Pith. sign in

REVIEW 3 major objections 6 minor 64 references

Multifold degeneracy points of quantum systems and singularities of matrix varieties

T0 review · 3 major / 6 minor · reviewed 2026-08-06 · deepseek-v4-flash

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

Pith's one-line read For a Hamiltonian whose k-fold degeneracy is isolated in the complex sense, every generic perturbation produces exactly k^2(k^2-1)/12 complex Weyl points, an upper bound on the number of real Weyl points that can be born.

desk verdict For a k-fold degeneracy, the upper bound ♯WP ≤ k^2(k^2−1)/12 is proved with care; the physics examples inherit a genuine, explicitly flagged caveat about complex isolation. read the letter →

arxiv 2507.17485 v1 pith:P3AIIAD5 submitted 2025-07-23 math-ph math.AGmath.MPquant-ph

György Frank , András Pályi , Gergő Pintér , Dániel Varjas This is my paper · ORCID
classification math-phmath.AGmath.MPquant-ph MSC 14B0514M1232S0515A1881Q70 PACS 02.40.-k03.65.Vf71.18.+y
keywords multifolddegeneracyWeylpointscomplexvarietydeterminantalmultiplicitySchrieffer-Wolfftransformationsingularitytheory
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

When the energy levels of a parameter-dependent quantum Hamiltonian coincide at a point of parameter space, that multifold degeneracy is dissolved by a generic perturbation into several ordinary two-fold degeneracies, the Weyl points. This paper proves that the number of Weyl points thus born is controlled by a number that depends only on the degeneracy order: for a linear Hamiltonian with a $k$-fold degeneracy that is isolated in the complex sense, every generic perturbation produces exactly $k^2(k^2-1)/12$ complex Weyl points, and hence at most that many real Weyl points. The count is an algebraic intersection number: the complexified Hamiltonian is a holomorphic map from three complex parameters into the space of complex matrices, and the born Weyl points are its transverse intersections with the geometric degeneracy variety, the set of matrices having some eigenvalue with at least two eigenvectors. The authors show this number is the same for every generic perturbation and does not depend on the size $n$ of the Hamiltonian matrix. The payoff is that a question about quantum band structure becomes a singularity-theory calculation with a clean, parameter-free answer.

What carries the argument

The load-bearing objects are three complex matrix varieties: the geometric degeneracy variety $\Sigma\subset\mathbb{C}^{n\times n}$ of matrices with some eigenvalue of geometric multiplicity at least two; the lifted variety $\widetilde{\Sigma}=\{(A,\lambda): A-\lambda 1\in\Sigma'\}$; and the determinantal variety $\Sigma'$ of matrices of rank at most $n-2$, whose vanishing ideal is generated by the $(n-1)\times(n-1)$ minors. The argument runs through four mechanisms. (1) The counting formula (Theorem 3.1.1): for a holomorphic map germ $f:(\mathbb{C}^3,0)\to(\mathbb{C}^{n\times n},A_0)$ isolated against $(\Sigma,A_0;\lambda_0)$, the number of complex Weyl points of a generic perturbation equals $\dim_{\mathbb{C}}\mathcal{O}_4/J$, where $J$ is generated by the minors $M_{ij}(f(x)-(\lambda+\lambda_0)1)$; finiteness of this dimension is equivalent to isolation in the complex sense (Remark 4.4.6). (2) The reduction: since $\widetilde{\Sigma}\cong\Sigma'\times\mathbb{C}$ is Cohen–Macaulay while $\Sigma$ itself is not, the count is computed through the lifted variety and the determinantal ideal, where the key perturbation-invariance principle (Proposition A.3.3) applies. (3) Local triviality by the complex Schrieffer–Wolff chart: a local biholomorphism $g(S,C,A_{\mathrm{eff}})$ puts any strictly $k$-fold degenerate matrix $A_0$ in the same normal form as the origin in $\mathbb{C}^{k\times k}$, yielding $(\Sigma^{(n)},A_0;\lambda_0)\cong(\Sigma^{(k)},0)\times\mathbb{C}^{n^2-k^2}$ and transferring all multiplicities to the origin (Corollary 2.3.15). (4) The evaluation $\mathrm{mult}(\Sigma'^{(k)},0)=k^2(k^2-1)/12$ via the graded Hilbert series of $\mathcal{O}_4/I_{k-1}(f)$ computed from the Gulliksen–Negård free resolution of the ideal of $(k-1)\times(k-1)$ minors of a generic linear $f$ (Section 3.5).

