An index theorem for one-dimensional gapless non-unitary quantum walks

Recent developments in the index theory of discrete-time quantum walks allow us to assign a certain well-defined supersymmetric index to a unitary time evolution U and a Z(2)-grading operator Gamma satisfying the chiral symmetry condition, U* = Gamma U Gamma. In the present article, we extend this index theory to encompass non-unitary time evolutions U. The existing literature for unitary U assumes that U is essentially gapped to define the associated index, that is, the essential spectrum of U contains neither -1 nor +1. We show this assumption is not necessary if U fails to be unitary. As a concrete example, we consider the well-known non-unitary quantum walk model on the integer lattice Z introduced by Mochizuki-Kim-Obuse.

Automated summary

Recent developments in the index theory of discrete-time quantum walks have allowed for the application of the theory to non-unitary time evolutions, showing that unitarity is not necessary for certain calculations in this context.