Topological phases in the non-Hermitian Su-Schrieffer-Heeger model

TL;DR

This paper explores topological phases in the non-Hermitian SSH model, demonstrating how a complex Berry phase serves as a quantized invariant predicting edge modes, with implications for non-Hermitian topological systems.

Contribution

It introduces the use of the complex Berry phase as a topological invariant in non-Hermitian SSH models, including chirally-broken and PT-symmetric cases, with a geometric interpretation.

Findings

Complex Berry phase is quantized and predicts edge modes.

Conjugated-pseudo-Hermiticity underpins topological classification.

Numerical validation supports theoretical predictions.

Abstract

We address the conditions required for a $\mathbb{Z}$ topological classification in the most general form of the non-Hermitian Su-Schrieffer-Heeger (SSH) model. Any chirally-symmetric SSH model will possess a "conjugated-pseudo-Hermiticity" which we show is responsible for a quantized "complex" Berry phase. Consequently, we provide the first example where the complex Berry phase of a band is used as a quantized invariant to predict the existence of gapless edge modes in a non-Hermitian model. The chirally-broken, $PT$-symmetric model is studied; we suggest an explanation for why the topological invariant is a global property of the Hamiltonian. A geometrical picture is provided by examining eigenvector evolution on the Bloch sphere. We justify our analysis numerically and discuss relevant applications.