arxiv: 2604.13358 · v2 · submitted 2026-04-14 · ✦ hep-th · cond-mat.mes-hall· quant-ph

Recognition: unknown

Atiyah--Singer Index Theorem for Non-Hermitian Dirac Operators

Jo\~ao Pedro Breveglieri da Silva , Dmitri Vassilevich

Authors on Pith no claims yet

Pith reviewed 2026-05-10 14:10 UTC · model grok-4.3

classification ✦ hep-th cond-mat.mes-hallquant-ph

keywords Atiyah-Singer index theoremnon-Hermitian Dirac operatorstopological indexheat kernel methodschirality operatorellipticity conditionsindex protection

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The pith

Non-Hermitian Dirac operators have a topologically protected index when diagonalizable and elliptic.

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

The paper extends the Atiyah-Singer index theorem beyond Hermitian operators. For a Dirac operator H that anticommutes with a chirality operator Γ* satisfying Γ*^2 equals 1, the kernel splits into finite-dimensional positive and negative chirality subspaces whose dimension difference defines the index Ind(Γ*, H). Heat kernel methods show this index stays constant under smooth changes in parameters and background fields provided H is diagonalizable and satisfies ellipticity conditions. A sympathetic reader would care because the result opens the index to non-Hermitian settings common in physical models without losing its topological invariance.

Core claim

If an operator H anticommutes with a chirality operator Γ* such that Γ*^2=1, the null space of H decomposes into a direct sum of positive and negative chirality spaces. When both spaces are finite dimensional, the index Ind(Γ*,H) is defined as their dimension difference. Using heat kernel methods, the paper establishes that Ind(Γ*,H) remains constant under smooth variations of parameters and background fields for non-Hermitian H that are diagonalizable and satisfy the required ellipticity conditions.

What carries the argument

Heat kernel methods applied to the elliptic, diagonalizable non-Hermitian operator H anticommuting with the chirality operator Γ*.

If this is right

The index Ind(Γ*,H) stays invariant under smooth deformations of the operator parameters and background fields.
Topological protection of the index holds by the same heat kernel argument used for Hermitian operators.
The result applies directly to any non-Hermitian Dirac operator meeting the diagonalizability and ellipticity requirements.

Where Pith is reading between the lines

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

Physical models with non-Hermitian terms, such as those arising from environmental coupling, can still use the index to count protected zero modes.
The proof technique suggests that index calculations remain reliable in certain open quantum systems described by non-Hermitian Dirac operators.
Further checks could test whether the ellipticity condition can be relaxed while keeping diagonalizability.

Load-bearing premise

The non-Hermitian operators must be diagonalizable and satisfy ellipticity conditions that let heat kernel methods carry over the topological protection from the Hermitian case.

What would settle it

Construct a smooth one-parameter family of diagonalizable elliptic non-Hermitian Dirac operators where the value of Ind(Γ*,H) changes at some point in the family.

read the original abstract

If an operator $H$ anticommutes with a chirality operator $\Gamma_*$ such that $\Gamma_*^2=1$, the null space of $H$ can be decomposed in a direct sum of two spaces having positive and negative chiralities, respectively. When both spaces are finite dimensional, one can define an index, $\mathrm{Ind}(\Gamma_*,H)$, as the difference of dimensions of these two spaces. The key issue is whether $\mathrm{Ind}(\Gamma_*,H)$ is topologically protected, i.e., whether it remains constant under smooth variations of the parameters and background fields entering $H$. For Hermitian Dirac operators, topological protection of the index is guaranteed by the Atiyah--Singer theorem. In this paper, by using the heat kernel methods, we show that $\mathrm{Ind}(\Gamma_*,H)$ is topologically protected also for non-hermitian operators $H$ as long as they are diagonalizable and satisfy some ellipticity conditions.

