arxiv: 2603.16838 · v3 · submitted 2026-03-17 · ❄️ cond-mat.mes-hall

Recognition: 2 theorem links

· Lean Theorem

Complex Wannier centers and drifting Wannier functions in non-Hermitian Hamiltonians

Pedro Fittipaldi de Castro , Yifan Wang , Wladimir A. Benalcazar

Authors on Pith no claims yet

Pith reviewed 2026-05-15 09:39 UTC · model grok-4.3

classification ❄️ cond-mat.mes-hall

keywords non-Hermitian topologyWannier centersWilson loopspseudo-Hermitian systemsfilling anomalyedge modesbiorthonormal quantum mechanicsKrein signatures

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The pith

Complex Wannier centers from nonunitary Wilson loops predict filling anomalies and whether edge modes gain or lose energy in non-Hermitian systems.

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

This paper develops a way to extract polarization and related topological features from non-Hermitian Hamiltonians that have line energy gaps. It defines complex Wannier centers as the gauge-invariant eigenvalues of the resulting nonunitary Wilson loops, using biorthonormal inner products. When these centers move off the real axis, the associated Wannier functions break reciprocity and acquire an imaginary momentum shift that produces observable directional drift over time. In the presence of pseudo-Hermiticity, conserved Krein signatures force the centers to lie on the real axis or appear in conjugate pairs, and this structure directly controls a bulk-boundary correspondence that links the centers to the existence of a filling anomaly and to the gain or loss experienced by edge modes.

Core claim

In non-Hermitian Hamiltonians with line gaps, nonunitary Wilson loops possess gauge-invariant eigenvalues that define complex Wannier centers. These centers carry physical content because the corresponding Wannier functions break reciprocity: an imaginary part in the center produces an effective momentum offset and therefore directional drift of the wave-packet centroid. When the Hamiltonian is pseudo-Hermitian, the projected metric operator supplies a Krein signature that restricts the centers to be real or to form complex-conjugate pairs. The Krein structure of the Wilson loop then establishes a bulk-boundary correspondence: the configuration of the centers determines whether the occupied,

What carries the argument

Complex Wannier centers, defined as the gauge-invariant eigenvalues of nonunitary Wilson loops under biorthonormal quantum mechanics, whose imaginary parts encode directional drift of the associated Wannier functions.

If this is right

Symmetries in pseudo-Hermitian systems force complex Wannier centers to be real or to appear in conjugate pairs, fixed by the Krein signatures of the projected metric.
The Krein structure of the Wilson loop determines the presence of a filling anomaly in the occupied bands.
The same Krein structure decides whether the resulting edge modes experience net gain or net loss.
A specific lattice implementation is proposed that makes these predictions experimentally testable.

Where Pith is reading between the lines

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

The directional drift implied by imaginary centers should appear in time-resolved wave-packet measurements, offering a direct dynamical signature.
The framework may generalize to other non-Hermitian topological markers that also become nonunitary.
The same Krein-signature logic could constrain complex centers in PT-symmetric or other symmetry classes not treated here.

Load-bearing premise

That the eigenvalues of nonunitary Wilson loops can be interpreted directly as physically meaningful complex Wannier centers whose imaginary parts produce observable directional drift, relying on biorthonormal quantum mechanics without additional verification.

What would settle it

A numerical or experimental check, in a concrete lattice model, of whether the predicted directional drift of a wave packet matches the imaginary part of the computed complex Wannier center, and whether the observed filling anomaly and edge-mode gain or loss follow the Krein signatures of the Wilson loop.

Figures

Figures reproduced from arXiv: 2603.16838 by Pedro Fittipaldi de Castro, Wladimir A. Benalcazar, Yifan Wang.

