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Topological phenomena in quantum walks; elementary introduction to the physics of topological phases

Takuya Kitagawa

arXiv:1112.1882v1quant-phcond-mat.quant-gas

TL;DR

The paper addresses how simple discrete quantum walks can make topological phases accessible while also revealing phenomena beyond static Hamiltonian classifications. It reviews one- and two-dimensional walks using effective Hamiltonians, winding and Chern numbers, and boundary-state physics, finding both static-like topological phases and driven-only phenomena. Quantum walks therefore serve as simulators of static topological phases and as platforms for studying periodically driven topology.

  • Problem

    Topological phases are difficult to explain and realize directly in condensed-matter systems, while some symmetry-classified phases lack physical realizations and their bound-state wavefunctions are hard to image.

  • Method

    The review analyzes one- and two-dimensional discrete quantum walks through effective Hamiltonians, symmetries, winding numbers, Chern numbers, and boundary-state constructions.

  • Results

    The review demonstrates boundary states and topological phases in quantum walks, including E = 0 and E = π bound-state flavors unique to periodically driven systems.

  • Takeaways & Limitations

    Quantum walks provide an accessible simulator for known static topological phases and an opportunity to study topological phenomena absent from static systems.

Abstract

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Discrete quantum walks are dynamical protocols for controlling a single quantum particle. Despite of its simplicity, quantum walks display rich topological phenomena and provide one of the simplest systems to study and understand topological phases. In this article, we review the physics of discrete quantum walks in one and two dimensions in light of topological phenomena and provide elementary explanations of topological phases and their physical consequence, namely the existence of boundary states. We demonstrate that quantum walks are versatile systems that simulate many topological phases whose classifications are known for static Hamiltonians. Furthermore, topological phenomena appearing in quantum walks go beyond what has been known in static systems; there are phenomena unique to quantum walks, being an example of periodically driven systems, that do not exist in static systems. Thus the quantum walks not only provide a powerful tool as a quantum simulator for static topological phases but also give unique opportunity to study topological phenomena in driven systems.

I. INTRODUCTION

Discrete quantum walks are simple protocols for controlling a single spin-1/2 particle, yet they provide an accessible setting for studying topological phases. The review develops effective-Hamiltonian, symmetry, and topological descriptions in one and two dimensions, including boundary states and periodically driven phenomena.

  • Discrete quantum walks: A conventional walk alternates spin rotation with spin-dependent translation, giving one-step evolution U = TRy(θ).The rotation acts around the y axis, while spin-up and spin-down components move in opposite directions.
  • Effective Hamiltonian description: The review uses effective Hamiltonians to explain quantum-walk dynamics, whose energies are quasi-energies defined modulo 2π/T.The walk reproduces the effective-Hamiltonian evolution at integer multiples of the step duration.
  • Topological structure: Spin-dependent translation produces spin-orbit coupling, which is central to realizing topological phases in discrete rather than continuous quantum walks.The coupling mixes quasi-momentum and the two internal spin degrees of freedom.
  • Band structure: For generic rotation angles, the conventional walk has two quasi-energy bands separated by a gap, while θ = 0 or 2π closes gaps at E = 0 and E = π.The gap closing occurs at distinct quasi-momenta for the two special quasi-energies.
  • Topological phases: The review connects quantum-walk topology to winding of eigenstates on the Bloch sphere and to robust bound states at boundaries between distinct phases.It also treats two-dimensional walks with Chern numbers and edge modes, plus bound states unique to periodically driven systems at E = 0 and E = π.

B. Hidden symmetry of quantum walks

Chiral symmetry pairs quasi-energy states at E and −E, with special unpaired possibilities at E = 0 and E = π. The same effective-Hamiltonian picture explains ballistic propagation and the resulting asymptotic position distribution.

