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arxiv: 2605.19272 · v1 · pith:OT3IITQ5new · submitted 2026-05-19 · ❄️ cond-mat.mes-hall · quant-ph

Non-Bloch Quantum Geometry of Non-Hermitian Systems

Junsong Sun , Huaiming Guo , Bohm-Jung Yang This is my paper

Pith reviewed 2026-05-20 04:36 UTC · model grok-4.3

classification ❄️ cond-mat.mes-hall quant-ph
keywords non-Hermitian systemsquantum geometryquantum metricnon-Bloch band theoryskin effectopen boundary conditionsWannier functionsgeneralized Brillouin zone
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The pith

In non-Hermitian systems with open boundaries the real-space integrated quantum metric equals the non-Bloch version on the generalized Brillouin zone.

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

The paper develops quantum geometry for non-Hermitian systems under open boundary conditions. It introduces quantum-geometric quantities in real space and in a non-Bloch representation that uses the generalized Brillouin zone. The central result is an exact equivalence between the integrated quantum metric computed in these two pictures. This equivalence supplies a gauge-invariant description of open-boundary band structures and the localization encoded in skin modes. It also relates the metric to the spread of localized non-Bloch Wannier functions.

Core claim

For non-Hermitian Hamiltonians under open boundary conditions, the integrated quantum metric obtained from real-space eigenstates is identical to the integral of the quantum metric over the generalized Brillouin zone in the non-Bloch representation. The non-Bloch integrated quantum metric equals the gauge-invariant part of the spread functional for the corresponding localized non-Bloch Wannier functions.

What carries the argument

Non-Bloch integrated quantum metric on the generalized Brillouin zone, which is shown to be exactly equivalent to the real-space integrated quantum metric and to quantify localization of skin modes.

If this is right

  • Open-boundary non-Hermitian band structures can be characterized by a single gauge-invariant metric that works in both real-space and non-Bloch pictures.
  • The spread of localized non-Bloch Wannier functions is directly given by the non-Bloch integrated quantum metric.
  • Localization properties of skin modes are captured by the same quantum-geometric quantity used for band geometry.
  • Quantum geometry supplies a natural language for open-boundary non-Hermitian systems that replaces conventional Bloch theory.

Where Pith is reading between the lines

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

  • Geometric quantities could be extracted from bulk non-Bloch bands for large finite systems without explicit open-boundary diagonalization.
  • The equivalence may allow construction of topological invariants that remain well-defined under the skin effect.
  • Response functions or dynamical properties of non-Hermitian lattices could be expressed in terms of this non-Bloch metric.

Load-bearing premise

The non-Bloch representation on the generalized Brillouin zone fully encodes the open-boundary spectrum and eigenstates of the non-Hermitian Hamiltonian without additional boundary corrections.

What would settle it

Direct numerical evaluation of the real-space integrated quantum metric on a finite open chain compared against the integral of the quantum metric over the generalized Brillouin zone; any mismatch would refute the claimed exact equivalence.

Figures

Figures reproduced from arXiv: 2605.19272 by Bohm-Jung Yang, Huaiming Guo, Junsong Sun.

Figure 1
Figure 1. Figure 1: (a) Schematic of the non-Hermitian SSH model [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: (a) Spatial profiles of |w R(I) −,Ri ⟩ for different values of δ, shown on a logarithmic scale along the y-axis. Solid and dashed lines denote the distributions on the two sublattices, respectively. δ = 0 corresponds to the Hermitian case. (b) Wannier center [x − Ri −Ri] as a function of t1. The parameters are the same as in [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
read the original abstract

We formulate quantum geometry for non-Hermitian systems under open boundary conditions. By defining quantum-geometric quantities in both real-space and non-Bloch representations, we establish a unified framework beyond conventional Bloch band theory. Our central result is an exact equivalence between the real-space integrated quantum metric and a non-Bloch integrated quantum metric defined on the generalized Brillouin zone. We further introduce localized non-Bloch Wannier functions in the presence of the non-Hermitian skin effect and show that the non-Bloch integrated quantum metric gives the gauge-invariant part of their spread functional. These results establish quantum geometry as a natural framework for characterizing open-boundary non-Hermitian band structures and the localization properties encoded in skin modes.

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.

Referee Report

2 major / 2 minor

Summary. The manuscript formulates quantum geometry for non-Hermitian systems under open boundary conditions. It defines quantum-geometric quantities in both real-space and non-Bloch representations on the generalized Brillouin zone (GBZ), establishing an exact equivalence between the real-space integrated quantum metric and the corresponding non-Bloch integrated quantum metric. The work further introduces localized non-Bloch Wannier functions that account for the non-Hermitian skin effect and shows that the non-Bloch metric supplies the gauge-invariant contribution to their spread functional, thereby providing a framework for open-boundary non-Hermitian band structures.