**Figure 2.** Figure 2: FIG. 2: Complex Wannier centers and nonreciprocal Wannier [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3: Dynamics of Wannier functions of the model [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

**Figure 5.** Figure 5: (a) shows the excellent agreement between the direct diagonalization of 𝑃𝑋𝑃 and the Γ-dependent expression. The remaining finite-size deviations originate from the discretization of the Brillouin zone and decay exponentially with ℓ, as FIG. 5: (a) Momentum distribution ∥𝑤𝑘 ∥ 2 of Wannier functions from direct diagonalization of 𝑃𝑋𝑃 (circles) and from equation (30) (red line) for system size ℓ = 800. (b) M… view at source ↗

**Figure 4.** Figure 4: FIG. 4: Dynamics of Wannier functions of the model [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗

**Figure 6.** Figure 6: FIG. 6: Protection of real WCs in the model [PITH_FULL_IMAGE:figures/full_fig_p011_6.png] view at source ↗

**Figure 7.** Figure 7: FIG. 7: Bulk boundary correspondence in regime (i) of model [PITH_FULL_IMAGE:figures/full_fig_p012_7.png] view at source ↗

**Figure 9.** Figure 9: FIG. 9: Schematic photonic implementation of model [PITH_FULL_IMAGE:figures/full_fig_p016_9.png] view at source ↗

read the original abstract

The extension of topological band theory to non-Hermitian Hamiltonians with line energy gaps remains largely unexplored, despite early indications of rich underlying physics. In these systems, Wilson loops, the objects characterizing polarization, become nonunitary. Yet, the physical consequences of this nonunitarity have remained unclear. Using biorthonormal quantum mechanics, we introduce the concept of complex Wannier centers, defined from the gauge-invariant eigenvalues of nonunitary Wilson loops. Complex Wannier centers acquire physical meaning through the breaking of reciprocity in their associated Wannier functions. When the centroid of a Wannier function shifts into the complex plane, it acquires an effective momentum offset that produces directional drift over time. We analyze how symmetries constrain complex Wannier centers and identify symmetry-protected Wannier configurations in pseudo-Hermitian Hamiltonians, where the centers are either real or form complex-conjugate pairs, as determined by conserved "Krein signatures" of the projected metric operator of pseudo-Hermiticity. We further show that the Krein structure of the Wilson loop can establish a bulk-boundary correspondence: in a system with anticommuting pseudo-Hermitian metric and (pseudo) inversion symmetries, the behavior of complex Wannier centers predicts the existence of a filling anomaly in the occupied bands and whether the resulting edge modes experience gain or loss. Finally, we propose an implementation of this system that enables experimental tests of our predictions.

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 defines complex Wannier centers from nonunitary Wilson loops and uses their Krein signatures to predict filling anomalies plus edge-mode gain or loss in pseudo-Hermitian systems.

read the letter

The core advance is the definition of complex Wannier centers as the gauge-invariant eigenvalues of nonunitary Wilson loops in line-gapped non-Hermitian bands. These centers acquire meaning because their imaginary part shifts the centroid of the associated Wannier function into the complex plane, which produces a directional drift under time evolution via the biorthonormal inner product. The authors show how pseudo-Hermiticity plus inversion symmetry constrains the centers to be either real or complex-conjugate pairs, with the Krein signature of the projected metric operator fixing which case occurs. From that structure they derive a bulk-boundary rule: the pattern of real versus paired centers determines whether a filling anomaly appears and whether the resulting edge modes sit in the gain or loss sector. The steps that map the bulk Krein data through the biorthonormal projector to the open-boundary spectrum are written out explicitly and do not rely on fitting or extra assumptions beyond the stated symmetries. The logic holds up on its own terms. The main limitation is that the numerical illustrations of the drift are still somewhat schematic; a few more concrete lattice examples with explicit time evolution would make the physical effect easier to visualize. Otherwise the derivations look reproducible from the given definitions. This work is aimed at researchers already comfortable with Hermitian Wannier theory who want to extend it to open systems in condensed matter or photonics. It is coherent and grounded enough to merit a full referee process rather than a desk rejection.

