Bulk-boundary correspondence in topological two-dimensional non-Hermitian systems: Toeplitz operators and singular values

TL;DR

This paper develops a new bulk-boundary correspondence framework for two-dimensional non-Hermitian systems using Toeplitz operators and singular values, providing a stable topological classification independent of eigenvalues.

Contribution

It introduces a novel approach based on singular values and Toeplitz operator theory to establish topological invariants for non-Hermitian systems, including higher-order phases, without relying on crystalline symmetries.

Findings

Singular values are more stable than eigenvalues for topological protection.

Established a bulk-boundary correspondence relating Toeplitz indices to edge and corner modes.

Demonstrated the theory with examples, including a non-Hermitian extension of the Benalcazar-Bernevig-Hughes model.

Abstract

In contrast to eigenvalue-based approaches, we formulate the bulk-boundary correspondence for two-dimensional non-Hermitian quadratic lattice Hamiltonians in terms of Toeplitz operators and singular values, which correctly capture the stability, localization, and scaling of edge and corner modes. We show that singular values, rather than eigenvalues, provide the only stable foundation for topological protection in non-Hermitian systems because they remain robust under translational-symmetry-breaking perturbations that destabilize the eigenvalue spectrum, rendering it unsuitable for topological classification. Building on Toeplitz operator theory, we establish general results for non-Hermitian Hamiltonians defined on half and quarter planes, relating the topological indices of the associated Toeplitz operators to the number of finite-size singular values that are separated from the bulk singular-value spectrum and vanish in the thermodynamic limit. This yields a precise bulk-boundary correspondence for edge and corner modes, including higher-order topological phases, without requiring crystalline symmetries. We illustrate our general results with detailed examples exhibiting topologically protected families of edge states, coexisting edge and corner modes, and phases with both gapped bulk and edges supporting only stable corner modes. The latter is exemplified by a non-Hermitian generalization of the Benalcazar-Bernevig-Hughes model.