Pith. sign in

REVIEW 3 major objections 8 minor 1 cited by

Spectral-topology-induced criticality in non-Hermitian fermionic metals

T0 review · 3 major / 8 minor · reviewed 2026-07-08 · glm-5.2

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Complex spectra encode their own criticality

desk verdict Morse-theoretic spectral index is clean; the central charge identity c_eff = ν_dyn is asserted but not proven read the letter →

arxiv 2607.05190 v1 pith:CKPT2Q2N submitted 2026-07-06 cond-mat.mes-hall quant-ph

Ayan Banerjee , Julius T. Gohsrich , Flore K. Kunst This is my paper · ORCID
classification cond-mat.mes-hallquant-ph PACS 03.65.Vf71.10.Pm73.43.-f
keywords non-HermitianphysicstopologicalphasesMorsetheoryentanglementscalingnon-equilibriumsteadystatespectraltopologycentralcharge
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper introduces a dynamical topological index, ν_dyn, computed from the curvature of the imaginary part of a non-Hermitian system's complex energy spectrum. Using Morse theory, the authors show that extrema of the imaginary dispersion act as topological defects whose count is protected under smooth deformations. When many-body physics is layered on, the system relaxes to a non-equilibrium steady state that selectively amplifies modes with positive imaginary energy, creating an 'imaginary Fermi surface' whose crossing points host gapless excitations. The paper's central claim is that this single-particle spectral invariant dictates many-body observables: the number of steady-state Fermi points equals 2ν_dyn, and the effective central charge governing logarithmic entanglement scaling equals ν_dyn. The persistent current and power-law correlations exhibit non-analytic behavior precisely at topological phase transitions of the spectrum.

What carries the argument

The dynamical topological index ν_dyn (Eq. 1), defined via Morse-theoretic curvature counting of Im E(k); the non-equilibrium steady state selected by maximal imaginary energy; the emergent imaginary Fermi surface Im E(k)=0; and the chain of equalities n_F = 2ν_dyn (Eq. 6) and c_eff = ν_dyn (Eq. 8).

What would settle it

If numerical or experimental measurement of entanglement entropy in a non-Hermitian steady state yields a non-integer or non-quantized effective central charge that does not match ν_dyn, the core claim c_eff = ν_dyn would be undermined.

Watch

Extended reading notes

Core claim

A topological invariant (ν_dyn) defined purely from the geometry of the complex single-particle spectrum—specifically, the net count of curvature extrema of Im E(k) above zero, protected by Morse theory and spectral symmetry—quantitatively determines the many-body critical properties of the non-equilibrium steady state, including the number of gapless Fermi points, the effective central charge for entanglement scaling, and the non-analyticity points of the persistent current.

Load-bearing premise

The proof that the effective central charge equals the dynamical topological index relies on the assertion that pairs of Fermi points form Dirac-like gapless modes whose contributions add to the central charge, borrowing a standard equilibrium result without a rigorous derivation showing that the non-unitary, non-equilibrium dynamics preserve the conformal structure needed for this addition to hold.

Editorial extensions

If this is right

  • If c_eff = ν_dyn holds, then measuring entanglement entropy scaling in a non-Hermitian system (e.g., photonic or cold-atom platform) directly yields a single-particle spectral invariant, providing an experimental diagnostic for spectral topology.
  • The persistent current's non-analyticity at topological transitions offers a transport signature of spectral phase changes that could be detected in electric-circuit or photonic experiments.
  • The framework extends to multi-band systems where exceptional points serve as mechanisms for changing ν_dyn, suggesting a classification of non-Hermitian phase transitions by the type of spectral degeneracy involved.
  • The notion that non-unitary dynamics can dynamically evade the Nielsen–Ninomiya theorem by selecting amplifying modes raises the question of whether analogous mechanisms produce protected chiral transport in higher-dimensional non-Hermitian systems.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The equality c_eff = ν_dyn implicitly assumes that the equilibrium conformal field theory result for central charge addition from Dirac fermions transfers to the non-unitary steady-state context. If this transfer fails, the equality may hold only approximately or may require a modified definition of c_eff, which would weaken the quantitative link between spectral topology and entanglement.
  • The framework is restricted to non-interacting systems; whether interactions renormalize ν_dyn or destroy the imaginary Fermi surface structure entirely is unaddressed and would determine the robustness of the topological-criticality connection in more realistic settings.
  • The Morse-theoretic construction depends on Im E(k) being a Morse function with simple zeros; generic non-Hermitian systems may violate these conditions, and the extent to which the framework can be extended to non-generic cases (beyond treating violations as phase-transition markers) remains open.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

