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.
Categories: quant-ph cond-mat.mes-hall cond-mat.quant-gas