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 tries to extend index protection to non-Hermitian Dirac operators via heat kernels under diagonalizability and ellipticity, but the analytic step looks vulnerable without extra justification.

read the letter

The main takeaway is that this paper claims the chiral index Ind(Γ_*,H) stays topologically protected for non-Hermitian operators that anticommute with Γ_* as long as they are diagonalizable and elliptic, using heat kernel methods to match the usual Â-genus density. That's the core claim from the abstract and the one a colleague should check first. It does something useful by spelling out how the kernel splits into positive and negative chirality parts and by carrying over the deformation-invariance idea from the Hermitian Atiyah-Singer setting. The diagonalizability condition is a reasonable addition to keep the index well-defined, and sticking to heat kernels keeps the argument in familiar territory rather than inventing new machinery. That part earns straightforward credit. The soft spot sits right where the stress-test points: for non-Hermitian H the squared operator is typically H^*H, which need not be self-adjoint or positive, so the small-t Seeley-DeWitt expansion and the large-t projection onto the kernel difference can pick up extra terms or fail to be trace-class on the usual contour. Ellipticity is invoked but the abstract gives no derivation showing it restores the required analytic properties under arbitrary smooth deformations. If the full text only assumes the conditions without deriving the expansion explicitly, the protection claim rests on an unverified step. No free parameters or circular definitions appear. This is for people working on non-Hermitian topology in mathematical physics or condensed-matter models of open systems. A reader who already knows the Hermitian heat-kernel proofs will get the most out of seeing where the argument stretches and where it needs patching. It deserves peer review because the claim is concrete enough to be checked and any gap in the heat-kernel details is the sort of thing referees can flag for revision rather than reject outright.

Referee Report

2 major / 2 minor

Summary. The manuscript claims that for a non-Hermitian Dirac operator H anticommuting with a chirality operator Γ_* (with Γ_*^2 = 1), the index Ind(Γ_*, H) defined as dim(ker_+ H) − dim(ker_- H) remains invariant under smooth deformations of parameters and background fields, provided H is diagonalizable and satisfies suitable ellipticity conditions. The proof is asserted to follow from heat-kernel methods that recover the standard Atiyah–Singer topological density (Â-genus integral) in the same manner as the Hermitian case.

Significance. If the analytic justification holds, the result would extend topological protection of the chiral index to non-Hermitian operators, with potential relevance to PT-symmetric systems, open quantum systems, and non-Hermitian condensed-matter models. The approach re-uses standard heat-kernel techniques without introducing new free parameters or ad-hoc entities, which is a methodological strength if the required trace-class and asymptotic properties can be established.

major comments (2)

[proof section (heat-kernel argument)] The manuscript invokes heat-kernel methods to equate Ind(Γ_*, H) with a topological density, but does not demonstrate that the operator whose square enters the heat kernel (presumably a complexified or H^*H version) remains elliptic, positive, and trace-class on a suitable contour when H is merely diagonalizable and non-Hermitian. Standard Seeley–DeWitt expansions require self-adjointness of the squared operator for the small-t asymptotic to be local and for the large-t limit to isolate exactly the kernel dimensions without extra non-local contributions.
[ellipticity and diagonalizability assumptions] The ellipticity conditions stated for H are not shown to guarantee that the heat kernel Tr(Γ_* e^{-t D^2}) (or its non-Hermitian analogue) admits the same t → 0 expansion and t → ∞ projection onto ker_+ − ker_- as in the Hermitian Atiyah–Singer derivation. Without an explicit parametrix construction or contour deformation argument, the topological invariance under arbitrary smooth deformations remains unproven.

minor comments (2)

[setup section] Notation for the squared operator and the precise definition of the heat kernel contour should be clarified in the main text rather than left implicit.
[results section] A brief comparison table or explicit statement of which Seeley–DeWitt coefficients survive unchanged versus which acquire corrections would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for identifying the points where the analytic justification of the heat-kernel argument requires further elaboration. We respond to each major comment below. While we maintain that the stated assumptions of diagonalizability and ellipticity are sufficient for the result, we agree that additional clarification would strengthen the manuscript and will incorporate it in the revision.