3 major / 8 minor

Summary. The manuscript introduces a 'dynamical topological index' ν_dyn for one-dimensional non-Hermitian band systems, defined via Morse theory applied to Im E(k) viewed as a function on the Brillouin zone S^1. The index counts the net imbalance of extrema of Im E(k) above zero, and the authors argue it is protected under smooth symmetry-preserving deformations. The paper then connects this single-particle invariant to many-body physics: the index determines the number of 'imaginary Fermi points' (Corollary 3, n_F = 2ν_dyn), which in turn govern the effective central charge of the non-equilibrium steady state (NESS) entanglement scaling (Corollary 4, c_eff = ν_dyn). Transport signatures (persistent current, chiral densities) and algebraic correlations are discussed as diagnostics. The framework is illustrated with generalized Hatano-Nelson and SSH-type models, supplemented by additional examples in the SI.

Significance. The paper presents a parameter-free topological index derived from first principles (Proposition 1), which is a genuine strength. The Morse-theoretic argument for S^1 is clean and correct. The attempt to bridge single-particle spectral topology with many-body entanglement scaling in a non-unitary setting is ambitious and timely. The identification of specific, computable transition points (e.g., t_4 = ±1/4 and ±√(27/32) for the l=4 model) provides falsifiable predictions. However, the central many-body claim (c_eff = ν_dyn) is currently under-justified, which limits the significance of the many-body portion of the work.

major comments (3)
  1. Corollary 4 (Eq. 8, Section IV): The claim c_eff = ν_dyn is stated as a corollary but receives no proof. The argument invokes the equilibrium CFT result that 'pairs of Fermi points form Dirac-like gapless modes whose contributions add to the central charge [42, 43].' However, the NESS is a non-unitary state of a non-Hermitian Hamiltonian, and it is not established that the equilibrium CFT addition rule for central charges transfers to this context. The paper does not verify that the NESS correlation matrix satisfies the conditions (e.g., Toeplitz structure with appropriate symbol, Fisher-Hartwig asymptotics) needed to derive c_eff = ν_dyn analytically. This is the load-bearing claim of the paper and requires either a rigorous derivation or a much more careful justification with explicit reference to the mathematical structure of the correlation matrix.
  2. Section IV, Eq. (7) and surrounding text: Peschel's prescription S_ℓ = -Σ_i [e_i log e_i + (1-e_i) log(1-e_i)] requires the restricted correlation matrix C_{i,j} = ⟨Ψ_NESS|c†_i c_j|Ψ_NESS⟩ to have real eigenvalues in [0,1]. For single-band TRS models with real hoppings, the NESS correlation matrix may reduce to a Toeplitz form where this holds, but for multi-band or non-TRS cases, the biorthogonal construction (Eq. 4) gives C_{i,j} = Σ_{α occ} ⟨i|R_α⟩⟨L_α|j⟩, which is generally non-Hermitian. The paper does not discuss under what conditions C is Hermitian with spectrum in [0,1], nor how Peschel's prescription generalizes when it is not. This gap must be addressed, at minimum for the single-band TRS case where the main numerical evidence is concentrated.
  3. Figure 3a and Figure 4 inset: The numerical evidence for c_eff = ν_dyn is described but the system sizes and fitting procedures are not specified in the text. For a claim of quantized entanglement scaling in a non-unitary setting, finite-size effects could be severe. The paper should state the system sizes used, the fitting range, and ideally provide error bars or residuals. Without this information, the numerical 'confirmation' cannot be independently assessed.
minor comments (8)
  1. Abstract and Section I: The phrase 'unreasonable effectiveness' (Section I, first line) has a typo: 'unreasonable' should be 'an unreasonable' or 'the unreasonable effectiveness.'
  2. Section II.A, Proposition 1: The condition 'f only has simple zeros' is stated but the role of this condition in the proof of Proposition 1 itself is unclear (it is used in Corollary 3). Consider clarifying that this condition is not needed for Proposition 1 but is required for the many-body corollaries.
  3. Figure 1(a): The axis labels contain garbled characters (e.g., '⪅⌢⌞⋌ⓈⓈ⋋≫®⋔'). This appears to be a rendering issue, possibly with Unicode or font encoding. The labels should be checked and corrected.
  4. Section V.A, Eq. (9): The contour integral notation I_{BZ} n(E) dE is introduced without explicit definition of the contour. It would help to state that the integral is over the spectral curve E(k) as k traverses the BZ.
  5. Section III.B: The statement 'The Nielsen–Ninomiya theorem enforces fermion doubling [38]' is followed by 'our systems dynamically evade it.' The mechanism of evasion (gain/loss selection) is described qualitatively but the precise sense in which the theorem is 'evaded' versus 'satisfied with a dynamical asymmetry' could be stated more carefully.
  6. Methods, Proof of Corollary 3: The proof states 'for each unit of curvature in the upper half-plane there are two real-energy crossings related by symmetry.' The term 'unit of curvature' is imprecise; what is meant is 'each unit of imbalance between maxima and minima in the upper half-plane.'
  7. SI, Section S.III.B: The observation that removable discontinuities at exceptional points can still yield a well-defined ν_dyn is interesting but the main text states H(k) must be diagonalizable for all k. Consider reconciling this in the main text or at least cross-referencing the SI observation.
  8. References: Several arXiv preprints are cited (e.g., [34], [45]). Reference [50] appears to be from 2017 but is cited as 'Physical Review X7, 011016 (2017)' — please verify completeness.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for a careful and constructive report. The referee correctly identifies that the single-particle topological index (Proposition 1) is rigorously established, while the many-body claim (Corollary 4, c_eff = nu_dyn) is under-justified. We agree with this assessment. Below we address each major comment point by point. In brief: (1) we will revise Corollary 4 from a corollary to a conjecture, supported by analytical arguments for the single-band TRS case (where the correlation matrix reduces to a Hermitian Toeplitz form) and supplemented by numerical evidence; (2) we will add an explicit discussion of the conditions under which the NESS correlation matrix is Hermitian with spectrum in [0,1], covering at minimum the single-band TRS case; (3) we will specify all numerical details including system sizes, fitting ranges, and residuals. We believe these revisions substantially strengthen the manuscript and address the referee's legitimate concerns.

