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arxiv: 2605.19740 · v1 · pith:LCTNWUBYnew · submitted 2026-05-19 · ❄️ cond-mat.str-el · cond-mat.mes-hall· cond-mat.stat-mech· physics.optics· quant-ph

Filling-Sensitive Spectral Complexity from Hilbert-Space Holonomy in Fragmented Non-Hermitian Systems

Jiong-Hao Wang , Maria Zelenayova , Christopher Ekman , Emil J. Bergholtz This is my paper

Pith reviewed 2026-05-20 02:31 UTC · model grok-4.3

classification ❄️ cond-mat.str-el cond-mat.mes-hallcond-mat.stat-mechphysics.opticsquant-ph
keywords non-Hermitian many-body systemsHilbert-space holonomyfragmented Hamiltoniansspectral realityKrylov graphnonreciprocal hoppinggauge fieldsimilarity transformation
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The pith

Hilbert-space holonomy restricts complex spectra to the most symmetric sectors in fragmented non-Hermitian systems.

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

The paper shows that Hilbert-space holonomy acts as a geometric principle organizing whether spectra are real or complex in fragmented non-Hermitian many-body systems. This is complementary to usual symmetry protections. Complex eigenvalues appear exclusively in sectors with maximum symmetry, like half filling for the fermion model and zero magnetization for the spin chain. A shift by one particle or one spin flip makes the entire spectrum real, even with the same boundary conditions. The mechanism treats nonreciprocal hopping as a discrete gauge field on the Krylov graph of connected states, where only trivial holonomy permits a similarity transformation to a Hermitian system.

Core claim

In two minimal fragmented models, complex spectra can arise only within the most symmetric sectors: half filling in the fermion model and zero magnetization in the spin chain. Nonreciprocal hopping amplitudes are viewed as a discrete gauge field on the Krylov graph such that trivial holonomy permits a diagonal similarity transformation to the Hermitian limit while nontrivial holonomy obstructs it and allows complex spectra. In certain regimes, trivial holonomy admits an emergent-boundary interpretation, and longer-range models exhibit finite real and complex regions governed by the same criterion.

What carries the argument

Hilbert-space holonomy induced by nonreciprocal hopping viewed as a discrete gauge field on the Krylov graph, determining whether a similarity transformation to Hermitian form exists.

If this is right

  • Complex spectra are possible only in the half-filling sector of the fermion model and the zero-magnetization sector of the spin chain.
  • Adding or removing one particle renders all eigenvalues real despite unchanged periodic boundary conditions.
  • Flipping a single spin in the chain model produces an entirely real spectrum.
  • Longer-range models display finite regions of real and complex spectra still governed by the holonomy criterion.
  • Trivial holonomy permits an emergent-boundary interpretation in certain regimes.

Where Pith is reading between the lines

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

  • The filling or magnetization sensitivity offers a route to control spectral properties simply by changing particle number without altering the Hamiltonian form.
  • The same gauge-field picture on Krylov graphs could extend to other fragmented non-Hermitian models where subspaces are disconnected.
  • The criterion suggests experiments that tune hoppings to move between trivial and nontrivial holonomy sectors and observe the resulting spectral transition.

Load-bearing premise

Nonreciprocal hopping amplitudes can be viewed as a discrete gauge field on the Krylov graph such that trivial holonomy permits a diagonal similarity transformation to the Hermitian limit while nontrivial holonomy obstructs it.

What would settle it

A direct calculation or numerical diagonalization showing complex eigenvalues outside half filling in the fermion model or outside zero magnetization in the spin chain, with the same nonreciprocal terms and boundaries, would falsify the claim.

