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

TL;DR

This work addresses gauge ambiguities in the Berry geometry of non-Hermitian systems arising from biorthogonal left/right eigenvectors and a noncompact GL$(N,\mathbb{C})$ frame freedom. By introducing a parameter-dependent Hilbert-space metric $\eta=S^{\dagger}S$ and a metric-compatible covariant derivative $D_{\lambda}=\partial_{\lambda}+\Gamma^{\lambda}$ with $\Gamma^{\lambda}=S^{-1}\partial_{\lambda}S$, the authors define a covariant Berry connection $\mathbb{A}$ that remains real under general frame changes and transforms as an affine gauge potential: $\mathbb{A}'^{\lambda}=T^{-1}\tilde{\mathbb{A}}^{\lambda}T+iT^{-1}\partial_{\lambda}T$, with $\tilde{\mathbb{A}}^{\lambda}=\mathbb{A}^{\lambda}+\Xi^{\lambda}$ and $\Xi^{\lambda}=i\langle R|\eta(\Gamma'^{\lambda}-\Gamma^{\lambda})|R\rangle$. The corresponding curvature $F=\mathrm{d}\mathbb{A}-i\mathbb{A}\wedge\mathbb{A}$ transforms covariantly, ensuring gauge-invariant topological invariants. An explicit two-level example shows that, in the norm-preserving quantum regime, the CBC yields vanishing Berry curvature away from exceptional points, illustrating that previously claimed emergent (2+1)-D fields are absent under strict quantum-state norm conservation. Overall, the framework provides a unique, real, and gauge-consistent geometric description of Berry phases, non-Abelian holonomies, and topological invariants for non-Hermitian quantum evolution.

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 unique 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.

Figures (1)

• Figure 1: Symbolic illustration of the vielbein map $S$, according to Eq. (\ref{['psi-psi']}), which relates the deformed (e.g., stretched) Hilbert space ${\cal H}_{\rm NH}$ of the non-Hermitian Hamiltonian, equipped with the metric $\eta=S^\dagger S$, and characterized by nonequivalent local frames at different points, to the flat Hilbert space ${\cal H}_{\rm Herm}$ associated with the corresponding Hermitian Hamiltonian, which admits a single global frame.