What would settle it

Take an explicit linear Hamiltonian $H:(\mathbb{R}^3,0)\to(\mathrm{Herm}(k),0)$ with $f^{-1}(\Sigma)=\{0\}$ and compute $\dim_{\mathbb{C}}\mathcal{O}_4/J$ for its minors ideal: any value different from $k^2(k^2-1)/12$, or two generic perturbations with different numbers of complex solutions, would refute the central formula. For the crystalline example of Section 4.6, finding a parameter value $\alpha$ where the degeneracy is isolated for real $k$ but $\dim_{\mathbb{C}}(\mathcal{O}_4^{\mathbb{C}}/J^{\mathbb{C}})$ is infinite would falsify the conjecture that real isolation implies complex isolation, while a proof of finiteness at the exceptional values $\alpha_2=\pm\sqrt{\alpha_0^2+\alpha_1^2}$ would extend the bound into the regime currently left open.

Watch

Extended reading notes

Core claim

The paper's central claim is that a strictly $k$-fold degeneracy of a linear Hamiltonian $H:(\mathbb{R}^3,0)\to(\mathrm{Herm}(k),0)$, $H(0)=0$, whose complexification $f$ is isolated against the geometric degeneracy variety $\Sigma\subset\mathbb{C}^{k\times k}$, dissolves under a generic perturbation into exactly $k^2(k^2-1)/12$ complex Weyl points, so that $\sharp\mathrm{WP}\le\sharp\mathrm{cWP}=k^2(k^2-1)/12$ holds for the number of real Weyl points (Eq. (1.4.3), Corollary 4.4.7). In the more general form (Corollary 3.2.6), any map germ with a strictly $k$-fold degenerate eigenvalue that is isolated in the complex sense satisfies $\sharp f_t^{-1}(\Sigma,A_0;\lambda_0)=k^2(k^2-1)/12$, independent of the perturbation and of the ambient matrix size $n$; the authors propose to call such degeneracies $k$-fold Weyl points. The supporting multiplicity identities (Theorem 3.2.1) state that $\mathrm{mult}(\Sigma^{(n)},A_0;\lambda_0)=\mathrm{mult}(\widetilde{\Sigma}^{(n)},(A_0,\lambda_0))=\mathrm{mult}(\Sigma'^{(n)},B_0)=k^2(k^2-1)/12$, with analogues $k(k^2-1)/6$ for complex-symmetric two-parameter families and $k(k-1)/2$ for diagonal one-parameter families. A byproduct is that $\Sigma$ is not Cohen–Macaulay for $k\ge 3$ (Theorem 3.7.11), which forces the counting argument to pass through the determinantal variety $\Sigma'$ and shows that pulling back the vanishing ideal of $\Sigma$ itself would give the wrong answer.

Load-bearing premise

The bounds count complex solutions and apply only when the degeneracy point is isolated in the complex sense: no point of the complexified parameter space arbitrarily close to zero, other than zero itself, may make some eigenvalue geometrically degenerate. For the crystalline band-structure example of Section 4.6 this condition is verified numerically for random parameters and otherwise left as a conjecture.