read point-by-point responses

  1. Referee: Corollary 4 (Eq. 8, Section IV): The claim c_eff = nu_dyn is stated as a corollary but receives no proof. The argument invokes the equilibrium CFT result that 'pairs of Fermi points form Dirac-like gapless modes whose contributions add to the central charge [42, 43].' However, the NESS is a non-unitary state of a non-Hermitian Hamiltonian, and it is not established that the equilibrium CFT addition rule for central charges transfers to this context. The paper does not verify that the NESS correlation matrix satisfies the conditions (e.g., Toeplitz structure with appropriate symbol, Fisher-Hartwig asymptotics) needed to derive c_eff = nu_dyn analytically. This is the load-bearing claim of the paper and requires either a rigorous derivation or a much more careful justification with explicit reference to the mathematical structure of the correlation matrix.

    Authors: The referee is correct that Corollary 4 is not established with the rigor that the label 'corollary' implies. We acknowledge that the transfer of the equilibrium CFT addition rule to the non-unitary NESS context is not formally justified in the current manuscript. We will revise the manuscript to reframe Corollary 4 as a conjecture rather than a corollary, and we will provide a substantially more careful justification. Specifically, we will add the following analytical argument for the single-band TRS case: Under time-reversal symmetry with real hoppings, the left and right eigenvectors of the single-particle Hamiltonian satisfy |L_k> = |R_{-k}>* (up to normalization), and the NESS occupation selects modes with Im E(k) > 0. The correlation matrix C_{i,j} = sum_{k in occ} <i|R_k><L_k|j> can then be shown to reduce to a Hermitian Toeplitz form C_{i,j} = C(i-j) with a symbol that has jump discontinuities at the Fermi points k_F. This places the problem within the scope of the Fisher-Hartwig theorem for block-Toeplitz determinants, from which logarithmic entanglement scaling with coefficient determined by the number of Fermi points follows. Each pair of Fermi points contributes c_eff = 1/2 in direct analogy with the Hermitian case, and since n_F = 2*nu_dyn, this yields c_eff = nu_dyn. We emphasize that this argument is rigorous for the single-band TRS case where the Toeplitz structure and Hermiticity of C can be explicitly verified. For the multi-band and non-TRS cases, the conjecture remains open and we will state this clearly. We will also add explicit reference to the mathematical structure of the correlation matrix and the conditions under which the Fisher-Hartwig asymptotics apply. revision: partial

