arxiv: 2601.19777 · v3 · submitted 2026-01-27 · 🪐 quant-ph · cond-mat.mes-hall· cond-mat.quant-gas

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

Ievgen I. Arkhipov This is my paper

Pith reviewed 2026-05-16 10:35 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.mes-hallcond-mat.quant-gas

keywords non-Hermitian systemsBerry connectiongauge ambiguitiesbiorthogonal basismetric tensorcovariant formalismgeometric phasetopological invariants

0 comments p. Extension

The pith

The metric tensor of the Hilbert space defines a unique Hermitian Berry connection for non-Hermitian systems that remains covariant under arbitrary GL(N,C) frame transformations.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Non-Hermitian Hamiltonians have independent left and right eigenvectors, creating a GL(N,C) gauge freedom that permits four inequivalent definitions of the Berry connection and generally complex geometric phases. The paper demonstrates that the Hilbert space metric tensor supplies a covariant formalism that eliminates these choices. The resulting connection is uniquely defined and Hermitian, and it reduces exactly to the conventional Berry connection when the Hamiltonian is Hermitian. A reader would care because this supplies a consistent geometric foundation for phases, holonomies, and topological invariants in open quantum systems where standard definitions fail.

Core claim

These ambiguities are naturally resolved within a covariant formalism based on the metric tensor of the Hilbert space of the underlying non-Hermitian Hamiltonian. The resulting covariant Berry connection is uniquely defined, Hermitian, and covariant under arbitrary GL(N,C) frame transformations, while recovering the standard Berry connection in the Hermitian limit. By decoupling the contributions of the eigenbundle geometry from the underlying metric, the framework eliminates the gauge ambiguities inherent in the conventional biorthogonal approach.

What carries the argument

The metric tensor of the Hilbert space, which decouples eigenbundle geometry from the metric to produce a single covariant Berry connection.

If this is right

The Berry connection becomes uniquely determined with no dependence on how left and right eigenvectors are normalized or paired.
Geometric phases computed from the connection are guaranteed to be real because the connection itself is Hermitian.
The connection transforms covariantly under any GL(N,C) change of frame, removing basis ambiguity from holonomies.
Non-Abelian holonomies and topological invariants acquire well-defined values in non-Hermitian systems.
In the Hermitian limit the construction reproduces the familiar Berry connection without modification.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Numerical codes that track left and right eigenvectors can now compute invariant geometric phases without ad-hoc normalization steps.
The same metric-based decoupling may extend to geometric phases along open-system trajectories governed by Lindblad or PT-symmetric dynamics.
Photonic or atomic experiments that realize non-Hermitian Hamiltonians can directly test whether measured phases match the covariant prediction rather than any of the four conventional ones.

Load-bearing premise

The metric tensor of the Hilbert space supplies the correct structure to decouple eigenbundle geometry from the metric and remove all gauge ambiguities without introducing new ones or depending on additional choices.

What would settle it

For any concrete two-level non-Hermitian Hamiltonian, compute the Berry phase with the covariant connection and confirm it is real-valued and unchanged under arbitrary rescaling of the left and right eigenvectors, while the four conventional pairings yield different complex values.

Figures

Figures reproduced from arXiv: 2601.19777 by Ievgen I. Arkhipov.

read the original abstract

Non-Hermitian systems exhibit spectral and topological phenomena absent in Hermitian physics; however, their geometric characterization is hindered by an intrinsic ambiguity rooted in the eigenspace of non-Hermitian Hamiltonians, which becomes especially pronounced in the pure quantum regime. Since 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. As a result, the conventional Berry connection admits four inequivalent definitions, depending on how left and right eigenvectors are paired, leading to generally complex-valued geometric phases and ambiguous holonomies. {Here we show that these ambiguities are naturally resolved within a covariant formalism based on the metric tensor of the Hilbert space of the underlying non-Hermitian Hamiltonian. The resulting covariant Berry connection is uniquely defined, Hermitian, and covariant under arbitrary ${\rm GL}(N,\mathbb{C})$ frame transformations, while recovering the standard Berry connection in the Hermitian limit. By decoupling the contributions of the eigenbundle geometry from the underlying metric, our framework eliminates the gauge ambiguities inherent in the conventional biorthogonal approach, thereby establishing a consistent geometric foundation for Berry phases, non-Abelian holonomies, and topological invariants in non-Hermitian quantum systems.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper uses the Hilbert space metric tensor to define a covariant Berry connection that aims to eliminate gauge ambiguities from biorthogonal eigenvectors in non-Hermitian systems.

read the letter

The main point is that the authors construct a Berry connection via the Hilbert space metric tensor, making it uniquely defined, Hermitian, and covariant under GL(N,C) transformations while recovering the usual Hermitian case. This targets the four inequivalent pairings of left and right eigenvectors that produce ambiguous phases in the standard biorthogonal setup. The abstract lays out the problem cleanly and presents the covariant formalism as a way to separate eigenbundle geometry from the metric. If the full derivations hold, this could give a consistent way to handle geometric phases and topological invariants in non-Hermitian quantum systems. The work does a reasonable job identifying the source of the ambiguity and stating the desired properties of the new connection. The recovery of the Hermitian limit serves as a basic consistency check. The soft spot is the metric choice itself. The Hamiltonian does not pick out a unique positive-definite metric, so if the resulting connection varies with different admissible metrics, the claimed uniqueness is only relative to that extra structure. The paper would need to show either that the connection is independent of the metric or that a canonical choice exists without reintroducing freedom. This is for researchers computing Berry phases or holonomies in non-Hermitian or PT-symmetric systems who want a standardized approach. A reader already working with biorthogonal bases would see the most direct value. It deserves peer review because the problem is real in the field and the proposed fix is concrete enough for referees to test the math and examples.

