Topological invariants of nonunitary quantum walk with chiral symmetry

Abstract

We propose an one-dimensional model of nonunitary quantum walk with chiral symmetry. The nonunitarity is caused by gain and loss that they stagger in time and act on different walking directions. The degenerate point of the quasi-energy on the dispersion-relation diagram becomes a line segment and the corresponding topological defect is characterized by the vorticity. The winding number is still well defined due to chiral symmetry. It is found that the vorticity, as a topological invariant, equals the difference of winding numbers within both sides of topological defect. Meanwhile, a pair of vorticities can be derived in two time symmetric frames and they count the number of topological defects with zero- and π- quasi-energy. At boundary in inhomogeneous system, appearing or disappearing of edge state is not affected by nonunitary action. The existing of the edge states can be measured by the average displacement calculated by the escape probability, and the bulk-boundary corresponding is shown to be invalid.