Boundary-dependent topological degeneracy in an Ising chain

TL;DR

This paper explores how boundary conditions in an Ising chain affect topological degeneracy, revealing a switch in degeneracy linked to boundary modifications and mapped onto Kitaev chains.

Contribution

It introduces an alternative boundary condition that maps the Ising chain to two Kitaev chains, showing a boundary-dependent topological degeneracy switch.

Findings

Boundary condition removes end coupling and transverse field on ends.

System maps onto two independent Kitaev chains with domain-wall fermions.

Topological degeneracy switch is explained via gauge dependence of the winding number.

Abstract

The topological degeneracy is a characteristic of quantum phase diagram in an Ising chain with transverse field. We revisit the phase diagram at nonzero temperature of an Ising chain with two types of open boundary conditions. In this work, we focus on an alternative boundary condition that not only removes the coupling between the two end sites but also eliminates the transverse field on them. We show that such a system can be exactly mapped onto two independent Kitaev chains, where spinless fermions correspond to domain-wall excitations. This results in a switch in the existence of the topological Kramers-like degeneracy in the phase diagram. The underlying mechanism is analyzed within the Majorana representation, which indicates that such a switch arises from the gauge dependence of the winding number in an SSH chain. The manifestation of bulk-boundary correspondence at nonzero temperature is demonstrated by numerical simulations on finite-size systems. This finding provides insight into the quantum spin chain.