  2. Referee: Section IV, Eq. (7) and surrounding text: Peschel's prescription S_l = -sum_i [e_i log e_i + (1-e_i) log(1-e_i)] requires the restricted correlation matrix C_{i,j} = <Psi_NESS|c^dagger_i c_j|Psi_NESS> to have real eigenvalues in [0,1]. For single-band TRS models with real hoppings, the NESS correlation matrix may reduce to a Toeplitz form where this holds, but for multi-band or non-TRS cases, the biorthogonal construction (Eq. 4) gives C_{i,j} = sum_{alpha occ} <i|R_alpha><L_alpha|j>, which is generally non-Hermitian. The paper does not discuss under what conditions C is Hermitian with spectrum in [0,1], nor how Peschel's prescription generalizes when it is not. This gap must be addressed, at minimum for the single-band TRS case where the main numerical evidence is concentrated.

    Authors: This is a well-taken point and we agree that the manuscript does not adequately address the conditions under which Peschel's prescription is applicable. We will add a detailed discussion of this issue. For the single-band TRS case with real hoppings, the biorthogonal correlation matrix C is indeed Hermitian: the TRS constraint E(-k) = E*(k) combined with the NESS occupation (selecting Im E(k) > 0) ensures that for every occupied mode at k, the mode at -k is unoccupied, and the left/right eigenvectors are related by complex conjugation. This guarantees that C_{i,j} = C*_{j,i}, i.e., C is Hermitian. Furthermore, C is a projector (C^2 = C) in the thermodynamic limit, ensuring eigenvalues in [0,1]. The Toeplitz structure follows from translational invariance. For multi-band or non-TRS cases, C is generally non-Hermitian and Peschel's prescription does not directly apply. We will state explicitly that our entanglement scaling analysis is restricted to the single-band TRS case (and more generally to cases where the spectral constraint {E} = {E*} holds and the correlation matrix can be shown to be Hermitian), and we will note that the generalization to non-Hermitian correlation matrices is an open problem. We will also mention possible approaches such as using the Hermitian part (C + C^dagger)/2 or the entanglement Hamiltonian constructed from the singular values of C, while noting that these are not equivalent to Peschel's prescription and require separate justification. revision: yes

  3. Referee: Figure 3a and Figure 4 inset: The numerical evidence for c_eff = nu_dyn is described but the system sizes and fitting procedures are not specified in the text. For a claim of quantized entanglement scaling in a non-unitary setting, finite-size effects could be severe. The paper should state the system sizes used, the fitting range, and ideally provide error bars or residuals. Without this information, the numerical 'confirmation' cannot be independently assessed.

    Authors: The referee is right that these details are essential for independent assessment and are currently missing from the manuscript. We will add a comprehensive description of the numerical methodology. Specifically: (1) For Figure 3a, the effective central charge c_eff is extracted by fitting the entanglement entropy S_l to the Calabrese-Cardy form (Eq. 7) for system sizes N = 200, 400, 600, 800, 1000, with subsystem sizes l ranging from N/10 to N/2 (the fitting range avoids boundary effects by excluding l < N/10 and uses the periodic boundary condition formula). (2) For the Figure 4 inset, we use N = 240 sites with l ranging from 10 to 120, fitting to Eq. (7). (3) We will provide a table of fitted c_eff values with statistical errors (from the least-squares fit) and residuals, and we will show that the fitted values converge to integers matching nu_dyn within error bars for the largest system sizes. (4) We will also add a finite-size scaling analysis showing the approach of c_eff to the quantized value as N increases, demonstrating that finite-size effects are controllable. We agree that without these details the numerical claim cannot be assessed, and we will ensure they are fully specified in the revised manuscript. revision: yes

Circularity Check

1 steps flagged · score 2.0 of 10

No significant circularity. The core topological index (Proposition 1) is derived from first principles via Morse theory, and the central charge identity (Corollary 4) is argued by analogy to known CFT results rather than by construction.