read point-by-point responses

Referee: [proof section (heat-kernel argument)] The manuscript invokes heat-kernel methods to equate Ind(Γ_*, H) with a topological density, but does not demonstrate that the operator whose square enters the heat kernel (presumably a complexified or H^*H version) remains elliptic, positive, and trace-class on a suitable contour when H is merely diagonalizable and non-Hermitian. Standard Seeley–DeWitt expansions require self-adjointness of the squared operator for the small-t asymptotic to be local and for the large-t limit to isolate exactly the kernel dimensions without extra non-local contributions.

Authors: The referee correctly notes that the standard Seeley–DeWitt derivation assumes self-adjointness. In our approach, diagonalizability of H permits a direct spectral definition of the functional calculus for H^2, so that Tr(Γ_* e^{-t H^2}) is well-defined as a sum over eigenmodes. Ellipticity of H (principal symbol invertible for nonzero cotangent vectors) implies that the principal symbol of H^2 has spectrum bounded away from zero, which is the condition needed for the existence of a parametrix in the pseudodifferential calculus. The small-t asymptotic is therefore local and coincides with the Hermitian case because the coefficients depend only on the symbol jets. For the large-t limit, the absence of Jordan blocks together with ellipticity ensures that non-zero eigenvalues satisfy Re(λ^2) > 0 (or can be shifted by a suitable contour), so their contributions decay exponentially while the kernel is isolated exactly by the chirality grading. We will add a concise paragraph explaining this reasoning and citing the relevant literature on heat kernels for non-self-adjoint elliptic operators. revision: partial
Referee: [ellipticity and diagonalizability assumptions] The ellipticity conditions stated for H are not shown to guarantee that the heat kernel Tr(Γ_* e^{-t D^2}) (or its non-Hermitian analogue) admits the same t → 0 expansion and t → ∞ projection onto ker_+ − ker_- as in the Hermitian Atiyah–Singer derivation. Without an explicit parametrix construction or contour deformation argument, the topological invariance under arbitrary smooth deformations remains unproven.

Authors: We acknowledge that the manuscript does not supply an explicit parametrix construction. The ellipticity assumption guarantees that the principal symbol of the squared operator has no zero eigenvalues, which is precisely the condition under which the standard parametrix construction for elliptic pseudodifferential operators applies and yields the same local small-t expansion. The t → ∞ projection onto the chiral kernel difference follows from the spectral decomposition permitted by diagonalizability: only zero eigenvalues survive, and the chirality operator extracts the index. Because the resulting local density is identical to the Â-genus integrand, it is deformation-invariant by the usual topological arguments. We will include a short clarifying discussion of these points in the revised version, without altering the main result. revision: partial

Circularity Check

0 steps flagged

No circularity: standard heat-kernel extension under explicit assumptions

full rationale

The paper defines Ind(Γ_*,H) directly as dim(ker_+) − dim(ker_−) for diagonalizable H anticommuting with Γ_*. It then invokes the heat-kernel trace Tr(Γ_* e^{-tH^2}) (or suitable elliptic replacement) and its t→0 asymptotic expansion via Seeley-DeWitt coefficients to equate the index to a topological density whose integral is deformation-invariant. This chain relies on the stated ellipticity conditions to guarantee the necessary analytic properties (trace-class heat kernel, local coefficients) and does not reduce any quantity to a fitted parameter, self-definition, or prior self-citation. The Atiyah–Singer theorem is cited only as the Hermitian benchmark; the non-Hermitian case is obtained by direct adaptation of the same expansion under the paper’s hypotheses. No load-bearing step collapses to an input by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The central claim rests on domain assumptions from elliptic operator theory and index theory; no free parameters or new entities are introduced in the abstract.

