No 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...

We discuss, for a two-dimensional Dirac Hamiltonian with a random scalar potential, the presence of a Z2 topological term in the nonlinear sigma model encoding the physics of Anderson localization in the symplectic symmetry class. The Z2 topological term realizes the sign of the Pfaffian of a family of Dirac operators. We compute the corresponding global anomaly, i.e., the change in the sign of the Pfaffian by studying a spectral flow numerically. This Z2 topological effect can be relevant to graphene when the impurity potential is long ranged and, also, to the two-dimensional boundaries of a three-dimensional lattice model of Z2 topological insulators in the symplectic symmetry class.

The effects of static disorder on the Z 2 quantum spin-Hall effect for non-interacting electrons propagating in two-dimensional space are studied numerically. A two-dimensional time-reversal symmetric network model is constructed to account for the effects of static disorder on the propagation of non-interacting electrons subjected to spin-orbit couplings. This network model is different from past network models belonging to the symplectic symmetry class in that the propagating modes along the links of the network can be arranged into an odd number of Kramers doublet. It is found that (1) a two-dimensional metallic phase of finite extent is embedded in a Z 2 insulating phase in parameter space and (2) the quantum phase transitions between the metallic and Z 2 insulating phases belong to the conventional symplectic universality class in two space dimensions.

We investigate one-dimensional (1D) discrete time quantum walks (QWs) with spatially or temporally random defects as a consequence of interactions with random environments. We focus on the QWs with chiral symmetry in a topological phase, and reveal that chiral symmetry together with bipartite nature of the QWs brings about intriguing behaviors such as coexistence of topologically protected edge states at zero energy and Anderson transitions in the 1D chiral class at non-zero energy in their dynamics. Contrary to the previous studies, therefore, the spatially disordered QWs can avoid complete localization due to the Anderson transition. It is further confirmed that the edge states are robust for spatial disorder but not for temporal disorder.