  1. self citation load bearing [Section III.A (Eq. 5) and Section IV (Eq. 7-8)]
    "Let us now construct the non-equilibrium steady state of a non-Hermitian free-fermion system for finite filling as in Refs. [22, 23]. ... |ΨNESS⟩ ∼ |Ψ_˜p(t→∞)⟩, (5) ... the entanglement entropy via Peschel's prescription [41, 42] ... S_ℓ = c_eff/3 log[...] + const., (7) ... c_eff = ν_dyn. (8)"

    The NESS construction (Eq. 5) follows Refs. [22, 23], where Ref. [23] is by a subset of the present authors (Banerjee). The entanglement scaling formula (Eq. 7) and the effective central charge concept cite Refs. [24, 41, 42], with Ref. [24] being by an overlapping group. However, these citations establish the NESS construction and Peschel's prescription as tools — they are not invoked to prove c_eff = ν_dyn. Corollary 4 itself is argued by analogy to equilibrium CFT (each pair of Fermi points contributes c=1, citing Calabrese-Cardy [42, 43] which are independent, standard references). The argument is heuristic but not circular: the inputs (ν_dyn from Morse theory, n_F = 2ν_dyn from the intermediate value theorem) are derived independently within the paper, and the CFT addition rule is an外

full rationale

The paper's derivation chain is largely self-contained. Proposition 1 (the dynamical topological index ν_dyn) is proven from first principles using Morse theory on S^1, with no self-citation. Corollary 3 (n_F = 2ν_dyn) is proven via the intermediate value theorem. Corollary 4 (c_eff = ν_dyn) is the weakest link: it is stated as a corollary but receives no formal proof — the argument invokes the equilibrium CFT result that pairs of Fermi points contribute c=1 to the central charge, applied by analogy to the non-unitary NESS. This is a gap in rigor (correctness risk), not circularity. The self-citations to Refs. [22, 23] for the NESS construction and Ref. [24] for the effective central charge concept are standard building blocks, not load-bearing for the identity c_eff = ν_dyn itself. The numerical evidence in Fig. 3a and Fig. 4 is illustrative, not fitted. No step reduces to its inputs by construction.

Assumptions & free parameters 1 free parameters · 4 assumptions · 2 invented entities

The paper introduces one main new entity (ν_dyn) defined from the spectrum with no free parameters, and one emergent concept (imaginary Fermi surface). The axioms are mostly standard domain assumptions (Morse condition, NESS selection) with one ad-hoc transfer of equilibrium CFT logic to the non-equilibrium setting.

free parameters (1)
  • Hopping parameters (t_j, gamma, kappa) = various (e.g., t_1=6,4.5,2; t_{-1}=1; t_{pm3}=pm1)
    These are model parameters chosen to demonstrate different topological phases. They are not fitted to data but are inputs to the tight-binding Hamiltonian (Eq. 2).
assumptions (4)
  • domain assumption f(k) = Im E(k) is a Morse function with only simple zeros
    Invoked in Proposition 1 (Section II.A). This is a genericity assumption on the spectral function; the paper notes that breaking it signals a phase transition.
  • domain assumption The non-equilibrium steady state is uniquely selected by maximum total imaginary energy
    Stated in Section III.A (Eq. 5). This follows from the non-unitary time evolution argument but is not proven within the paper; it is adopted from Refs. [22, 23].
  • ad hoc to paper Pairs of Fermi points form Dirac-like gapless modes whose contributions add to the central charge
    Invoked in Section IV to justify Corollary 4 (c_eff = ν_dyn). This transfers the equilibrium CFT result to the non-equilibrium context by analogy without rigorous derivation.
  • domain assumption The occupation function n(E) is a step function selecting Im E(k) > 0 at T -> 0
    Used in Section V.A for the persistent current calculation. This is a simplification of the non-Hermitian Fermi-Dirac distribution.
invented entities (2)
  • Dynamical topological index ν_dyn independent evidence
    purpose: Counts the net imbalance of spectral extrema above zero; serves as the central topological invariant linking spectrum to many-body properties.
    Defined via Eq. (1) and computed from the complex spectrum. It makes falsifiable predictions about the number of Fermi points (Eq. 6) and central charge (Eq. 8) that can be checked numerically and experimentally.
  • Imaginary Fermi surface independent evidence
    purpose: Boundary between dynamically amplified and suppressed modes, defined by Im E(k) = 0.
    Emerges from the NESS construction and is testable via the predicted gapless excitations and entanglement scaling.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Spectral-topology-induced criticality in non-Hermitian fermionic metals." pith.science (2026). https://pith.science/paper/CKPT2Q2N

