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International Journal of Modern Physics A

GEOMETRIC METHODS AND THE GAUGE STRUCTURE OF BERRY'S PHASE

Bernard Francis Bednarz

Published Oct 10, 1989• 2 citations
Publisher

Abstract

The use of geometric methods to obtain the connection associated with Berry's phase is discussed. Failures and successes in past attempts at using only geometry are presented. It is argued that this approach is most useful for Hamiltonians rotated on the parameter space.
DOI
10.1142/s0217751x89001734

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Non-Hermitian systems exhibit spectral and topological phenomena absent in Hermitian physics; however, their geometric characterization remains subtle due to the intrinsic ambiguity of biorthogonal eigenspaces. Since left and right eigenvectors are not related by Hermitian conjugation, the associated Berry connection is generally nonunique, leading to complex geometric phases and ambiguously defined holonomies. Here, we formulate a covariant geometric framework for non-Hermitian quantum systems based on the metric structure of the underlying Hilbert space. We show that, in the quantum regime with continuous state evolution, the conventional Berry connection and the associated Berry holonomy over closed parameter-space loops can be consistently defined only in the pseudo-Hermitian limit, where the spectrum is real. For generic non-Hermitian Hamiltonians with complex spectra, the relevant geometric object is instead the Aharonov-Anandan holonomy associated with cyclic evolution in projective Hilbert space. Within the pseudo-Hermitian regime, we construct a unique Hermitian Berry connection that is covariant under arbitrary GL(N,C) frame transformations and reduces to the standard Berry connection in the Hermitian limit. The resulting formalism separates the intrinsic geometry of the Hamiltonian eigenspace from contributions arising from the parameter dependence of the Hilbert-space metric, revealing that the conventional biorthogonal formulation generally mixes these distinct geometric effects. Consequently, geometric phases, synthetic gauge fields, and topological characteristics commonly attributed to non-Hermitian eigenspace geometry may, in part, originate from the underlying metric structure. Our framework therefore provides a consistent geometric foundation for Berry phases, non-Abelian holonomies, and topological invariants in non-Hermitian quantum systems.