Editorial extensions

If this is right

  • The number of complex Weyl points born from a strictly $k$-fold degeneracy depends only on $k$, never on the perturbation, the matrix size $n$, or the physical details of the Hamiltonian, so the bound $\sharp\mathrm{WP}\le k^2(k^2-1)/12$ is universal for linear Hamiltonians with an isolated complex degeneracy.
  • For a spin-$s$ particle in a magnetic field ($k=2s+1$), the two-sided bound is $k(k^2-1)/6\le\sharp\mathrm{WP}\le k^2(k^2-1)/12$, and the spin-1 case attains both ends: one perturbation produces 4 real Weyl points, another produces all 6.
  • For the crystalline fourfold degeneracies of Section 4.6, the formula predicts exactly $4^2(4^2-1)/12=20$ complex Weyl points, so a symmetry-breaking perturbation of such a fourfold crossing creates at most 20 Weyl points when the complex-isolation conjecture holds.
  • For ordinary two-fold degeneracies, the same algebra recovers the classical description: the local algebra $\mathcal{O}_4/J$ is isomorphic to the local algebra of the effective map germ, so the degeneracy type is read off directly from the Hamiltonian without performing a Schrieffer–Wolff transformation (Section 4.7).
  • Because $\Sigma$ is not Cohen–Macaulay, any counting method that pulls back the vanishing ideal of $\Sigma$ itself will generically miscount; the route through the lifted variety and the determinantal ideal is essential rather than optional (Theorem 3.7.12).

Reading between the lines

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

  • The paper leaves open whether the upper bound is generically sharp for real Weyl points: for spin-1 it exhibits one perturbation reaching the full 6 real points and notes that constant (translation) perturbations of that Hamiltonian cannot reach 6. A direct numerical test would be to search the crystalline fourfold example for a perturbation producing all 20 real Weyl points.
  • Remark 4.4.9 raises the open question of which Chern-number patterns a $k$-fold Weyl point can carry; if those patterns were classified, the integers reachable between the Chern lower bound and the upper bound $k^2(k^2-1)/12$ would likely follow, yielding a sharper prediction for each symmetry class.
  • Since the count is a purely holomorphic intersection number, the same formulas should transfer to other matrix ensembles with the same local geometry, such as real-symmetric or antisymmetric families, and to degeneracies at boundaries of parameter regions, where proving the paper's conjectured equivalence between real and complex isolation would make the bound rigorous.
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, and a circularity audit.

Referee Report

3 major / 6 minor

Summary. The paper derives an upper bound on the number of Weyl points produced by perturbing a multifold degeneracy point of a parameter-dependent Hamiltonian. After introducing the geometric degeneracy variety Sigma of complex matrices having an eigenvalue of geometric multiplicity at least two, its lift eSigma, and the determinantal variety Sigma', the authors prove that for a holomorphic map germ f:(C^3,0)->(C^{n x n},A0) with f^{-1}(Sigma,A0;lambda0)={0}, the number of preimages of a transverse perturbation equals dim O4/J, where J is generated by the (n-1)x(n-1) minors of f(x)-(lambda+lambda0)1 (Theorem 3.1.1). For a strictly k-fold degenerate eigenvalue this number is evaluated as k^2(k^2-1)/12 (Theorem 3.2.1), giving the bound WP <= cWP = k^2(k^2-1)/12 for linear Hamiltonians whose complexification is isolated with respect to Sigma (Corollary 4.4.7). The paper also computes the analogous multiplicities for complex symmetric and diagonal families, proves that Sigma is not Cohen-Macaulay, gives Chern-number lower bounds, and discusses spin and crystalline examples.

Significance. If the main theorem is fully established, the formula k^2(k^2-1)/12 is a striking parameter-free prediction: the number of complex Weyl points born from a k-fold degeneracy depends only on k and not on the perturbation. The paper supports this with detailed proofs, including an explicit Gulliksen-Negaard free resolution and Hilbert-series computation, and it gives concrete perturbations for the spin-1 case attaining both the lower bound 4 and the upper bound 6. The non-Cohen-Macaulay result for Sigma is a valuable caution that the vanishing ideal of Sigma cannot be used directly for these counts. The authors are also transparent about what is proven versus conjectured, which is a strength.

