Biorthogonal Bulk-Boundary Correspondence in Non-Hermitian Systems

TL;DR

This paper develops a biorthogonal framework to understand the breakdown and generalization of bulk-boundary correspondence in non-Hermitian systems, linking boundary states to phase transitions through biorthogonal polarization.

Contribution

It introduces a biorthogonal quantum mechanics approach to unify bulk-boundary correspondence in non-Hermitian systems, including open boundary models.

Findings

Biorthogonal density reveals phase transitions in non-Hermitian systems.

Generalized bulk-boundary correspondence with quantized biorthogonal polarization.

Phase diagrams derived for non-Hermitian extensions of known models.

Abstract

Non-Hermitian systems exhibit striking exceptions from the paradigmatic bulk-boundary correspondence, including the failure of bulk Bloch band invariants in predicting boundary states and the (dis)appearance of boundary states at parameter values far from those corresponding to gap closings in periodic systems without boundaries. Here, we provide a comprehensive framework to unravel this disparity based on the notion of biorthogonal quantum mechanics: While the properties of the left and right eigenstates corresponding to boundary modes are individually decoupled from the bulk physics in non-Hermitian systems, their combined biorthogonal density penetrates the bulk precisely when phase transitions occur. This leads to generalized bulk-boundary correspondence and a quantized biorthogonal polarization that is formulated directly in systems with open boundaries. We illustrate our general insights by deriving the phase diagram for several microscopic open boundary models, including exactly solvable non-Hermitian extensions of the Su-Schrieffer-Heeger model and Chern insulators.