  • Hidden symmetry: A π rotation about the axis A maps the effective Hamiltonian to its negative, defining the unitary sublattice or chiral symmetry.The eigenstate vector n(k) is perpendicular to A, so the rotation sends n(k) to −n(k).
  • Hidden symmetry: Chiral symmetry pairs states with quasi-energies E and −E, except at the special values E = 0 and E = π.These special energies satisfy E = −E under quasi-energy periodicity.
  • Hidden symmetry: The E = π state is a novel feature of periodically driven systems, whereas an E = 0 state is known for non-driven systems with chiral symmetry.The review links these special energies to topological protection.
  • Ballistic propagation: Quantum-walk particles propagate ballistically with group velocity v_k = ∂E(k)/∂k rather than diffusively as in a classical random walk.A momentum-localized state propagates according to the slope of the quasi-energy band.
  • Asymptotic distribution: The rescaled coordinate X = x/N approaches a finite asymptotic distribution, obtained by summing contributions from momentum states and dependent on the initial spin state.For θ = π/2 with an initial spin along z, the distribution can be evaluated analytically and plotted.

D. Experimental realizations of quantum walks

Quantum walks have been realized across cold-atom, photonic, and ion platforms, enabling distinct controls and studies of topological phases. The review connects these implementations to winding-based topology, symmetry constraints, engineered split-step phases, and experimentally observed localized states.

  • Experimental realizations: Quantum walks have been implemented with cold atoms, photons, and ions, using different physical controls and spatial boundary capabilities.These platforms support choices such as rotation operations, controlled dephasing, and spatially varying rotations.
  • Experimental realizations: Ion implementations can encode quantum walks in abstract phase space and maintain coherence for up to 23 steps.The walk uses internal ionic states as spin and motional excitations as spatial degrees of freedom.
  • Experimental consequences: Proposed and realized experiments use quantum walks to image and test topological bound states, including a localized state near the ground state in an energy-space realization.The review presents these systems as experimentally accessible simulators of topological phases.
  • Topological structure: Quantum walks realize one-dimensional chiral topological classes associated with the Su-Schrieffer-Heeger and Jackiw-Rebbi models.The split-step protocol actively engineers phases within this symmetry class through rotation and translation operations.
  • Topological structure: Under chiral symmetry, the winding number is robust to small symmetry-preserving perturbations but is not topological when the symmetry constraint is absent.Split-step walks provide phases with winding numbers W = 0 and W = 1, while the simpler family has a fixed nonzero winding number.

C. Physical manifestations of topological band structure

Spatially varying split-step quantum walks create interfaces between regions with different winding numbers. Because the winding number can change only through gap closure, these interfaces support localized states near quasi-energy E = 0, as confirmed by simulations.

  • Interface construction: A spatially varying second rotation creates a boundary between regions with winding numbers W = 0 and W = 1.The first rotation remains homogeneous while θ2 changes between asymptotic values across a finite region near the origin.
  • Interface construction: The bands are gapped far from the interface, so the winding-number change requires a gap closing near the origin.For slowly varying rotations, local band structures provide an intuitive description of the inhomogeneous walk.
  • Boundary states: A gap state near E = 0 is confined around the interface because this energy lies in the bulk gap on both sides.The resulting localized state is the boundary manifestation of distinct topological phases.
  • Simulation: After 60 steps, a smooth W = 0-to-W = 1 interface retains a probability peak near the origin, indicating a topological bound state.The simulation initializes a spin-up particle at x = 0 and varies θ2 over a length scale of approximately 6 sites.
  • Simulation: After 60 steps, a walk with W = 0 throughout shows probability near the origin decaying close to zero, indicating no localized boundary state.The comparison uses the same initial state and spatial functional dependence as the topologically distinct case.

D. Quantum walks with a reflecting boundary

A reflecting boundary can terminate a topological quantum walk and support a localized edge state when the full evolution preserves chiral symmetry. The bound state can occur at quasi-energy E = 0 or E = π, depending on the boundary phase and rotation angle.

