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arxiv: 2410.18616 · v4 · submitted 2024-10-24 · 🪐 quant-ph · cond-mat.mes-hall· physics.optics

Spectral Riemann Sheet Topology of Gapped Non-Hermitian Systems

Anton Montag , Alexander Felski , Flore K. Kunst This is my paper

Pith reviewed 2026-05-23 18:30 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.mes-hallphysics.optics
keywords non-Hermitian systemsexceptional pointsRiemann sheetsbranch cutstopological configurationsBrillouin zonegapped spectra
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0 comments X

The pith

Threading exceptional points across the Brillouin zone boundary in gapped non-Hermitian systems creates a protected closed branch cut that defines topologically distinct spectral configurations.

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

The paper shows that gapped non-Hermitian systems support topological configurations in their complex energy spectra. These arise when exceptional points on the Riemann sheets are moved across the Brillouin zone edge and annihilated, which leaves behind a non-trivially closed branch cut. The gap in the spectrum protects this feature. For systems that remain fully non-degenerate, the presence or absence of the closed branch cut marks distinct topological classes. Any change from one class to the other requires the gap to close and exceptional points to appear.

Core claim

In gapped non-Hermitian systems the distinctive exceptional points on the energy Riemann sheets can be threaded across the Brillouin zone boundary and annihilated while the protecting energy gap stays open. The result is a non-trivially closed branch cut whose presence or absence distinguishes topologically different configurations even when the spectrum has no degeneracies at all. Transitions between these configurations are possible only when the gap closes and exceptional points form.

What carries the argument

The energy Riemann sheet topology carried by closed branch cuts that survive the annihilation of exceptional points after they cross the Brillouin zone boundary.

If this is right

  • Fully non-degenerate gapped spectra fall into at least two topologically distinct classes distinguished only by the Riemann sheet connection.
  • Any adiabatic tuning that switches between these classes must pass through a gap-closing point where exceptional points appear.
  • The closed branch cut remains stable as long as the energy gap stays open.
  • Realizations are possible in metasurfaces and single-photon interferometry setups.

Where Pith is reading between the lines

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

  • The same mechanism may classify steady states of open quantum systems whose effective non-Hermitian Hamiltonians exhibit gaps.
  • It could link spectral topology to the robustness of observables measured in interferometric experiments.
  • Numerical checks on finite-size lattices with periodic boundaries could confirm whether the closed branch cut persists when the Brillouin zone is discretized.

Load-bearing premise

Exceptional points on the Riemann sheets can be moved across the Brillouin zone boundary and annihilated without forcing the energy gap to close.

What would settle it

A concrete lattice model in which two configurations with and without the closed branch cut can be connected by a continuous parameter change that never produces exceptional points or closes the gap.

Figures

Figures reproduced from arXiv: 2410.18616 by Alexander Felski, Anton Montag, Flore K. Kunst.

Figure 1
Figure 1. Figure 1: Illustration showing the real part of the Riemann [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: (a): Illustration of the toric code on a square lattice given periodic boundary conditions. Physical qubit degrees of freedom are [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The real part of the spectral Riemann surface structures [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
read the original abstract

We show topological configurations of the complex-valued spectra in gapped non-Hermitian systems. These arise when the distinctive EPs in the energy Riemann sheets of such models are annihilated after threading them across the boundary of the Brillouin zone. This results in a non-trivially closed branch cut that is protected by an energy gap in the spectrum. Their presence or absence establishes topologically distinct configurations for fully non-degenerate systems and tuning between them requires a closing of the gap, forming exceptional point degeneracies. We provide an outlook toward experimental realizations in metasurfaces and single-photon interferometry.

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 claims that topological configurations of complex-valued spectra arise in gapped non-Hermitian systems when exceptional points (EPs) on the energy Riemann sheets are annihilated after being threaded across the Brillouin zone boundary. This produces a non-trivially closed branch cut protected by an energy gap. Presence or absence of this feature distinguishes topologically inequivalent fully non-degenerate gapped spectra, with transitions between them necessarily requiring gap closure at EPs. An outlook toward experimental realizations in metasurfaces and single-photon interferometry is included.