major comments (3)
  1. [Section 3.1, Theorem 3.1.1 and its proof (Eqs. (3.1.6)-(3.1.9))] The proof passes from f_t^{-1}(Sigma,A0;lambda0) to e f_t^{-1}(eSigma) by invoking the fact that the projection eSigma->Sigma is generically one-to-one (Corollary 2.3.33). However, transversality to the branch (Sigma,A0;lambda0) at non-singular points does not exclude the possibility that f_t(x) has a second geometric degenerate eigenvalue near another eigenvalue mu0 of A0. At such a point the same x appears twice in e f_t^{-1}(eSigma), once for each degenerate eigenvalue, so the equality #f_t^{-1}(Sigma,A0;lambda0)=#e f_t^{-1}(eSigma) can fail under the stated definition of a generic perturbation. The proof needs a stronger genericity condition (for example, transverse and avoiding the singular locus of Sigma) or a transversality argument showing that such bad intersections can be avoided by a one-parameter perturbation. This issue is load-bearing because Theorem 3.1.1 and all subsequent corollaries depend on this equality.
  2. [Section 4.6 and Remark 4.4.6] For the crystalline Hamiltonian (4.6.1), the complex-isolation hypothesis f^{-1}(Sigma)={0} is verified only numerically for random values of alpha and is left as a conjecture. Consequently the statement near the end of Section 4.6 that cWP=20 is an upper bound in all regions of alpha is not a theorem: Corollary 4.4.7 and Eq. (1.4.3) apply only once the isolation conjecture is proved. The paper should either provide a proof (for example, by showing symbolically that dim O4/J=20 for all non-exceptional alpha) or explicitly label the crystalline application as conjectural in the abstract and in the summary of results in Section 1.4.
  3. [Remark A.3.2 and Appendix A.3] The paper defines 'generic perturbation' as transverse in the sense of Section A.3, but it does not prove that such perturbations form an open dense set or even that they exist for every Hamiltonian germ. The physical statements in the abstract and in Section 4.4 use 'generic' in the usual sense of 'almost all' perturbations. The authors should either prove or cite a stratified transversality theorem that supplies such perturbations, or consistently state all results for 'every transverse perturbation with respect to Sigma' rather than for 'generic perturbations'. This is not merely a terminology issue, because the proof of Theorem 3.1.1 currently relies on an implicit genericity assumption to avoid the overcounting described above.
minor comments (6)
  1. [Section 2.1] The word 'subvariaty' should be 'subvariety'.
  2. [Remark 4.2.2] The word 'strictrly' should be 'strictly'.
  3. [Section 3.2] The word 'Therorem' in the introduction to Section 3.2 should be 'Theorem'.
  4. [Lemma 2.3.30 proof] The word 'neighborhhod' should be 'neighborhood'.
  5. [Appendix A.3, proof of Proposition A.3.3] The word 'coomplete' should be 'complete'.
  6. [Section 4 (introductory paragraph)] The phrase 'can be apllied' should be 'can be applied'.

Circularity Check

0 steps flagged · score 2.0 of 10

No meaningful circularity: the upper-bound formula is derived from determinantal variety theory, and self-citations are auxiliary.

full rationale

The paper's central formula, ♯cWP = k^2(k^2−1)/12, is not presupposed or fitted. It is obtained through Theorem 3.2.1, whose proof reduces to the Reduced Multiplicity Theorem and then computes the Hilbert series using the Gulliksen–Negård free resolution in Section 3.5; the symmetric and diagonal cases are proved directly in Sections 3.4 and 3.3. No parameter is fitted to Weyl-point counts, and no 'prediction' is defined in terms of the final formula. The complex Schrieffer–Wolff chart (Theorem 2.3.7) is proved in the paper for complex matrices and is used to show that the local branch (Σ, A0; λ0) is a trivial deformation of (Σ(k), 0); the citation to the authors' earlier [50] for the hermitian analogue is auxiliary and does not carry the central equality. References to [51] for two-fold degeneracy facts enter only in the lower-bound and two-fold discussion, not in the k-fold upper-bound derivation. The main limitation flagged by the paper itself is the crystalline example in Section 4.6, where the complex-isolation hypothesis f^{-1}(Σ) = {0} is only verified numerically for random parameters and then conjectured; this is a stated conditionality of the physical claim, not a circular reduction, because the paper explicitly presents the formula as conditional on that hypothesis. Similarly, Remark A.3.2 notes that 'generic perturbation' is taken to mean transverse without a topological-genericity proof; this is a completeness caveat rather than a self-referential derivation. The self-citations are therefore present but not load-bearing, and the derivation itself is self-contained against standard determinantal variety theory.