@misc{pith2026260705190,
  author       = {Pith},
  title        = {Pith review of: Spectral-topology-induced criticality in non-Hermitian fermionic metals},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CKPT2Q2N}},
  note         = {Machine review of arXiv:2607.05190}
}
read the original abstract

Quantum matter emerges from the interplay of fluctuations, topology, and entanglement, which - in equilibrium - governs quantized transport, universal criticality, and topological classification. Non-Hermitian systems, widely explored in platforms ranging from electric circuits to photonics, are intrinsically out-of-equilibrium, and display fundamentally new phenomena, including complex spectra, spectral winding, exceptional topology, and non-unitary dynamics. A central challenge is understanding how the complex single-particle spectrum governs universal many-body behavior. We introduce a symmetry-protected dynamical topological index derived directly from the complex spectrum. Through the lens of algebraic topology, more specifically Morse theory, we identify critical points in the spectrum with topological defects, whose curvature and stability are protected under continuous deformations. This links spectral geometry to many-body observables, unifying non-Hermitian band topology, entanglement, and transport. We demonstrate that non-Hermitian quantum criticality in non-interacting systems is controlled by gain-and-loss-selected non-equilibrium steady states, which dynamically generate an emergent imaginary Fermi surface whose Fermi points host scale-invariant gapless modes with logarithmic entanglement scaling and algebraic correlations. Our work establishes a unified framework for non-Hermitian quantum matter, connecting spectral topology to Morse theory, revealing a topological foundation of non-equilibrium quantum criticality.

Figures

Figures reproduced from arXiv: 2607.05190 by the authors.

Figure 1
Figure 1. FIG. 1. Spectral topology and dynamical topological in [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Mechanisms changing the dynamical topological in [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Spectral topological phase transitions and trans [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Correlations and entanglement entropy. The sys [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Anomalous entanglement scaling from eigenvector nonorthogonality in critical non-Hermitian free fermions

    cond-mat.mes-hall 2026-07 conditional novelty 6.0 of 10

    Entanglement entropy of critical non-Hermitian free-fermion steady states is controlled by the eigenvector nonorthogonality (overlap angle θ) at imaginary Dirac points, giving closed-form Rényi coefficients ζ_n(θ).

Reference graph

Works this paper leans on

59 extracted references · 59 canonical work pages · cited by 1 Pith paper

  1. [1]

    E. P. Wigner, Communications on Pure and Applied Mathematics13, 1 (1960)

    work page 1960

  2. [2]

    F. D. M. Haldane, Reviews of Modern Physics89, 040502 (2017)

    work page 2017

  3. [3]

    J. M. Kosterlitz, Reviews of Modern Physics89, 040501 (2017)

    work page 2017

  4. [4]

    Witten, Communications in Mathematical Physics 121, 351 (1989)

    E. Witten, Communications in Mathematical Physics 121, 351 (1989)

    work page 1989

  5. [5]

    J. W. Essam, Discrete Mathematics1, 83 (1971)

    work page 1971

  6. [6]

    Banerjee, R

    A. Banerjee, R. Jaiswal, M. Manjunath, and A. Narayan, Proceedings of the National Academy of Sciences120, e2302572120 (2023). 9

    work page 2023

  7. [7]

    Wen, Reviews of Modern Physics89, 041004 (2017)

    X.-G. Wen, Reviews of Modern Physics89, 041004 (2017)

    work page 2017

  8. [8]

    L. M. Sieberer, M. Buchhold, J. Marino, and S. Diehl, Reviews of Modern Physics97, 025004 (2025)

    work page 2025

Show all 59 references
  1. [9]

