Bulk-boundary correspondence for chiral symmetric quantum walks

Abstract

Discrete-time quantum walks (DTQW) have topological phases that are richer than those of time-independent lattice Hamiltonians. Even the basic symmetries, on which the standard classification of topological insulators hinges, have not yet been properly defined for quantum walks. We introduce the key tool of time frames, i.e., we describe a DTQW by the ensemble of time-shifted unitary time-step operators belonging to the walk. This gives us a way to consistently define chiral symmetry (CS) for DTQW's. We show that CS can be ensured by using an "inversion symmetric" pulse sequence. For one-dimensional DTQW's with CS, we identify the bulk Z × Z topological invariant that controls the number of topologically protected 0 and π energy edge states at the interfaces between different domains, and give simple formulas for these invariants. We illustrate this bulk-boundary correspondence for DTQW's on the example of the "4-step quantum walk," where tuning CS and particle-hole symmetry realizes edge states in various symmetry classes. The realization that band insulators can have nontrivial topological properties that determine the low-energy physics at their boundary has been a rich source of new physics in the last decade. The general theory of topological insulators and superconductors 1,2 classifies gapped Hamiltonians according to their dimension and their symmetries.3 As very few real-life materials are topological insulators, there is a strong push to develop model systems, "artificial materials," that simulate topological phases. 4 One of the promising approaches is to use discrete-time quantum walks (DTQW), 5-8 which can simulate topological insulators from all symmetry classes in 1D and 2D.9-11 DTQW's with particle-hole symmetry (PHS) go beyond simulating topological insulating Hamiltonians: they have topological phases with no counterpart in standard solid-state setups. In 1D DTQW's with PHS, edge states, "Majorana modes" can have two protected quasienergies: ε = 0 or π (time is measured in units of the time step andh = 1). Building on the results for periodically driven systems, 12 one of us has defined the corresponding Z 2 × Z 2 topological invariant.13 Both 0 and π energy Majorana edge states have been experimentally observed in a quantum walk. 14 The situation of chiral symmetry (CS) of DTQW's is much less clear. Even for the simplest one-dimensional DTQW, it is disputed whether it even has CS 9 or not. 13 Although it is expected that CS should imply a Z × Z bulk topological invariant, this has not yet been found for DTQW's. As opposed to the case of PHS, there is also not much to draw on from periodically driven systems. What DTQW's have CS? How can the bulk "winding number" be expressed for DTQW's with CS? These are the problems we tackle in this Rapid Communication.A DTQW concerns the dynamics of a particle, "walker," whose wave function is given by a vector, | = N x=1 s=−1,1 (x,s)|x,s . Here, x = 1, . . . ,N is the discrete position, and s = ±1 indexes the two orthogonal internal states of the walker, t...

Localization, delocalization, and topological phase transitions in the one-dimensional split-step quantum walk

“…We remark that for a periodically driven particle-hole symmetric Hamiltonian, the symmetry is inherited by the effective Hamiltonian in all time frames [34]. To see chiral symmetry of a quantum walk explicitly, it is necessary to go to a chiral symmetric time frame [28]. In the case of the split-step walk, there are two such time frames, specified by Eqs.…”

“…The split-step walk has both particle-hole symmetry, represented by complex conjugation K, and chiral symmetry, which places the system in Cartan class BDI [28]. To see particle-hole symmetry of the quantum walk, note that all matrix elements of the time-step operator U (θ 1 ,θ 2 ) are real (in a position and σ z basis).…”

“…There is considerable freedom in specifying a quantum walk, i.e., a periodic sequence of operations, corresponding to shifting the starting time of the period, which we refer to as changing the time frame [28]. As an example, consider the operators…”

“…However, unlike particle-hole symmetry, the chiral symmetry requirement is nonlocal in time [28]. The addition of a phase-shift operation to the time step, as in Eq.…”

“…These invariants can be expressed as winding numbers of the bulk time-step operator over a part of the tim step [28][29][30] or using a generalization of the scattering theory of topological phases [31,32]. In this respect quantum walks are representative of the extreme limit of periodically driven systems [33][34][35][36], as is the (closely related) quantum kicked rotator [37].…”