Index Theory of Chiral Unitaries and Split-Step Quantum Walks

Abstract

Building from work by Cedzich et al. and Suzuki et al., we consider topological and index-theoretic properties of chiral unitaries, which are an abstraction of the time evolution of a chiral-symmetric self-adjoint operator. Split-step quantum walks provide a rich class of examples. We use the index of a pair of projections and the Cayley transform to define topological indices for chiral unitaries on both Hilbert spaces and Hilbert $C^{\ast }$-modules. In the case of the discrete time evolution of a Hamiltonian-like operator, we relate the index for chiral unitaries to the index of the Hamiltonian. We also prove a double-sided winding number formula for anisotropic split-step quantum walks on Hilbert $C^{\ast }$-modules, extending a result by Matsuzawa.

References in corpus (5)

• On the noncommutative spectral flow
• An algorithm to factorize quantum walks into shift and coin operations
• The bulk-edge correspondence for the split-step quantum walk on the one-dimensional integer lattice
• Index Theorems for One-dimensional Chirally Symmetric Quantum Walks with Asymptotically Periodic Parameters
• The Witten index for one-dimensional split-step quantum walks under the non-Fredholm condition

Cited by in corpus (2)

• Exact mobility edges for almost-periodic CMV matrices via gauge symmetries
• Interfaces of discrete systems - spectral and index properties