Topological characterization of special edge modes from the winding of relative phase

TL;DR

This paper introduces a new topological invariant based on the winding of the relative phase between components of a spinor to explain special edge modes in symmetry-broken systems, extending the bulk-boundary correspondence.

Contribution

It proposes a novel topological characterization using relative phase winding, applicable to systems lacking discrete symmetries, and extends the concept to two-dimensional cases.

Findings

Relative phase winding correlates with the presence of one-sided edge modes.

The method applies to both one-dimensional and two-dimensional systems.

Topology depends on whether the projection includes or excludes the origin.

Abstract

The symmetry-constrained topological invariant fails to explain the emergence of the special edge modes when system does not preserve discrete symmetries. The inversion or chiral symmetry broken SSH model is an example of one such system where one-sided edge state with finite energy appears at one end of the open chain. To investigate whether this special edge mode is of topological origin or not, we introduce a concept of relative phase between the components of a two-component spinor and define a winding number by the change of this relative phase over the one-dimensional Brillouin zone. The relative phase winds non-trivially (trivially) in accord with the presence (absence) of the one-sided edge mode inferring the bulk boundary correspondence. We extend this analysis to a two dimensional case where we characterize the non-trivial phase, hosting gapped one-sided edge mode, by the winding in relative phase only along a certain axis in the Brillouin zone. We demonstrate all the above findings from a generic parametric representation while topology is essentially determined by whether the underlying lower-dimensional projection includes or excludes the origin. Our study thus reveals a new paradigm of symmetry broken topological phases for future studies.