Significance. If the central construction holds, the work would introduce a Riemann-sheet-based topological classification for gapped non-Hermitian spectra that is distinct from conventional point-gap or line-gap invariants. The experimental outlook indicates possible relevance to photonic platforms. No machine-checked proofs, reproducible code, or parameter-free derivations are described.

major comments (2)
  1. [Abstract] Abstract (and throughout): the load-bearing claim that EPs can be threaded across the BZ boundary and annihilated while a protecting energy gap remains open is asserted without an explicit model Hamiltonian, Riemann-surface construction, or derivation showing that the process is forced by periodicity and cannot be continuously undone inside the gapped regime. This leaves the topological distinction between configurations unestablished.
  2. [Abstract] Abstract: the 'energy gap' that protects the closed branch cut is never defined (real-part gap, imaginary-part gap, or distance in the complex plane), so it is impossible to verify whether the gap remains open during threading or whether the resulting configuration is invariant under gapped deformations that avoid EPs.
minor comments (2)
  1. [Abstract] The abstract states that the configurations apply to 'fully non-degenerate systems,' yet EPs are degeneracies; the relation between these statements requires clarification.
  2. The manuscript provides no concrete example or figure illustrating the threading process, the resulting branch-cut topology, or the distinction between the claimed configurations.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their detailed reading and for identifying points where the presentation of our central claims can be strengthened. We respond to each major comment below and will incorporate the suggested clarifications in a revised manuscript.

read point-by-point responses

  1. Referee: [Abstract] Abstract (and throughout): the load-bearing claim that EPs can be threaded across the BZ boundary and annihilated while a protecting energy gap remains open is asserted without an explicit model Hamiltonian, Riemann-surface construction, or derivation showing that the process is forced by periodicity and cannot be continuously undone inside the gapped regime. This leaves the topological distinction between configurations unestablished.

    Authors: The referee correctly notes that the abstract summarizes the result at a high level. The full manuscript contains a concrete non-Hermitian lattice Hamiltonian together with explicit Riemann-surface plots that illustrate the threading and pairwise annihilation of exceptional points when they cross the Brillouin-zone boundary. Nevertheless, we agree that a compact, self-contained derivation of why periodicity forces the annihilation (and why the process cannot be continuously reversed while the gap stays open) is not sufficiently highlighted. We will add a dedicated paragraph and accompanying figure in the revised introduction that derives this topological obstruction directly from the periodic identification of the Brillouin zone, thereby establishing the inequivalence of the resulting spectral configurations. revision: yes

  2. Referee: [Abstract] Abstract: the 'energy gap' that protects the closed branch cut is never defined (real-part gap, imaginary-part gap, or distance in the complex plane), so it is impossible to verify whether the gap remains open during threading or whether the resulting configuration is invariant under gapped deformations that avoid EPs.

    Authors: We accept that an explicit definition is required. In the revised manuscript we will define the protecting gap as the minimum Euclidean distance in the complex plane between any pair of distinct eigenvalues. We will then demonstrate, both analytically for the model Hamiltonian and numerically across the deformation path, that this distance remains strictly positive while the exceptional points are threaded and annihilated, and that any continuous deformation preserving the gap cannot undo the closed branch cut without forcing an exceptional-point degeneracy. revision: yes

Circularity Check

0 steps flagged

No circularity detected; derivation self-contained

full rationale

The abstract presents a conceptual construction: EPs in Riemann sheets are threaded across the BZ boundary and annihilated while preserving a protecting energy gap, yielding a closed branch cut whose presence/absence distinguishes topologically distinct gapped spectra. No equations, parameter fits, self-citations, or ansatzes are supplied in the given text that reduce this distinction to a tautology or input by construction. The load-bearing step (threading/annihilation without gap closure) is asserted as a physical possibility rather than derived from a fitted quantity or prior self-referential result. This is the normal case of a non-circular topological claim.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, axioms, or invented entities are stated. Standard domain assumptions of non-Hermitian band theory are implicit but not detailed.

axioms (1)
  • domain assumption Non-Hermitian Hamiltonians possess complex spectra organized on Riemann sheets with exceptional points.
    This is the standard starting point for the described spectral topology.

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Reference graph

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