  • Boundary construction: Terminating a W = 1 quantum walk against the trivial vacuum creates an edge phase boundary expected to support a bound state.The vacuum has topological number zero, so the boundary separates distinct topological classes.
  • Boundary construction: A unitary reflecting boundary rotates the spin, translates spin down inward, and leaves spin up at the edge with phase accumulation.The boundary operation is defined for a walk extending from x = −∞ to x = 0.
  • Symmetry condition: Chiral symmetry of the full walk with an edge requires the reflecting phase to be ϕ = 0 or π.This condition is necessary and sufficient for the edge evolution operator to possess the symmetry.
  • Bound-state solutions: For θ = π/2 and ϕ = 0, the analytical edge state has quasi-energy E = π and localization length λ ≈ 1.1.The small localization length makes the state tightly localized around the boundary.
  • Bound-state solutions: For chiral-symmetric quantum walks, bound-state quasi-energies are generally pinned to E = 0 or E = π.At θ = π with ϕ = 0, the localized edge eigenstate gains a minus sign after one step and therefore has E = π.

E. Topological protection of the bound states: topological invariants

Chiral symmetry protects quantum-walk bound states at quasienergies 0 and π, while integer invariants classify their numbers and constrain how they can change.

  • Topological bound states remain at E = 0 or E = π under small protocol changes, provided the corresponding bulk gap stays open.Their energies cannot shift continuously because chiral symmetry pairs states at E and −E; removal requires mixing with bulk states after a gap closing.
  • Chiral symmetry prevents a single 0 or π bound state from disappearing unless the bulk gap closes at the same quasienergy.A lone state cannot split into the ±ε pair required to move away from E = 0 or E = π.
  • The classification of chiral-symmetric quantum walks is Z × Z, allowing any integer numbers of protected E = 0 and E = π states.These bound-state invariants differ from the winding numbers assigned to bulk quantum-walk bands.
  • The quantities Q0 and Qπ are integer topological invariants assigned to bound states at E = 0 and E = π, respectively.They are constructed from the chiral-symmetry eigenvalues of the corresponding bound states.
  • Chiral-symmetry-preserving perturbations cannot mix same-eigenvalue bound states, so Q0 and Qπ remain unchanged under small Hamiltonian deformations.The invariants can change only when bound states at 0 or π mix with bulk states.

F. Breaking of sublattice (chiral) symmetry

Breaking sublattice (chiral) symmetry removes the topological protection that pins bound states to quasi-energies 0 or π, although small perturbations cannot eliminate them without bulk-state mixing. Periodically driven walks can nevertheless support both 0 and π bound states, reflecting a classification beyond static winding numbers.

  • Breaking sublattice symmetry removes the topological protection of bound states because no topological structure remains definable without that symmetry.
  • Small symmetry-breaking perturbations preserve bound states while the states remain separated from bulk states, but they can shift their quasi-energy away from 0 or π.
  • A boundary gap closing at E = 0 or E = π determines whether the corresponding 0 or π bound state appears.
  • Even phases with the same winding number can host two topological bound states at quasi-energies E = 0 and E = π.
  • The coexistence of 0 and π bound states also occurs when separated phases involve gap closings at E = 0 and E = π.
  • Quantum-walk topology is classified by Z × Z rather than the single Z winding-number classification of static Hamiltonians.

V. QUANTUM WALKS IN TWO DIMENSION

The review constructs a two-dimensional spin-1/2 quantum walk from alternating rotations and spin-dependent translations, then analyzes its effective Hamiltonian and Chern numbers. These walks realize symmetry-independent topological phases whose boundaries support unidirectional edge modes without backscattering.

