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

• URL: http://arxiv.org/abs/2601.19777v2
• Date: Mon, 02 Feb 2026 09:41:19 GMT
• Title: Resolving Gauge Ambiguities of the Berry Connection in Non-Hermitian Systems
• Authors: Ievgen I. Arkhipov,
• Abstract summary: We introduce a covariant-derivative formalism built from the metric tensor of the Hilbert space of the underlying non-Hermitian Hamiltonian.<n>The resulting uniquely defined Berry connection remains real-valued under an arbitrary $rm GL(N,mathbb C)$ frame change.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Non-Hermitian systems display spectral and topological phenomena absent in Hermitian physics; yet, their geometric characterization can be hindered by an intrinsic ambiguity rooted in the eigenspace of non-Hermitian Hamiltonians, which becomes especially pronounced in the pure quantum regime. Because 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. Consequently, the standard Berry connection admits four inequivalent definitions depending on how left and right eigenvectors are paired, giving rise to distinct Berry phases and generally complex-valued holonomies. Here we show that these ambiguities and the emergence of complex phases are fully resolved by introducing a covariant-derivative formalism built from the metric tensor of the Hilbert space of the underlying non-Hermitian Hamiltonian. The resulting uniquely defined Berry connection remains real-valued under an arbitrary ${\rm GL}(N,{\mathbb C})$ frame change, and transforms as an affine gauge potential, while reducing to the conventional Berry (or Wilczek-Zee) connection in the Hermitian limit. This establishes an unambiguous and gauge-consistent geometric framework for Berry phases, non-Abelian holonomies, and topological invariants in quantum systems described by non-Hermitian Hamiltonians.

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