t) = ψL/R(x±a, t) = ψL/R(x, t)±a∂xψL/R(x, t) + O(a2), 8 with x=na , we obtain to rst order ψL(x, t)+∆t ∂tψL(x, t) = ψL(x, t) + c∆t ∂xψL(x, t)−imc2 ℏ∆t ψR(x, t) + O(∆t2), ψR(x, t)+∆t ∂tψR(x, t) = ψR(x, t)−c∆t ∂xψR(x, t)−imc2 ℏ∆t ψL(x, t) + O(∆t2). Subtracting ψL/R(x, t) from both sides and dividing by ∆t yields ∂tψL−c∂xψL=−imc2 ℏψR+O(∆t), ∂tψR+c∂xψR=−imc2 ℏψL+O(∆t). Multiplying by iℏ and collecting terms in spinor form Ψ = (ψL, ψR)T , we nd iℏ∂tΨ = −iℏc σz∂x+mc2σxΨ + O(∆t), which is the desired Dirac equation up to O(∆t) corrections. A more careful analysis using Fourier methods and norm estimates shows that the error remains O(∆t) in operator norm on any xed time interval for wave packets with bounded momentum support, in line with rigorous continuum-limit results for quantum walks. 4.2 Proof of Theorem 2 (chiral symmetry and winding number) We rst verify chiral symmetry. Using Γ = σx and the explicit form of e U(k) , e U(k) = e−ikcos θ−ie−iksin θ ieiksin θeikcos θ, we compute Γe U(k)Γ = σxe U(k)σx=eikcos θie−iksin θ ieiksin θe−ikcos θ=e U†(k), where the last equality follows from unitarity and complex conjugation. Thus the chiral symmetry condition holds. To compute the winding number, we write e U(k)=e−iE(k)ˆ n(k)·σ with cos(E∆t) = cos θcos k, and a Bloch vector ˆ n(k) lying on the Bloch sphere. A convenient parametrization, used in Lam's analysis of the Hadamard quantum walk, is ˆ n(k) = 1 sin(E∆t)  sin θsin k sin θcos k cos θsin k , valid away from gap closing points where sin(E∆t)= 0 . The chiral symmetry implies that the relevant winding is that of the projection of ˆ n(k) onto the plane orthogonal to Γ , which here is the (yz) -plane. As k runs from −π 9 4. In the Dirac limit the velocity operator has eigenvalues ±c , implying that any excitation satises v2 ext +v2 int =c2 , where vext is the average group velocity and vint quanties internal uctuations associated with Zitterbewegung. Massive excitations thus divert part of the xed information speed budget into internal motion. 5. Interfaces where the eective mass changes sign support topologically protected bound states, naturally interpreted as defects in the pattern of topological impedance. Experimental platforms based on photonic and cold-atom quantum walks can test these predictions and directly probe the interplay between mass, topology and information ow. Taken together, these results suggest that within a QCA ontology, mass need not be an arbitrary parameter inserted into the continuum eld theory. Instead, it can emerge as a topologically protected property of discrete information dynamics, constraining how excitations allocate their nite information speed budget between external propagation and internal self-referential motion.  9 Acknowledgements, Code Availability The author acknowledges the existing body of work on quantum walks, quantum cellular automata and topological phases that underpins this study, in particular the contributions of Strauch, Childs, Kitagawa, Asbóth, Bisio, D'Ariano, Tosini, Farrelly and many others. No numerical simulations beyond standard analytical calculations were required for the derivations presented here. Simple quantum-walk simulators sucient to reproduce the dispersion relations, Zitterbewegung and domain-wall bound states discussed in this paper can be implemented straightforwardly in standard scientic computing environments; no dedicated code repository is provided.  A Appendix A: Detailed Derivation of the Dirac Continuum Limit A.1 A.1 Scaling and smoothness assumptions We adopt the scaling a=c∆t, θ =mc2 ℏ∆t, and assume that the lattice wave function Ψ(n, t) at t= 0 is obtained by sampling a smooth continuum spinor Ψ(x, 0) at positions x=na . Explicitly, Ψ(n, 0) = Ψ(x=na, 0), with Ψ twice continuously dierentiable and decaying suciently fast at innity. The goal is to construct a continuum spinor Ψ(x, t) satisfying the Dirac equation such that Ψ(n, t) = Ψ(x=na, t) + O(∆t2) for t in a bounded interval. The error estimate can be made precise in Sobolev norms, following methods used in rigorous continuum-limit analyses of quantum walks. 