  • Chern-number phases: The walk realizes Chern-number phases associated with integer quantum Hall effects, with upper- and lower-band Chern numbers related by opposite signs.
  • Two-dimensional quantum-walk protocol: The two-dimensional walk acts on a spin-1/2 particle on a square lattice through three rotations and three alternating translations.
  • Two-dimensional quantum-walk protocol: Spin-dependent translations move the two spin components along different lattice directions, defining the walk's spatial dynamics.
  • Effective Hamiltonian: Because one step preserves even and odd coordinate parity, the effective lattice constant is 2 and the Brillouin zone satisfies −π/2 ≤ kx ≤ π/2 and −π/2 ≤ ky ≤ π/2.
  • Effective Hamiltonian: The Chern number counts how many times the map from the two-dimensional Brillouin-zone torus to the Bloch sphere wraps around the sphere.
  • Chern-number phases: Unlike the one-dimensional winding number, the Chern number does not require a symmetry and can exist without any symmetry.
  • Boundary states: Chern-number phase boundaries host bound states because the band gap must close where the topological number changes.
  • Boundary states: Nonzero-Chern-number bound states propagate unidirectionally along edges without backscattering.

B. Unidirectionally propagating modes in quantum walks without Chern numbers

A simple two-dimensional quantum walk can host unidirectionally propagating boundary modes even though every bulk phase has zero Chern number. These modes exhibit energy winding and persist under continuous protocol changes unless the bulk gap closes.

  • Protocol: The walk uses two y-axis spin rotations, each followed by a spin-dependent translation along one spatial direction.The first translation moves spin-up right and spin-down left; the second moves spin-up up and spin-down down.
  • Boundary modes: Although the quantum walk has Chern number zero throughout its phase diagram, an inhomogeneous interface supports two unidirectionally propagating boundary modes.The two regions use different rotation angles and are separated by a single gapless phase.
  • Boundary modes: At special rotation angles, spin-up and spin-down states near the interface propagate to the right, producing two chiral modes.The boundary evolution exchanges the states at y = 0 and y = −1 while acquiring momentum-dependent phases.
  • Protection: The edge states wind from quasi-energy E = −π to E = π as kx runs from −π to π.This nonzero energy winding is distinct from the bulk Chern number and characterizes the protected edge propagation.
  • Protection: Continuous changes of the quantum-walk protocols cannot remove the edge states unless the bulk gap closes.The protection applies when the two interface phases remain continuously connected to the stated limiting protocols without crossing a gapless phase.

VI. OTHER TOPOLOGICAL PHASES

The review places quantum walks within the broader classification of topological phases while emphasizing phenomena unique to periodically driven systems. It also identifies three-dimensional and interacting extensions as open directions.

  • Scope: Variations of quantum-walk protocols can realize the topological phases classified for non-interacting static Hamiltonians in one and two dimensions.The review presents quantum walks as a route to studying both established static classifications and driven-system phenomena.
  • Driven phenomena: Periodically driven quantum walks exhibit 0 and π energy bound states in zero-winding phases and energy-winding unidirectional edge states in zero-Chern phases.The review identifies both effects as absent from static systems and notes that they can extend to other driven systems.
  • Open questions: A non-trivial three-dimensional topological phase has not yet been realized with quantum walks.The review highlights possible three-dimensional topological-insulator and Hopf-insulator realizations as open problems.
  • Open questions: Other proposed directions include quantum walks on hexagonal lattices and few- to many-particle systems with strong correlations.The latter may support phenomena such as spin-liquid phases in the presence of frustrated hopping.

Appendix A: Asymptotic distribution of quantum walk

The appendix derives the asymptotic position distribution of a quantum walk from its effective Hamiltonian. It uses a characteristic function, quasi-momentum diagonalization, and a large-step expansion.

  • Setup: The asymptotic analysis starts with a particle localized at x = 0 and an arbitrary initial spin state.The initial state is |i⟩ = |x = 0⟩⊗|s⟩.
  • Setup: Because the effective Hamiltonian is non-interacting, the particle is expected to propagate ballistically, motivating the scaled coordinate X = x/N.The asymptotic distribution is written as P(X).
  • Characteristic function: The derivation evaluates the characteristic function of the rescaled position after N steps in the limit N →∞.The target quantity is the expectation of e−isx/N.
  • Evaluation: Diagonalizing the effective Hamiltonian in quasi-momentum space makes the characteristic-function evaluation straightforward.The large-N expression is expanded to lowest order in s/N, introducing the group-velocity contribution.
  • Evaluation: The appendix obtains the final asymptotic expression by evaluating the resulting expectation in the initial state.The initial state is represented as an integral over quasi-momentum states combined with the spin state.