Figures

Figures reproduced from arXiv: 2605.19740 by Christopher Ekman, Emil J. Bergholtz, Jiong-Hao Wang, Maria Zelenayova.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗
read the original abstract

We show that Hilbert-space holonomy provides a geometric organizing principle for spectral reality in fragmented non-Hermitian many-body systems, complementary to conventional symmetry protection. In two minimal fragmented models, complex spectra can arise only within the most symmetric sectors: half filling in the fermion model and zero magnetization in the spin chain. Adding or removing a single particle, or flipping a single spin, renders the spectra entirely real despite unchanged periodic boundary conditions, reminiscent of boundary-condition sensitivity in systems with a non-Hermitian skin effect. We explain this by viewing nonreciprocal hopping amplitudes as a discrete gauge field on the Krylov graph: trivial holonomy permits a diagonal similarity transformation to the Hermitian limit, whereas nontrivial holonomy obstructs it and allows complex spectra. In certain regimes, trivial holonomy admits an emergent-boundary interpretation, and longer-range models exhibit finite real and complex regions governed by the same criterion.

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

0 major / 3 minor

Summary. The manuscript claims that Hilbert-space holonomy provides a geometric organizing principle for spectral reality in fragmented non-Hermitian many-body systems, complementary to conventional symmetry protection. In two minimal models, complex spectra arise only in the most symmetric sectors (half filling for fermions, zero magnetization for the spin chain). Nonreciprocal hopping is interpreted as a discrete gauge field on the Krylov graph such that trivial holonomy permits a diagonal similarity transformation rendering fragments Hermitian (real spectra), while nontrivial holonomy obstructs this and permits complex eigenvalues. The filling/magnetization dependence follows from which sectors close nontrivial loops; extensions to longer-range models and emergent-boundary interpretations are discussed.

Significance. If the geometric criterion holds, the work supplies a useful organizing principle that accounts for the observed filling sensitivity of spectral reality in fragmented non-Hermitian systems and links it to gauge consistency on the Krylov graph. This perspective is complementary to symmetry-based arguments and may generalize to other non-Hermitian many-body settings with fragmentation. The internal consistency of the holonomy-to-similarity-transformation link and the direct connection to sector-dependent loop structure are strengths that could stimulate further analytic and numerical studies.

minor comments (3)
  1. The abstract states that the holonomy-to-similarity link is explanatory, yet the main text would benefit from a short, self-contained derivation or explicit loop-product calculation for at least one filling/magnetization value to make the obstruction to the diagonal transformation fully transparent.
  2. Notation for the Krylov graph and the discrete gauge field could be introduced with a small diagram or table early in the manuscript to aid readers unfamiliar with the fragmentation literature.
  3. A brief comparison paragraph with existing non-Hermitian skin-effect literature would help situate the emergent-boundary interpretation mentioned for certain regimes.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading and positive evaluation of our work. The referee's summary accurately captures the central claim that Hilbert-space holonomy supplies a geometric organizing principle for spectral reality in fragmented non-Hermitian systems, complementary to symmetry protection. We appreciate the recognition of the internal consistency of the holonomy-to-similarity-transformation link and the connection to sector-dependent loop structure on the Krylov graph. We have prepared a revised manuscript addressing the recommendation for minor revision.

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper's central organizing principle—that nonreciprocal hopping acts as a discrete gauge field on the Krylov graph, with holonomy controlling the existence of a diagonal similarity transformation to a Hermitian limit—is introduced as a conceptual framework applied to the fragmented sectors of the minimal models. This framework directly accounts for the observed filling/magnetization dependence via graph connectivity and loop products without reducing any claimed prediction or first-principles result to a fitted parameter, self-definition, or self-citation chain. The fragmentation itself, the preservation under similarity transformations, and the sector-specific holonomy follow from the model Hamiltonians and boundary conditions as stated, rendering the derivation self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the interpretive step that nonreciprocal terms define a discrete gauge field whose holonomy directly controls the existence of a similarity transformation to Hermitian form. No free parameters or new entities are introduced in the abstract; the gauge-field view is a modeling choice rather than a derived result.

axioms (1)
  • domain assumption Nonreciprocal hopping amplitudes admit a discrete gauge-field representation on the Krylov graph of the fragmented Hilbert space.
    This premise is invoked to link holonomy to the possibility of a similarity transformation.

pith-pipeline@v0.9.0 · 5715 in / 1237 out tokens · 39348 ms · 2026-05-20T02:31:35.502085+00:00 · methodology

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Reference graph

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