Referee Report

1 major / 1 minor

Summary. The manuscript extends topological band theory to non-Hermitian Hamiltonians possessing line gaps by defining complex Wannier centers from the gauge-invariant eigenvalues of nonunitary Wilson loops within biorthonormal quantum mechanics. These centers are argued to acquire physical content through reciprocity breaking in the associated Wannier functions, producing directional drift. In pseudo-Hermitian systems, Krein signatures of the projected metric operator constrain the centers to be real or to appear in complex-conjugate pairs. The Krein structure is further claimed to furnish a bulk-boundary correspondence that predicts the existence of a filling anomaly together with the gain or loss character of the resulting edge modes. An experimental realization is proposed.

Significance. If the derivations are completed rigorously, the work supplies a concrete dictionary between nonunitary Wilson-loop data and observable edge-mode properties in non-Hermitian systems, extending the polarization-anomaly paradigm beyond Hermitian settings and offering testable predictions for directional drift and gain/loss imbalance.

major comments (1)

[Bulk-boundary correspondence (section developing the Krein-structure argument)] The bulk-boundary correspondence asserted in the final paragraph of the abstract (and developed in the main text) requires an explicit derivation showing how the Krein signatures (real eigenvalues versus complex-conjugate pairs) of the nonunitary Wilson-loop eigenvalues translate, via the biorthonormal projector and the pseudo-Hermitian metric operator, into a concrete boundary condition such as a jump in the integrated density of states or a specific pairing of left/right eigenvectors. Without this step the prediction of filling anomaly and edge-mode gain/loss remains an assertion rather than a theorem.

minor comments (1)

[Abstract] The abstract is information-dense; a brief sentence separating the definition of complex Wannier centers from the bulk-boundary claim would improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and constructive feedback on our manuscript. We address the single major comment below and will revise the manuscript accordingly to strengthen the presentation.

read point-by-point responses

Referee: [Bulk-boundary correspondence (section developing the Krein-structure argument)] The bulk-boundary correspondence asserted in the final paragraph of the abstract (and developed in the main text) requires an explicit derivation showing how the Krein signatures (real eigenvalues versus complex-conjugate pairs) of the nonunitary Wilson-loop eigenvalues translate, via the biorthonormal projector and the pseudo-Hermitian metric operator, into a concrete boundary condition such as a jump in the integrated density of states or a specific pairing of left/right eigenvectors. Without this step the prediction of filling anomaly and edge-mode gain/loss remains an assertion rather than a theorem.

Authors: We agree that the link between Krein signatures and boundary observables can be made fully explicit. In the revised manuscript we will insert a new subsection that derives the correspondence step by step: (i) we start from the gauge-invariant eigenvalues of the nonunitary Wilson loop and their Krein signatures under the projected metric operator; (ii) we apply the biorthonormal projector to these eigenvalues and show how real centers versus complex-conjugate pairs produce a discontinuity in the integrated density of states when the system is terminated; (iii) we demonstrate that the same signatures enforce a specific pairing of left and right eigenvectors at the edge, directly determining the gain or loss character of the resulting modes. This derivation will be carried out for the anticommuting pseudo-Hermitian metric plus (pseudo)inversion symmetry case highlighted in the abstract. We believe the added steps convert the correspondence from an outline into a self-contained argument. revision: yes

Circularity Check

0 steps flagged

No significant circularity: definitions and symmetry analysis remain independent of target predictions.

full rationale

The paper introduces complex Wannier centers explicitly as the gauge-invariant eigenvalues of nonunitary Wilson loops and then derives their physical interpretation and symmetry constraints from biorthonormal quantum mechanics and the pseudo-Hermitian metric. The bulk-boundary claim is presented as a consequence of these definitions plus the anticommuting symmetries, without any reduction of the predicted filling anomaly or edge-mode gain/loss back to a fitted parameter or self-referential definition. No self-citation chain is load-bearing for the central result, and no ansatz is smuggled in. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The framework rests on biorthonormal quantum mechanics for non-Hermitian operators and the existence of line energy gaps; pseudo-Hermiticity is invoked to constrain centers via Krein signatures.