Assumptions & free parameters 0 free parameters · 5 assumptions · 2 invented entities

The central claim rests on standard theorems of commutative algebra and singularity theory (Nullstellensatz, Cohen-Macaulayness of determinantal varieties, exactness of Gulliksen-Negaard and Jozefiak resolutions, proper mapping theorem) and on the domain assumption that physical Hamiltonians are real analytic. There are no fitted parameters, no ad hoc constants, and no newly postulated physical entities; the only new constructions (eSigma, complex SW chart) are mathematical objects defined in the paper.

assumptions (5)
  • standard math Hilbert-Rückert Nullstellensatz over the complex numbers
    Used in Appendix A to relate vanishing ideals and set germs, e.g., to identify reduced structures.
  • standard math Determinantal varieties defined by r x r minors are irreducible, Cohen-Macaulay, with prime vanishing ideal (Bruns-Vetter [11])
    Used in Sections 2.2 and 3 to ensure eSigma and Sigma' behave well under perturbation (Proposition A.3.3).
  • standard math Exactness of the Gulliksen-Negaard free resolution for matrix germs isolated with respect to Sigma'
    Used in Section 3.5 to compute the Hilbert series and multiplicity of O4/I_{k-1}(f).
  • standard math Exactness of the Jozefiak free resolution for symmetric matrix germs isolated with respect to Sigma'_sym
    Used in Section 3.6 to prove linear independence of pulled-back minors in the symmetric case.
  • domain assumption The Hamiltonian and complexified map germs are analytic
    Section 4.1 requires analyticity to define the complexification; this holds for the cited physical models (spin Hamiltonian, k.p Bloch Hamiltonians).
invented entities (2)
  • Lifted geometric degeneracy variety eSigma = {(A, lambda) : A - lambda 1 in Sigma'}
    purpose: To reduce the non-Cohen-Macaulay variety Sigma to the Cohen-Macaulay determinantal variety Sigma' via a finite, generically one-to-one projection, enabling the use of Proposition A.3.3.
    A mathematical construction internal to the paper; it makes no falsifiable prediction outside the paper, so independent_evidence is false.
  • Complex Schrieffer-Wolff chart (local biholomorphism g fitting (Sigma, A0; lambda0))
    purpose: To show that the germ (Sigma(n), A0; lambda0) is an analytic trivial deformation of (Sigma(k), 0), reducing the multiplicity computation to the zero matrix case.
    A local parametrization tool, not a physical entity; it is fully defined in Theorem 2.3.7 and does not predict new observables by itself.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Multifold degeneracy points of quantum systems and singularities of matrix varieties." pith.science (2026). https://pith.science/paper/P3AIIAD5

@misc{pith2026250717485,
  author       = {Pith},
  title        = {Pith review of: Multifold degeneracy points of quantum systems and singularities of matrix varieties},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/P3AIIAD5}},
  note         = {Machine review of arXiv:2507.17485}
}
read the original abstract

Parameter-dependent quantum systems often exhibit energy degeneracy points, whose comprehensive description naturally lead to the application of methods from singularity theory. A prime example is an electronic band structure where two energy levels coincide in a point of momentum space. It may happen, and this case is the focus of our work, that three or more levels coincide at a parameter point, called multifold degeneracy. Upon a generic perturbation, such a multifold degeneracy point is dissolved into a set of Weyl points, that is, generic two-fold degeneracy points. In this work, we provide an upper bound to the number of Weyl points born from the multifold degeneracy point. To compute this upper bound, we describe the geometric degeneracy variety in the space of complex matrices. We compute its multiplicity at certain singular points corresponding to a multifold degeneracy, and the multiplicity of holomorphic map germs with respect to this variety. Our work covers physics and mathematics aspects in detail, and attempts to bridge the two disciplines and communities. For self-containedness, we survey examples of multi-fold degeneracies in quantum systems and condensed-matter physics, as well as the established tools of local algebraic geometry that we use to identify the upper bound.

