Resolving Gauge Ambiguities of the Berry Connection in Non-Hermitian Systems

TL;DR

This paper introduces a covariant formalism to resolve gauge ambiguities in the Berry connection of non-Hermitian systems, providing a consistent geometric framework for topological phenomena.

Contribution

It develops a unique, Hermitian covariant Berry connection based on the metric tensor, eliminating gauge ambiguities in non-Hermitian quantum systems.

Findings

The covariant Berry connection is uniquely defined and Hermitian.

The formalism is covariant under GL(N,C) transformations.

It recovers the standard Berry connection in the Hermitian limit.

Abstract

Non-Hermitian systems exhibit spectral and topological phenomena absent in Hermitian physics; however, their geometric characterization is hindered by an intrinsic ambiguity rooted in the eigenspace of non-Hermitian Hamiltonians, which becomes especially pronounced in the pure quantum regime. Since left and right eigenvectors are not related by conjugation, their norms are not fixed, giving rise to a biorthogonal ${\rm GL}(N,\mathbb{C})$ gauge freedom. As a result, the conventional Berry connection admits four inequivalent definitions, depending on how left and right eigenvectors are paired, leading to generally complex-valued geometric phases and ambiguous holonomies. {Here we show that these ambiguities are naturally resolved within a covariant formalism based on the metric tensor of the Hilbert space of the underlying non-Hermitian Hamiltonian. The resulting covariant Berry connection is uniquely defined, Hermitian, and covariant under arbitrary ${\rm GL}(N,\mathbb{C})$ frame transformations, while recovering the standard Berry connection in the Hermitian limit. By decoupling the contributions of the eigenbundle geometry from the underlying metric, our framework eliminates the gauge ambiguities inherent in the conventional biorthogonal approach, thereby establishing a consistent geometric foundation for Berry phases, non-Abelian holonomies, and topological invariants in non-Hermitian quantum systems.