16 A.2 A.2 Expansion of the discrete update We write the discrete update equations as Ψ(n, t + ∆t) = UΨ(·, t)(n), where U acts on lattice spinors according to 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 interpret Ψ(n, t) as sampling of a continuum eld Ψ(x, t) , and Taylor-expand around x=na . Using Ψ(x±a, t) = Ψ(x, t)±a∂xΨ(x, t) + a2 2∂2 xΨ(x, t) + O(a3), and cos θ= 1 −θ2 2+O(θ4)=1− O(∆t2), sin θ=θ+O(θ3) = mc2 ℏ∆t+O(∆t3), we obtain ψL(x, t + ∆t) = ψL(x, t) + a∂xψL(x, t)−imc2 ℏ∆t ψR(x, t) + O(∆t2), ψR(x, t + ∆t) = ψR(x, t)−a∂xψR(x, t)−imc2 ℏ∆t ψL(x, t) + O(∆t2). Here we used a=c∆t and neglected terms of order ∆t2 or higher. Rewriting this as a rst-order time discretization of a continuum equation, Ψ(x, t + ∆t) = Ψ(x, t)+∆t ∂tΨ(x, t) + O(∆t2), we identify ∂tΨ = −cσz∂xΨ−imc2 ℏσxΨ + O(∆t), and hence iℏ∂tΨ = −iℏcσz∂x+mc2σxΨ + O(∆t). Standard stability and consistency arguments for one-step schemes then show that the discrete evolution converges to the Dirac evolution with error O(∆t) over nite times. B Appendix B: Zitterbewegung in Dirac and QCA Dynamics Here we give the standard derivation of Zitterbewegung for the Dirac Hamiltonian and outline how it appears in the QCA. 17 B.1 B.1 Zitterbewegung in the Dirac theory Consider the free Dirac Hamiltonian in (1 + 1) dimensions, H=cpσz+mc2σx, acting on spinors Ψ(x) . In the Heisenberg picture, dX(t) dt=i ℏ[H, X(t)],dσz(t) dt=i ℏ[H, σz(t)]. Using [p, X] = −iℏ , one nds dX(t) dt=cσz(t) = ˆv(t), and dσz(t) dt=2mc2 ℏσy(t),dσy(t) dt=−2 ℏcpσx(t) + mc2σz(t). Solving these coupled equations yields ˆv(t) = c2pH−1+ e2iHt/ℏˆv(0) −c2pH−1, and integrating over time, X(t) = X(0) + c2pH−1t+iℏc 2He2iHt/ℏ−1ˆv(0) −c2pH−1. The rst term represents uniform motion with group velocity vext =c2p/E , while the second is an oscillatory term with frequency ωZB = 2E/ℏ and amplitude of order ℏc/(2E) . For wave packets consisting purely of positive energy eigenstates, ˆv(0) coincides with c2pH−1 , and the oscillatory term vanishes; Zitterbewegung arises only when both positive and negative energy components are present. B.2 B.2 Zitterbewegung in the Dirac-QCA For the QCA, the Heisenberg equations must be formulated with respect to the eective Hamiltonian Heff(k) or, more directly, via the discrete-time Heisenberg evolution Xm+1 =U†XmU, with m labelling timesteps. Working in momentum space, one nds that for wave packets peaked around small momenta and small masses, the discrete evolution of ⟨Xm⟩ reproduces the Dirac behavior to good accuracy, including an oscillatory contribution with frequency close to 2E/ℏ . Detailed calculations and numerical simulations for the Dirac automaton have been carried out in previous work, conrming that Zitterbewegung is an intrinsic feature of the QCA dynamics. This supports the interpretation that Zitterbewegung, both in the continuum and in the QCA, is the manifestation of the internal component vint of the xed information speed budget, while the group velocity vext accounts for the net transport. C Appendix C: Computation of the Winding Number We briey detail the computation of the winding number for the one-coin quantum walk considered here. 18 C.1 C.1 Chiral decomposition With chiral symmetry Γ = σx and Floquet unitary e U(k) , we can switch to a basis in which Γ is diagonal: Γ = 1 0 0−1, and e U(k) can be written in block form e U(k) = A(k)B(k) C(k)D(k). Chiral symmetry implies Γe U(k)Γ = e U†(k), which leads to constraints A=D† , B=−B† , C=−C† . In particular, the o-diagonal block B(k) encodes the non-trivial topology; its phase as a function of k winds around the origin in the complex plane. For our two-component model, one can work directly with the Bloch vector ˆ n(k) and its projection onto the plane orthogonal to Γ , as described in the main text. C.2 C.2 Explicit phase winding The crucial object is z(k) = sin θcos k+ i cos θsin k=psin2θcos2k+ cos2θsin2keiφ(k), whose argument φ(k) = arg (sin θcos k+ i cos θsin k) is a continuous function of k for 0< θ < π , with φ(−π) = −arctan (cot θ), φ(π) = φ(−π)+2π. Thus as k runs from −π to π , the phase φ(k) increases by 2π , and the winding number W=1 2πZπ −π ∂φ(k) ∂k dk is equal to 1 . At θ= 0 or θ=π , the quantity z(k) collapses to the real axis, the gap closes at quasienergy 0 or π/∆t , and the winding becomes ill-dened; in those cases the system is topologically trivial with W= 0 . This computation aligns with general classications of one-dimensional chiral-symmetric quantum walks and with explicit evaluations of winding numbers in related models. References [1] R. P. Feynman, "Simulating physics with computers," International Journal of Theoretical Physics 21, 467488 (1982). [2] A. Bisio, G. M. D'Ariano, and A. 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