Scattering theory of topological phases in discrete-time quantum walks

Abstract

One-dimensional discrete-time quantum walks show a rich spectrum of topological phases that have so far been exclusively analysed based on the Floquet operator in momentum space. In this work we introduce an alternative approach to topology which is based on the scattering matrix of a quantum walk, adapting concepts from time-independent systems. For quantum walks with gaps in the quasienergy spectrum at 0 and π, we find three different types of topological invariants, which apply dependent on the symmetries of the system. These determine the number of protected boundary states at an interface between two quantum walk regions. Unbalanced quantum walks on the other hand are characterised by the number of perfectly transmitting unidirectional modes they support, which is equal to their non-trivial quasienergy winding. Our classification provides a unified framework that includes all known types of topology in one dimensional discrete-time quantum walks and is very well suited for the analysis of finite size and disorder effects. We provide a simple scheme to directly measure the topological invariants in an optical quantum walk experiment.

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

“…We have introduced an effective numerical tool (the cloning trick) which allows us to calculate the scattering amplitudes for the split-step walk and efficiently calculate the topological invariants proposed in Ref. [32]. We then showed theoretically and investigated numerically various localization-delocalization transitions that occur whenever this system is tuned to a critical point at a topological phase transition.…”

“…Since disorder breaks translation invariance, the bulk topological invariants can no longer be obtained as k-space winding numbers. There are two alternative approaches to the topological invariants for the disordered case: one based on the scattering matrix [31,32] and one based on a reformulation of the winding number in real space, recently used for the disordered SSH model [39]. In the following we detail the first approach.…”

“…To define a scattering matrix, we have to use open boundary conditions, with two translationally invariant leads attached to the scattering region [32]. To obtain leads that host the right number of propagating modes, we omit the rotations in semi-infinite parts of the system, so the time-step operator there simply reads U = S ↓ S ↑ , as shown in Fig.…”

“…Chiral symmetry ensures that the reflection amplitudes at these energies are real and the topological invariants for a bulk are given by [32] …”

“…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].…”