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

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.

💡 Research Summary

The paper addresses a long‑standing problem in the geometric description of non‑Hermitian quantum systems: the Berry connection, which underlies Berry phases, holonomies and topological invariants, is ambiguous because left and right eigenvectors of a non‑Hermitian Hamiltonian are independent and can be rescaled independently. This freedom is mathematically described by the non‑compact general linear group GL(N,ℂ). As a result, the conventional Berry connection defined as
(A^{\alpha\beta}\mu = i\langle\psi\alpha|\partial_\mu\psi_\beta\rangle)
with (\alpha,\beta\in{L,R}) yields four inequivalent connections. Under a generic GL(N,ℂ) transformation the connections acquire different gauge‑dependent imaginary parts, so even when the original connection is real the transformed one becomes complex. In the pure quantum regime this is unacceptable because the norm of the state vector represents a probability and must stay unity during adiabatic evolution.

To resolve the issue the authors introduce a positive‑definite metric operator (\eta) on the Hilbert space of the non‑Hermitian Hamiltonian. The metric satisfies the dynamical equation (\dot\eta = i\eta H - i H^\dagger \eta) and can be factorised as (\eta = S^\dagger S) with an invertible “vielbein’’ operator (S). Mapping the right eigenvectors (|\psi\rangle) to (|\phi_H\rangle = S|\psi\rangle) produces an equivalent Hermitian problem with Hamiltonian (H_H = S H S^{-1} + i(\partial_t S)S^{-1}). In the metric‑deformed Hilbert space ordinary derivatives cannot compare vectors at different parameter points because the inner product itself changes. Therefore a covariant derivative is defined: \