Figures

Figures reproduced from arXiv: 2507.17485 by the authors.

Figure 1
Figure 1. Fourfold degeneracy point of a 1-parameter linear family of 4 [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Threefold degeneracy point of a 2-parameter linear family of 3 [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

64 extracted references · 49 canonical work pages

  1. [50]

    url: https://scipost.org/10.21468/SciPostPhysCodeb.50

    work page doi:10.21468/scipostphyscodeb.50

  2. [51]

    ¨Uber das Verhalten von Eigenwerten bei adiabatischen Prozessen

    J. von Neumann and E. P. Wigner. “ ¨Uber das Verhalten von Eigenwerten bei adiabatischen Prozessen”. In: Physikalische Zeitschrift 30 (1929), 467. doi: 10.1007/978- 3- 662- 02781- 3_20. url: https://link.springer.com/chapter/10.1007/978-3-662-02781-3_20 . 56

    doi:10.1007/978- 1929

  3. [1]

    Fundamental laws of chiral band crossings: Local constraints, global con- straints, and topological phase diagrams

    Kirill Alpin et al. “Fundamental laws of chiral band crossings: Local constraints, global con- straints, and topological phase diagrams”. In: Phys. Rev. Res. 5 (4 Nov. 2023), 043165. doi: 10 . 1103 / PhysRevResearch . 5 . 043165. url: https : / / link . aps . org / doi / 10 . 1103 / PhysRevResearch.5.043165

    work page 2023

  4. [2]

    Weyl and Dirac semimetals in three- dimensional solids

    N. P. Armitage, E. J. Mele, and Ashvin Vishwanath. “Weyl and Dirac semimetals in three- dimensional solids”. In: Rev. Mod. Phys. 90 (1 Jan. 2018), 015001. doi: 10.1103/RevModPhys. 90.015001. url: https://link.aps.org/doi/10.1103/RevModPhys.90.015001

    doi:10.1103/revmodphys 2018

  5. [3]

    Remarks on Eigenvalues and Eigenvectors of Hermitian Matrices, Berry Phase, Adiabatic Connections and Quantum Hall Effect

    V. I. Arnold. “Remarks on Eigenvalues and Eigenvectors of Hermitian Matrices, Berry Phase, Adiabatic Connections and Quantum Hall Effect”. In: Selecta Mathematica 1 (1995), 1. doi: 10.1007/BF01614072

    work page doi:10.1007/bf01614072 1995

  6. [4]

    “Singularities of differentiable maps, Volume

    VI Arnold, SM Gusein-Zade, and AN Varchenko. “Singularities of differentiable maps, Volume

    work page

  7. [5]

    Classification of critical points, caustics and wave fronts. Transl. from the Russian by Ian Porteous, edited by VI Arnold”. In: Monographs in Mathematics 82 (2012), 382

    work page 2012

  8. [6]

    J. K. Asb´ oth, L. Oroszl´ any, and A. P´ alyi.A Short Course on Topological Insulators. Heidelberg: Springer, 2016. doi: 10.1007/978-3-319-25607-8

    doi:10.1007/978-3-319-25607-8 2016

Show all 64 references
  1. [7]

    Solid state physics

    Neil W Ashcroft and N David Mermin. Solid state physics . New York, NY: Holt, Rinehart and Winston, 1976. url: https://cds.cern.ch/record/102652

    1976

  2. [8]

    Beyond Dirac and Weyl fermions: Unconventional quasiparticles in conventional crystals

    Barry Bradlyn et al. “Beyond Dirac and Weyl fermions: Unconventional quasiparticles in conventional crystals”. In: Science 353.6299 (2016), aaf5037. doi: 10.1126/science.aaf5037. eprint: https : / / www . science . org / doi / pdf / 10 . 1126 / science . aaf5037. url: https : ...