Appendix C: Analytic solution of the bound state for quantum walks with reflecting boundary condition

The appendix solves reflecting-boundary bound states by reducing the eigenvalue problem to a recursive transfer-matrix relation and imposing normalizability. It determines when localized states exist, their quasi-energies, and their localization lengths.

  • Method: The bound-state calculation begins by solving the eigenvalue problem for a quantum walk with a reflecting boundary.The quasi-energy Eb and normalizability conditions determine the allowed boundary-state solutions.
  • Method: A recursive matrix K relates wavefunction coefficients at neighboring sites and controls the behavior toward the bulk.The eigenvalues of K determine the asymptotic behavior as x →−∞.
  • Existence conditions: Normalizability requires selecting the decaying eigenvector according to the sign of cos(Eb), while no normalizable state exists when both transfer eigenvalues have unit magnitude.The excluded condition is cos^2(Eb) − cos^2(θ/2) < 0.
  • Quasi-energy: The bound-state quasi-energy switches between Eb = 0 and Eb = π depending on the phase ϕ and the interval containing θ.The appendix lists the corresponding eigenvector choices for ϕ = 0 and ϕ = π.
  • Localization: The localization length scales as ∼ 1/|log(λ−)| and diverges as θ approaches 0 or 2π.The bound state is most localized at θ = π.

Appendix D: Spectrum of two dimensional quantum walk

The appendix develops a general method for computing the spectrum of a two-dimensional quantum walk in quasi-momentum space. It uses the effective Hamiltonian and the trace of the evolution operator to identify the quasi-energy spectrum.

  • The method computes the spectrum of a two-dimensional quantum walk from its one-step evolution operator in quasi-momentum space.The approach is described as general and extendable to other quantum-walk protocols and higher dimensions.
  • For a spin-1/2 system, the effective Hamiltonian is expressed using the identity and Pauli matrices.This follows because these matrices generate two-by-two unitary matrices.
  • The spectrum is identified by evaluating the trace of the evolution operator U(kx, ky).The explicit trace evaluation yields E0(k) = 0 for all k.

Appendix E: gapless phase of two dimensional

The appendix derives the parameter conditions under which the two-dimensional quantum walk becomes gapless at quasi-energies 0 or π. Maximizing a component of the evolution operator yields the phase-boundary conditions plotted in Fig. 10(a).

  • Gap closing occurs when the two Hamiltonian bands become degenerate at quasi-energy E(k) = 0 or π.Because quasi-energy is periodic, degeneracy requires E(k) = −E(k), which occurs at these two values.
  • At E(k) = 0 or π, the evolution operator equals 1 or −1, respectively.The calculation therefore searches for parameter values where U(kx, ky) reaches these values for some momentum.
  • The gapless conditions are obtained by maximizing the (1, 1) component a11 of the evolution operator over kx and ky.The right-hand side has maximum magnitude 1, allowing the analysis to separate into cases.
  • For nonzero sin θ1 sin(θ2/2), extrema occur at cos kx = ±1, leading to parameter relations involving θ1 ± θ2/2.The cos kx = 1 and cos kx = −1 cases yield the same gapless phases.
  • When sin(θ2/2) = 0, an additional gapless line occurs at θ2 = nπ.This case satisfies the gap-closing condition by setting cos(kx + 2ky) = 0 without fixing θ1.
  • For quasi-energy E = π, gap closing occurs when θ1 + θ2/2 = 2πn + π, θ1 − θ2/2 = 2πn, or θ2 = nπ.These conditions produce the gapless phases plotted in Fig. 10(a).
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