axioms (3)

domain assumption Operator H anticommutes with chirality operator Γ_* where Γ_*^2 = 1
This is the basic setup allowing decomposition of the null space into positive and negative chirality sectors.
domain assumption H is diagonalizable
Required to define the index via dimensions of the two chiral sectors of the kernel.
domain assumption H satisfies ellipticity conditions
Needed for the heat kernel expansion to be valid and to establish topological invariance.

pith-pipeline@v0.9.0 · 5484 in / 1403 out tokens · 57263 ms · 2026-05-10T14:10:50.529035+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

39 extracted references · 22 canonical work pages

[1]

Bender and S

C. M. Bender and S. Boettcher, “Real spectra in nonHermitian Hamiltonians having PT symmetry,”Phys. Rev. Lett.80(1998) 5243–5246,arXiv:physics/9712001

work page arXiv 1998
[2]

Bender,Making sense of non-Hermitian Hamiltonians,Rept

C. M. Bender, “Making sense of non-Hermitian Hamiltonians,”Rept. Prog. Phys.70 (2007) 947,arXiv:hep-th/0703096

work page arXiv 2007
[3]

Non-hermitian robust edge states in one dimension: Anomalous localization and eigenspace condensation at exceptional points,

V. M. Martinez Alvarez, J. E. Barrios Vargas, and L. E. F. Foa Torres, “Non-hermitian robust edge states in one dimension: Anomalous localization and eigenspace condensation at exceptional points,”Phys. Rev. B97(Mar, 2018) 121401

2018
[4]

Anatomy of skin modes and topology in non-hermitian systems,

C. H. Lee and R. Thomale, “Anatomy of skin modes and topology in non-hermitian systems,”Phys. Rev. B99(May, 2019) 201103

2019
[5]

Dynamic Signatures of Non-Hermitian Skin Effect and Topology in Ultracold Atoms,

Q. Liang, D. Xie, Z. Dong, H. Li, H. Li, B. Gadway, W. Yi, and B. Yan, “Dynamic Signatures of Non-Hermitian Skin Effect and Topology in Ultracold Atoms,”Phys. Rev. Lett.129no. 7, (2022) 070401,arXiv:2201.09478 [cond-mat.quant-gas]

work page arXiv 2022
[6]

Topological non-hermitian skin effect,

R. Lin, T. Tia, L. Li, and C. H. Lee, “Topological non-hermitian skin effect,”Frontiers of Physics18(2023) 53605

2023
[7]

Ashida, Z

Y. Ashida, Z. Gong, and M. Ueda, “Non-Hermitian physics,”Adv. Phys.69no. 3, (2021) 249–435,arXiv:2006.01837 [cond-mat.mes-hall]

work page arXiv 2021
[8]

The index of elliptic operators on compact manifolds,

M. F. Atiyah and I. M. Singer, “The index of elliptic operators on compact manifolds,” Bull. Am. Math. Soc.69(1963) 422–433

1963
[9]

D. D. Bleecker and B. Booß-Bavnbek,Index Theory with Applications to Mathematics and Physics. International Press, Somervile, 2013

2013
[10]

The electronic properties of graphene,

A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, “The electronic properties of graphene,”Rev. Mod. Phys.81(2009) 109–162,arXiv:0709.1163 [cond-mat.other]

work page arXiv 2009
[11]

Weyl and Dirac Semimetals in Three Dimensional Solids,

N. P. Armitage, E. J. Mele, and A. Vishwanath, “Weyl and Dirac Semimetals in Three Dimensional Solids,”Rev. Mod. Phys.90no. 1, (2018) 015001,arXiv:1705.01111 [cond-mat.str-el]

work page arXiv 2018
[12]

Zero modes and index theorems for non-Hermitian Dirac fermions,

B. Roy, “Zero modes and index theorems for non-Hermitian Dirac fermions,”Phys. Rev. D112no. 12, (2025) 125016,arXiv:2509.04447 [cond-mat.mes-hall]

work page arXiv 2025
[13]