    2016 doi

  3. [9]

    Schrieffer–Wolff transformation for quantum many-body systems

    Sergey Bravyi, David P. DiVincenzo, and Daniel Loss. “Schrieffer–Wolff transformation for quantum many-body systems”. In: Annals of Physics 326.10 (2011), 2793–2826. doi: 10 . 1016/j.aop.2011.06.004

    2011

  4. [10]

    On the Geometry of the Set of Sym- metric Matrices with Repeated Eigenvalues

    Paul Breiding, Khazhgali Kozhasov, and Antonio Lerario. “On the Geometry of the Set of Sym- metric Matrices with Repeated Eigenvalues”. In:Arnold Mathematical Journal 4.3 (2018), 423– 443

    2018

  5. [11]

    Berry phase, topology, and degeneracies in quantum nanomagnets

    Patrick Bruno. “Berry phase, topology, and degeneracies in quantum nanomagnets”. In: Phys- ical review letters 96.11 (2006), 117208. doi: 10.1103/PhysRevLett.96.117208. url: https: //journals.aps.org/prl/abstract/10.1103/PhysRevLett.96.117208

    2006 doi

  6. [12]

    Determinantal Rings

    Winfried Bruns and Udo Vetter. Determinantal Rings. Springer-Verlag Berlin Heidelberg, 1988

    1988

  7. [13]

    Complex symmetric matrices

    B. D. Craven. “Complex symmetric matrices”. In: Journal of the Australian Mathematical Society 10.3–4 (1969), 341–354. doi: 10.1017/S1446788700007588

    1969 doi

  8. [14]

    Codebase release 2.1 for Pymablock

    Isidora Araya Day et al. “Codebase release 2.1 for Pymablock”. In: SciPost Phys. Codebases (2025), 50–r2.1. doi: 10.21468/SciPostPhysCodeb.50-r2.1 . url: https://scipost.org/ 10.21468/SciPostPhysCodeb.50-r2.1. 54

    2025 doi

  9. [15]

    Pymablock: An algorithm and a package for quasi-degenerate pertur- bation theory

    Isidora Araya Day et al. “Pymablock: An algorithm and a package for quasi-degenerate pertur- bation theory”. In: SciPost Phys. Codebases (2025), 50. doi: 10.21468/SciPostPhysCodeb

    2025 doi

  10. [16]

    Hermitian matrices with a bounded number of eigenvalues

    M. Domokos. “Hermitian matrices with a bounded number of eigenvalues”. In: Linear Algebra and its Applications 439.12 (2013), 3964–3979. issn: 0024-3795. doi: https://doi.org/10. 1016/j.laa.2013.09.040. url: https://www.sciencedirect.com/science/article/pii/ S0024379513006083

    2013

  11. [17]

    Local analytic geometry: Basic theory and applications

    Theo De Jong and Gerhard Pfister. Local analytic geometry: Basic theory and applications . Springer Science & Business Media, 2013. isbn: 978-3-322-90159-0

    2013

  12. [18]

    Discriminant of symmetric matrices as a sum of squares and the orthogonal group

    M´ aty´ as Domokos. “Discriminant of symmetric matrices as a sum of squares and the orthogonal group”. In: Communications on Pure and Applied Mathematics 64.4 (2011), 443–465. doi: https://doi.org/10.1002/cpa.20353 . eprint: https://onlinelibrary.wiley.com/doi/ pdf/10.1002/cpa...