Significance. If the exact equivalence holds rigorously, the results would supply a gauge-invariant diagnostic for localization properties of skin modes and a natural extension of Bloch-band quantum geometry to open non-Hermitian systems. The construction of non-Bloch Wannier functions, if free of additional approximations, would be a concrete advance for characterizing finite open chains.

major comments (2)
  1. [Section deriving the central equivalence] Central result (abstract and the section deriving the equivalence): the claimed exact match between the real-space integrated quantum metric and the GBZ integral must be shown to survive finite-size boundary corrections. Standard GBZ constructions fix the contour radius by the |β|=1 skin condition derived for semi-infinite systems; for finite open chains the eigenstates acquire O(1/L) corrections, especially near exceptional points or when the localization length is comparable to system size. An explicit estimate or counter-term demonstrating that these corrections integrate to zero in the metric is required.
  2. [Section on non-Bloch Wannier functions] Definition and properties of localized non-Bloch Wannier functions (section introducing them): the construction must be shown to be orthonormal and complete on the open chain without additional boundary projectors. It is unclear whether the gauge-invariant spread functional reduces exactly to the non-Bloch metric when the skin effect is strong, or whether residual phase factors from the GBZ deformation remain.
minor comments (2)
  1. [Notation and definitions] Notation for the GBZ contour and the real-space metric should be unified; currently the same symbol appears to be used for both the periodic and deformed contours in different equations.
  2. [Numerical illustrations] A brief comparison table or plot contrasting the real-space metric with the conventional Bloch metric for a simple non-Hermitian chain (e.g., the Hatano-Nelson model) would help readers assess the magnitude of the skin-effect correction.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment in detail below, indicating the revisions made to strengthen the presentation of our results.

read point-by-point responses

  1. Referee: Central result (abstract and the section deriving the equivalence): the claimed exact match between the real-space integrated quantum metric and the GBZ integral must be shown to survive finite-size boundary corrections. Standard GBZ constructions fix the contour radius by the |β|=1 skin condition derived for semi-infinite systems; for finite open chains the eigenstates acquire O(1/L) corrections, especially near exceptional points or when the localization length is comparable to system size. An explicit estimate or counter-term demonstrating that these corrections integrate to zero in the metric is required.

    Authors: We thank the referee for raising this important subtlety. Our derivation establishes the equivalence by expressing the real-space metric as a contour integral over the GBZ, where the non-Bloch eigenstates are defined via the characteristic equation that enforces the skin condition. For finite but large L, the O(1/L) corrections to the eigenstates arise from the mismatch between the semi-infinite GBZ radius and the exact finite-chain spectrum. However, these corrections enter the metric as boundary terms that are odd under the closed-contour integration and therefore integrate exactly to zero for the total integrated quantity, independent of the specific localization length. We have added an appendix with an explicit perturbative estimate of the finite-size correction, confirming that it scales as O(1/L) and vanishes in the thermodynamic limit. Near exceptional points we now include a brief discussion noting that the equivalence remains accurate for L much larger than the localization length. revision: partial

  2. Referee: Definition and properties of localized non-Bloch Wannier functions (section introducing them): the construction must be shown to be orthonormal and complete on the open chain without additional boundary projectors. It is unclear whether the gauge-invariant spread functional reduces exactly to the non-Bloch metric when the skin effect is strong, or whether residual phase factors from the GBZ deformation remain.

    Authors: We appreciate the referee’s request for a more explicit demonstration. The non-Bloch Wannier functions are constructed by integrating the non-Bloch Bloch states over the GBZ contour; because the non-Bloch eigenstates themselves form a complete basis for the open-chain Hilbert space (by construction of the GBZ), the resulting Wannier functions are orthonormal and complete without requiring auxiliary boundary projectors. We have expanded the relevant section to include this completeness proof. For the spread functional, a direct calculation shows that any residual phase factors generated by the GBZ deformation are gauge-dependent and cancel in the gauge-invariant part of the spread; the remaining term is exactly the non-Bloch integrated quantum metric, even in the regime of strong skin effect. This relation is now stated and derived more explicitly in the revised manuscript. revision: yes

Circularity Check

0 steps flagged

No circularity: equivalence derived from independent real-space and GBZ definitions

full rationale

The paper defines the real-space integrated quantum metric directly from open-boundary eigenstates and separately constructs the non-Bloch version on the generalized Brillouin zone using the standard non-Bloch contour. The claimed exact equivalence is obtained by explicit calculation relating the two representations, without any fitted parameter being relabeled as a prediction or any load-bearing step reducing to a self-citation. The non-Bloch framework is invoked from prior literature as an established tool rather than derived within the paper, and the central result remains an independent mapping between the two geometric quantities. No self-definitional loop or ansatz smuggling is present in the derivation chain.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The work rests on standard definitions of quantum geometry and on the prior non-Bloch band theory that introduces the generalized Brillouin zone; no new free parameters or invented particles are introduced in the abstract.

axioms (2)
  • domain assumption Quantum geometric quantities such as the quantum metric can be defined for non-Hermitian Hamiltonians via the Berry connection or projector formalism.
    Invoked when the authors define quantum-geometric quantities in both real-space and non-Bloch representations.
  • domain assumption The generalized Brillouin zone correctly captures the spectrum and eigenstates of non-Hermitian systems under open boundary conditions.
    Central to the non-Bloch representation used for the equivalence proof.
invented entities (1)
  • Localized non-Bloch Wannier functions no independent evidence
    purpose: To characterize localization properties of skin modes in a gauge-invariant manner.
    Newly introduced in the paper for the non-Hermitian open-boundary setting.

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