axioms (2)

domain assumption Biorthonormal quantum mechanics provides the correct inner product for defining physical quantities in non-Hermitian systems
Used to assign meaning to complex Wannier centers and their drift.
domain assumption Non-Hermitian Hamiltonians with line gaps admit well-defined nonunitary Wilson loops
Central to extracting eigenvalues as complex centers.

invented entities (1)

Complex Wannier centers no independent evidence
purpose: Gauge-invariant characterization of polarization and drift in non-Hermitian bands
Defined directly from Wilson loop eigenvalues; no independent experimental signature provided in abstract.

pith-pipeline@v0.9.0 · 5565 in / 1420 out tokens · 29720 ms · 2026-05-15T09:39:30.166706+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Complex Wannier centers defined from gauge-invariant eigenvalues of nonunitary Wilson loops; Krein signatures of the projected metric operator of pseudo-Hermiticity
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Krein collisions and bulk-boundary correspondence via anticommuting pseudo-Hermitian metric and (pseudo) inversion symmetries

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

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Under the gauge transformation (B2), 𝐺 𝑘 transforms as 𝐺 𝑘 →𝐺 ′ 𝑘 =𝐿 ′† 𝑘+Δ 𝑅′ 𝑘 =(𝑈 −1 𝑘+Δ)𝐺 𝑘 𝑈𝑘,(B6) i.e., it is gauge-covariant

Wilson line elements and Wilson loops The biorthogonal Wilson line element connecting neighbor- ing momenta is 𝐺 𝑘 ≡𝐺 𝑘+Δ,𝑘 =𝐿 † 𝑘+Δ 𝑅𝑘,(B5) where Δ =2𝜋/ℓ for a discretization of the Brillouin zone intoℓ points. Under the gauge transformation (B2), 𝐺 𝑘 transforms as 𝐺 𝑘 →𝐺 ′ 𝑘 =𝐿 ′† 𝑘+Δ 𝑅′ 𝑘 =(𝑈 −1 𝑘+Δ)𝐺 𝑘 𝑈𝑘,(B6) i.e., it is gauge-covariant. The biorthog...

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Under Eq

Gauge dependence of the Berry connection In the continuum limit, the Wilson line element may be written as 𝐺 𝑘 =I−𝑖Δ𝐴(𝑘) + O (Δ 2), where 𝐴(𝑘)=−𝑖𝐿 † 𝑘 𝜕𝑘 𝑅𝑘 is the biorthogonal Berry connection. Under Eq. (B2), one finds the standard non-Abelian transformation law 𝐴(𝑘) →𝐴 ′ (𝑘)=𝑈 −1 𝑘 𝐴(𝑘)𝑈 𝑘 −𝑖𝑈 −1 𝑘 𝜕𝑘𝑈𝑘 .(B10) Hence, 𝐴(𝑘) is gauge dependent pointwise, ...

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The periodic (unitary) position operator in the crystal- momentum basis acts as a discrete shift, 𝑋= ∑︁ 𝑘 |𝑘+Δ⟩ ⟨𝑘| ⊗I 𝑁 ,Δ = 2𝜋 ℓ ,(E3) where𝑘+Δis understood modulo the Brillouin zone. Since 𝑃|𝑊 𝑥⟩=|𝑊 𝑥⟩, we may expand the Wannier function in the (right) Bloch basis as |𝑊𝑥⟩= ∑︁ 𝑘 |𝑘⟩ ⊗ |𝑤 𝑘⟩ = ∑︁ 𝑘 |𝑘⟩ ⊗𝑎 𝑘 |𝑢 𝑅 𝑘 ⟩,(E4) where |𝑤 𝑘⟩ is the orbital conten...

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