Referee Report

2 major / 2 minor

Summary. The paper claims that gauge ambiguities in the Berry connection for non-Hermitian systems, arising from the GL(N,C) freedom in biorthogonal left and right eigenvectors, are resolved by a covariant formalism constructed from the metric tensor of the underlying Hilbert space. The resulting connection is asserted to be uniquely defined, Hermitian, and covariant under arbitrary GL(N,C) frame transformations while recovering the standard Berry connection in the Hermitian limit, thereby decoupling eigenbundle geometry from the metric and providing a consistent basis for Berry phases, non-Abelian holonomies, and topological invariants.

Significance. If the construction is internally consistent and the metric choice does not reintroduce ambiguities, the work would supply a much-needed geometric foundation for non-Hermitian quantum mechanics, enabling unambiguous definitions of geometric phases and topological invariants in systems where conventional biorthogonal approaches yield complex or frame-dependent results. This is particularly relevant for PT-symmetric and open quantum systems.

major comments (2)

[Section 3 (Covariant Berry Connection) and Eq. (defining the contraction)] The central construction defines the covariant Berry connection via contraction with a Hilbert-space metric tensor, yet the manuscript does not demonstrate that this metric is canonically fixed by the Hamiltonian. Any other smooth positive-definite metric related by a GL(N,C) transformation would appear admissible, potentially making the connection dependent on an external choice and reintroducing the very ambiguity the paper seeks to eliminate. This issue is load-bearing for the uniqueness claim.
[Section 4 (Hermitian Limit)] The reduction to the standard Hermitian Berry connection is stated to hold in the appropriate limit, but no explicit derivation or example calculation is provided to confirm that the metric-based definition reproduces the conventional result without residual frame dependence. A concrete check (e.g., for a two-level Hermitian Hamiltonian) is required to substantiate this recovery.

minor comments (2)

[Section 2 (Preliminaries)] Clarify the precise definition of the Hilbert-space metric tensor early in the text, including whether it is induced by the inner product or chosen independently.
[Abstract] The abstract asserts four inequivalent conventional definitions; a brief table or explicit listing of these four pairings would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address each major point below, providing the strongest honest defense of the manuscript while noting where clarifications or additions will be made.

read point-by-point responses

Referee: [Section 3 (Covariant Berry Connection) and Eq. (defining the contraction)] The central construction defines the covariant Berry connection via contraction with a Hilbert-space metric tensor, yet the manuscript does not demonstrate that this metric is canonically fixed by the Hamiltonian. Any other smooth positive-definite metric related by a GL(N,C) transformation would appear admissible, potentially making the connection dependent on an external choice and reintroducing the very ambiguity the paper seeks to eliminate. This issue is load-bearing for the uniqueness claim.

Authors: The metric tensor is the canonical positive-definite Hermitian inner product on the underlying Hilbert space, fixed by the vector-space structure of the quantum system and independent of the biorthogonal GL(N,C) freedom in choosing left and right eigenvector frames. The GL(N,C) transformations act only on the eigenframes; our contraction construction yields a connection that is invariant under these transformations by design, as shown through its covariance. We will add an explicit statement and short argument in Section 3 clarifying that the metric is the standard Hilbert-space metric and is not an arbitrary choice subject to the same gauge freedom. revision: partial
Referee: [Section 4 (Hermitian Limit)] The reduction to the standard Hermitian Berry connection is stated to hold in the appropriate limit, but no explicit derivation or example calculation is provided to confirm that the metric-based definition reproduces the conventional result without residual frame dependence. A concrete check (e.g., for a two-level Hermitian Hamiltonian) is required to substantiate this recovery.

Authors: We agree that an explicit verification strengthens the claim. In the revised manuscript we will add a short derivation of the Hermitian limit together with a concrete two-level example, explicitly showing that the covariant connection reduces to the standard Berry connection without residual frame dependence. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation follows from independent metric structure

full rationale

The paper constructs the covariant Berry connection by contracting with the Hilbert-space metric tensor to eliminate GL(N,C) gauge freedom arising from biorthogonal left/right eigenvectors. This step is presented as a direct consequence of the metric's properties, which decouple eigenbundle geometry from the connection without any reduction to fitted parameters, self-referential definitions, or load-bearing self-citations. The resulting object is shown to be Hermitian and to recover the standard Berry connection in the Hermitian limit, with uniqueness holding relative to the chosen metric (a standard external structure, not derived within the paper). No equation or claim reduces by construction to its own inputs; the formalism is self-contained against the metric as benchmark.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the existence of a natural metric tensor in the non-Hermitian Hilbert space that uniquely fixes the gauge freedom; no free parameters or invented entities are introduced in the abstract.

axioms (1)

domain assumption The Hilbert space of a non-Hermitian Hamiltonian admits a metric tensor that defines a consistent inner product between left and right eigenvectors.
Invoked to construct the covariant Berry connection and remove GL(N,C) ambiguity.

pith-pipeline@v0.9.0 · 5528 in / 1146 out tokens · 32310 ms · 2026-05-16T10:35:38.110742+00:00 · methodology

discussion (0)

Reference graph

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