Topological Phases in the Non-Hermitian Su-Schrieffer-Heeger Model

We address the conditions required for a Z topological classification in the most general form of the non-Hermitian Su-Schrieffer-Heeger (SSH) model. While chiral symmetry ensures a topological transition, we show that it also results in a “conjugated-pseudo-Hermiticity” which is responsible for a quantized “complex” Berry phase. We comment on the PT-symmetric problem, where previous studies have demonstrated that pseudo-anti-Hermiticity ensures a topological transition. 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. An intuitive picture is provided by examining eigenvector evolution on the Bloch sphere. We verify our claims numerically and discuss relevant experimental set-ups. Our work sheds light on the general problem of extending topological classification to non-Hermitian models.