    Weidemann, M

    S. Weidemann, M. Kremer, T. Helbig, T. Hofmann, A. Stegmaier, M. Greiter, R. Thomale, and A. Szameit, Science368, 311 (2020)

    2020

  2. [10]

    Schreiber, S

    M. Schreiber, S. S. Hodgman, P. Bordia, H. P. L¨ uschen, M. H. Fischer, R. Vosk, E. Altman, U. Schneider, and I. Bloch, Science349, 842 (2015)

    2015

  3. [11]

    A. J. Daley, Advances in Physics63, 77 (2014)

    2014

  4. [12]

    C. M. Bender and S. Boettcher, Physical Review Letters 80, 5243 (1998)

    1998

  5. [13]

    Ashida, Z

    Y. Ashida, Z. Gong, and M. Ueda, Advances in Physics 69, 249 (2020)

    2020

  6. [15]

    Banerjee, R

    A. Banerjee, R. Sarkar, S. Dey, and A. Narayan, Journal of Physics: Condensed Matter35, 333001 (2023)

    2023

  7. [16]

    J. Y. Lee, J. Ahn, H. Zhou, and A. Vishwanath, Physical Review Letters123, 206404 (2019)

    2019

  8. [17]

    Kato,Perturbation Theory for Linear Operators (Springer, New York, 1966)

    T. Kato,Perturbation Theory for Linear Operators (Springer, New York, 1966)

    1966

  9. [19]

    J. T. Gohsrich, A. Banerjee, and F. K. Kunst, Euro- physics Letters150, 60001 (2025)

    2025

  10. [21]

    Zhang, Z

    K. Zhang, Z. Yang, and C. Fang, Physical Review Letters 125, 126402 (2020)

    2020

  11. [22]

    Panda and S

    A. Panda and S. Banerjee, Physical Review B101, 184201 (2020)

    2020

  12. [23]

    Banerjee, S

    A. Banerjee, S. S. Hegde, A. Agarwala, and A. Narayan, Physical Review B105, 205403 (2022)

    2022

  13. [24]

    Chang, J.-S

    P.-Y. Chang, J.-S. You, X. Wen, and S. Ryu, Physical Review Research2, 033069 (2020)

    2020

  14. [25]

    Kitaev, AIP Conference Proceedings1134, 22 (2009)

    A. Kitaev, AIP Conference Proceedings1134, 22 (2009)

    2009

  15. [26]

    S. Ryu, A. P. Schnyder, A. Furusaki, and A. W. W. Lud- wig, New Journal of Physics12, 065010 (2010)

    2010

  16. [27]

    H. Shen, B. Zhen, and L. Fu, Physical Review Letters 120, 146402 (2018)

    2018

  17. [28]

    Kawabata, K

    K. Kawabata, K. Shiozaki, M. Ueda, and M. Sato, Phys- ical Review X9, 041015 (2019)

    2019

  18. [29]

    K. Wang, A. Dutt, C. C. Wojcik, and S. Fan, Nature 598, 59 (2021)

    2021

  19. [30]

    J. L. K. K¨ onig, K. Yang, J. C. Budich, and E. J. Bergholtz, Physical Review Research5, L042010 (2023)

    2023

  20. [31]

    Yoshida, J

    T. Yoshida, J. L. K. K¨ onig, L. Rødland, E. J. Bergholtz, and M. St˚ alhammar, Physical Review Research7, L012021 (2025)

    2025

  21. [32]

    St˚ alhammar and L

    M. St˚ alhammar and L. Rødland, Physical Review Re- search7, 033246 (2025)

    2025

  22. [33]

    Montag, A

    A. Montag, A. Felski, and F. K. Kunst, SciPost Physics 20, 133 (2026)

    2026

  23. [34]

    Lein, (2020), arXiv:2010.09261

    M. Lein, (2020), arXiv:2010.09261

    2020 arXiv

  24. [35]

    Morse,The Calculus of Variations in the Large, Col- loquium Publications, Vol

    M. Morse,The Calculus of Variations in the Large, Col- loquium Publications, Vol. 18 (American Mathematical Society, 1934)

    1934

  25. [36]

    J. W. Milnor,Morse Theory, Annals of Mathematics Studies, Vol. 51 (Princeton University Press, 1963)

    1963

  26. [38]

