Topological Defects in Floquet Circuits

TL;DR

This paper introduces a Floquet circuit with topological defects in the driven Ising chain, analyzing their effects on return amplitudes, boundary conditions, and Majorana zero modes, with explicit constructions and entanglement entropy calculations.

Contribution

It presents a novel Floquet circuit model incorporating topological defects, including duality and spin-flip types, and explores their implications on system dynamics and symmetries.

Findings

Return amplitudes match defect fusion rules.

A Majorana zero mode appears with duality-twisted boundary conditions.

Analytic entanglement entropy expressions are derived for various defect configurations.

Abstract

We introduce a Floquet circuit describing the driven Ising chain with topological defects. The corresponding gates include a defect that flips spins as well as the duality defect that explicitly implements the Kramers-Wannier duality transformation. The Floquet unitary evolution operator commutes with such defects, but the duality defect is not unitary, as it projects out half the states. We give two applications of these defects. One is to analyze the return amplitudes in the presence of "space-like" defects stretching around the system. We verify explicitly that the return amplitudes are in agreement with the fusion rules of the defects. The second application is to study unitary evolution in the presence of "time-like" defects that implement anti-periodic and duality-twisted boundary conditions. We show that a single unpaired localized Majorana zero mode appears in the latter case. We explicitly construct this operator, which acts as a symmetry of this Floquet circuit. We also present analytic expressions for the entanglement entropy after a single time step for a system of a few sites, for all of the above defect configurations.