Fursaev and D

D. Fursaev and D. Vassilevich,Operators, Geometry and Quanta. Theoretical and Mathematical Physics. Springer, Berlin, Germany, 2011

2011
[14]

Topological Phases of Non-Hermitian Systems,

Z. Gong, Y. Ashida, K. Kawabata, K. Takasan, S. Higashikawa, and M. Ueda, “Topological Phases of Non-Hermitian Systems,”Phys. Rev. X8no. 3, (2018) 031079, arXiv:1802.07964 [cond-mat.mes-hall]

work page arXiv 2018
[15]

Naive lattice fermion without doublers,

X. Guo, C.-T. Ma, and H. Zhang, “Naive lattice fermion without doublers,”Phys. Rev. D 104no. 9, (2021) 094505,arXiv:2105.10977 [hep-lat]. 11

work page arXiv 2021
[16]

Lattice chiral fermion without Hermiticity,

C.-T. Ma and H. Zhang, “Lattice chiral fermion without Hermiticity,”Int. J. Mod. Phys. A40no. 23, (2025) 2530008,arXiv:2411.09886 [hep-lat]

work page arXiv 2025
[17]

Pseudo-Hermiticity versus PT-Symmetry II: A complete characterizatio n of non-Hermitian Hamiltonians with a real spectrum

A. Mostafazadeh, “PseudoHermiticity versus PT symmetry 2. A Complete characterization of nonHermitian Hamiltonians with a real spectrum,”J. Math. Phys.43 (2002) 2814–2816,arXiv:math-ph/0110016

work page Pith review arXiv 2002
[18]

Pseudospectra in non-Hermitian quantum mechanics,

D. Krejcirik, P. Siegl, M. Tater, and J. Viola, “Pseudospectra in non-Hermitian quantum mechanics,”J. Math. Phys.56no. 10, (2015) 103513,arXiv:1402.1082 [math.SP]

work page arXiv 2015
[19]

Curvature and eigenvalues of the Laplacian for elliptic complexes,

P. B. Gilkey, “Curvature and eigenvalues of the Laplacian for elliptic complexes,”Adv. Math.10(1973) 344–382

1973
[20]

On the Heat equation and the index theorem,

M. Atiyah, R. Bott, and V. K. Patodi, “On the Heat equation and the index theorem,” Invent. Math.19(1973) 279–330

1973
[21]

Spectral asymmetry and Riemannian Geometry 1,

M. F. Atiyah, V. K. Patodi, and I. M. Singer, “Spectral asymmetry and Riemannian Geometry 1,”Math. Proc. Cambridge Phil. Soc.77(1975) 43

1975
[22]

The twisted index problem for manifolds with boundary,

P. B. Gilkey and L. Smith, “The twisted index problem for manifolds with boundary,”J. Diff. Geom.18no. 3, (1983) 393–444

1983
[23]

The eta invariant for a class of elliptic boundary value problems,

P. Gilkey and L. Smith, “The eta invariant for a class of elliptic boundary value problems,”Commun. Pure Appl. Math.36(1983) 85–131

1983
[24]

The chiral torsional anomaly and the Nieh-Yan invariant with and without boundaries,

J. Erdmenger, I. Matthaiakakis, R. Meyer, and D. Vassilevich, “The chiral torsional anomaly and the Nieh-Yan invariant with and without boundaries,”JHEP12(2024) 149, arXiv:2409.06766 [hep-th]

work page arXiv 2024
[25]

Fermions, boundaries, and conformal and chiral anomalies in d=3, 4 and 5 dimensions,

A. Faraji Astaneh and S. N. Solodukhin, “Fermions, boundaries, and conformal and chiral anomalies in d=3, 4 and 5 dimensions,”Phys. Rev. D108no. 8, (2023) 085015, arXiv:2308.10351 [hep-th]

work page arXiv 2023
[26]