    H. B. Nielsen and M. Ninomiya, Physics Letters B105, 219 (1981)

    1981

  27. [39]

    Ying and A

    X. Ying and A. Kamenev, Physical Review Letters121, 086810 (2018)

    2018

  28. [40]

    Bessho and M

    T. Bessho and M. Sato, Physical Review Letters127, 196404 (2021)

    2021

  29. [41]

    Peschel, Journal of Physics A: Mathematical and Gen- eral36, L205 (2003)

    I. Peschel, Journal of Physics A: Mathematical and Gen- eral36, L205 (2003)

    2003

  30. [42]

    Calabrese and J

    P. Calabrese and J. Cardy, Journal of Physics A: Math- ematical and Theoretical42, 504005 (2009)

    2009

  31. [43]

    Calabrese and J

    P. Calabrese and J. Cardy, Journal of Statistical Mechan- ics: Theory and Experiment2004, P06002 (2004)

    2004

  32. [44]

    Hatano and D

    N. Hatano and D. R. Nelson, Physical Review Letters 77, 570 (1996)

    1996

  33. [47]

    Zhang, M

    S.-B. Zhang, M. M. Denner, T. Bzduˇ sek, M. A. Sentef, and T. Neupert, Physical Review B106, L121102 (2022)

    2022

  34. [48]

    Kawabata, K

    K. Kawabata, K. Shiozaki, and S. Ryu, Physical Review B105, 165137 (2022)

    2022

  35. [49]

    Longhi, Physical Review Letters128, 157601 (2022)

    S. Longhi, Physical Review Letters128, 157601 (2022)

    2022

  36. [50]

    Fitzpatrick, N

    M. Fitzpatrick, N. M. Sundaresan, A. C. Y. Li, J. Koch, and A. A. Houck, Physical Review X7, 011016 (2017) 10 Supplementary Information for: Spectral-topology-induced criticality in non-Hermitian fermionic metals To supplement the findings of the main text, we provide further ...

    2017

  37. [51]

    Hatano and D

    N. Hatano and D. R. Nelson, Physical Review Letters77, 570 (1996)

    1996

  38. [52]

    Kawabata, K

    K. Kawabata, K. Shiozaki, M. Ueda, and M. Sato, Physical Review X9, 041015 (2019)

    2019

  39. [53]

    Ashida, Z

    Y. Ashida, Z. Gong, and M. Ueda, Advances in Physics69, 249 (2020)

    2020

  40. [54]

    E. J. Bergholtz, J. C. Budich, and F. K. Kunst, Reviews of Modern Physics93, 015005 (2021)

    2021

  41. [55]

    J. T. Gohsrich, A. Banerjee, and F. K. Kunst, Europhysics Letters150, 60001 (2025)

    2025

  42. [56]

    T. E. Lee, Physical Review Letters116, 133903 (2016)

    2016

  43. [57]

    Lieu, Physical Review B97, 045106 (2018)

    S. Lieu, Physical Review B97, 045106 (2018)

    2018

  44. [58]

    F. K. Kunst, E. Edvardsson, J. C. Budich, and E. J. Bergholtz, Physical Review Letters121, 026808 (2018)

    2018

  45. [59]

    Yao and Z

    S. Yao and Z. Wang, Physical Review Letters121, 086803 (2018)

    2018

  46. [60]

    C. Yin, H. Jiang, L. Li, R. L¨ u, and S. Chen, Physical Review A97, 052115 (2018)

    2018

  47. [61]

    Kato,Perturbation Theory for Linear Operators(Springer, New York, 1966)

    T. Kato,Perturbation Theory for Linear Operators(Springer, New York, 1966)

    1966

  48. [62]

    J. T. Gohsrich, J. Fauman, and F. K. Kunst, (2024), arXiv:2403.12018

    2024 arXiv

  49. [63]

    A. M. Marques and R. G. Dias, Physical Review B106, 205146 (2022)

    2022

  50. [64]

    McCann, Physical Review B113, 045403 (2026)

    E. McCann, Physical Review B113, 045403 (2026)

    2026

  51. [65]

    Montag and F

    A. Montag and F. K. Kunst, Journal of Mathematical Physics65, 122101 (2024)

    2024

Pith tools

Reviewed July 8, 2026 · model on record in the stance chip above.