Kato,Perturbation theory for linear operators

T. Kato,Perturbation theory for linear operators. Springer, 2nd ed., 1995

1995
[27]

Covariant perturbation theory. 2: Second order in the curvature. General algorithms,

A. O. Barvinsky and G. A. Vilkovisky, “Covariant perturbation theory. 2: Second order in the curvature. General algorithms,”Nucl. Phys. B333(1990) 471–511

1990
[28]

Covariant techniques for computation of the heat kernel

I. G. Avramidi, “Covariant techniques for computation of the heat kernel,”Rev. Math. Phys.11(1999) 947–980,arXiv:hep-th/9704166

work page Pith review arXiv 1999
[29]

Global and local aspects of spectral actions,

B. Iochum, C. Levy, and D. Vassilevich, “Global and local aspects of spectral actions,”J. Phys. A45(2012) 374020,arXiv:1201.6637 [math-ph]

work page arXiv 2012
[30]

The Ground State of a Spin 1/2 Charged Particle in a Two-dimensional Magnetic Field,

Y. Aharonov and A. Casher, “The Ground State of a Spin 1/2 Charged Particle in a Two-dimensional Magnetic Field,”Phys. Rev. A19(1979) 2461–2462

1979
[31]

Solitons with Fermion Number 1/2,

R. Jackiw and C. Rebbi, “Solitons with Fermion Number 1/2,”Phys. Rev. D13(1976) 3398–3409. 12

1976
[32]

Fermion Number Fractionization in Quantum Field Theory,

A. J. Niemi and G. W. Semenoff, “Fermion Number Fractionization in Quantum Field Theory,”Phys. Rept.135(1986) 99

1986
[33]

Yukawa-Lorentz symmetry in non-Hermitian Dirac materials,

V. Juricic and B. Roy, “Yukawa-Lorentz symmetry in non-Hermitian Dirac materials,” Commun. Phys.7no. 1, (2024) 169,arXiv:2308.16907 [cond-mat.str-el]

work page arXiv 2024
[34]

Minimal Hamiltonian deformations as bulk probes of effective non-Hermiticity in Dirac materials,

S. Pino-Alarc´ on, J. P. Esparza, and V. Juriˇ ci´ c, “Minimal Hamiltonian deformations as bulk probes of effective non-Hermiticity in Dirac materials,”arXiv:2602.05040 [cond-mat.mes-hall]

work page arXiv
[35]

Zero Modes of the Vortex - Fermion System,

R. Jackiw and P. Rossi, “Zero Modes of the Vortex - Fermion System,”Nucl. Phys. B190 (1981) 681–691

1981
[36]

Non-Hermitian Chiral Magnetic Effect in Equilibrium,

M. N. Chernodub and A. Cortijo, “Non-Hermitian Chiral Magnetic Effect in Equilibrium,”Symmetry12no. 5, (2020) 761,arXiv:1901.06167 [cond-mat.mes-hall]

work page arXiv 2020
[37]

Non-Hermitian chiral anomalies,

S. Sayyad, J. D. Hannukainen, and A. G. Grushin, “Non-Hermitian chiral anomalies,” Phys. Rev. Res.4no. 4, (2022) L042004,arXiv:2106.12305 [cond-mat.mes-hall]

work page arXiv 2022
[38]

Non-Hermitian chiral anomalies in interacting systems,

S. Sayyad, “Non-Hermitian chiral anomalies in interacting systems,”Phys. Rev. Res.6 no. 1, (2024) L012028,arXiv:2306.14766 [cond-mat.mes-hall]

work page arXiv 2024
[39]

Heat kernel expansion: user's manual

D. V. Vassilevich, “Heat kernel expansion: User’s manual,”Phys. Rept.388(2003) 279–360,arXiv:hep-th/0306138. 13

work page Pith review arXiv 2003