    2011 doi

  13. [19]

    Invariant theoretic characterization of subdiscriminants of matrices

    M. Domokos. “Invariant theoretic characterization of subdiscriminants of matrices”. In: Linear and Multilinear Algebra 62.1 (2014), 63–73. doi: 10.1080/03081087.2012.762714 . eprint: https://doi.org/10.1080/03081087.2012.762714 . url: https://doi.org/10.1080/ 03081087.2012.762714

    2014

  14. [20]

    An Algebraic Approach to the Topological Degree of a Smooth Map

    David Eisenbud. “An Algebraic Approach to the Topological Degree of a Smooth Map”. In: Bulletin of the American Mathematical Society 84.5 (1978), 751–752

    1978

  15. [21]

    A.R. Edmonds. Angular Momentum in Quantum Mechanics . Investigations in Physics Series. Princeton University Press, 1996. isbn: 9780691025896. url: https://books.google.hu/ books?id=0BSOg0oHhZ0C

    1996

  16. [22]

    An Algebraic Formula for the Degree of a C ∞ Map Germ / Sur une in´ egalit´ e ` a la Minkowski pour les Multiplicit´ es

    David Eisenbud, Harold I. Levine, and Bernard Teissier. “An Algebraic Formula for the Degree of a C ∞ Map Germ / Sur une in´ egalit´ e ` a la Minkowski pour les Multiplicit´ es”. In:Annals of Mathematics 106.1 (1977), 19–44

    1977

  17. [23]

    Commutative Algebra with a View Toward Algebraic Geometry

    David Eisenbud. Commutative Algebra with a View Toward Algebraic Geometry . Vol. 150. Graduate Texts in Mathematics. New York: Springer-Verlag, 1995

    1995

  18. [24]

    Magnetic degeneracy points in interacting two-spin systems: Geometrical patterns, topological charge distributions, and their stability

    Gy¨ orgy Frank et al. “Magnetic degeneracy points in interacting two-spin systems: Geometrical patterns, topological charge distributions, and their stability”. In: Physical Review B 101.24 (2020), 245409. doi: 10.1103/PhysRevB.101.245409 . url: https://journals.aps.org/ prb/a...

    2020 doi

  19. [25]

    Singularity theory of Weyl-point creation and annihilation

    Gy¨ orgy Frank, Gerg˝ o Pint´ er, and Andr´ as P´ alyi. “Singularity theory of Weyl-point creation and annihilation”. In: arXiv preprint arXiv:2309.05506 (2023). url: https://arxiv.org/ abs/2309.05506

    2023 arXiv

  20. [26]

    Weyl-point teleportation

    Gy¨ orgy Frank et al. “Weyl-point teleportation”. In: Phys. Rev. B 109 (20 May 2024), 205415. doi: 10 . 1103 / PhysRevB . 109 . 205415. url: https : / / link . aps . org / doi / 10 . 1103 / PhysRevB.109.205415

    2024

  21. [27]

    Topological charge distributions of an interacting two-spin system

    Gy¨ orgy Frank et al. “Topological charge distributions of an interacting two-spin system”. In: Physical Review B 105.3 (2022), 035414. doi: 10.1103/PhysRevB.105.035414 . url: https: //journals.aps.org/prb/abstract/10.1103/PhysRevB.105.035414

    2022 doi

  22. [28]

    Fulton and R

    W. Fulton and R. Weiss. Algebraic Curves: An Introduction to Algebraic Geometry. Benjamin, 1969

    1969

  23. [29]

    Fulton and J

    W. Fulton and J. Harris. Representation Theory: A First Course . Graduate Texts in Math- ematics. Springer New York, 2013. isbn: 9781461209799. url: https://books.google.hu/ books?id=6TwmBQAAQBAJ

    2013

  24. [30]

    Tjurina and Milnor numbers of matrix singularities

    Victor Goryunov and David Mond. “Tjurina and Milnor numbers of matrix singularities”. In: Journal of the London Mathematical Society 72.1 (2005), 205–224

    2005

  25. [31]

    Teoriya Matrits

    F. R. Gantmacher. Theory of Matrices . Trans. by K. A. Hirsch. English translation of the Russian original “Teoriya Matrits”. New York: Chelsea Publishing Co., 1960. 55

    1960

  26. [32]

    Principles of Algebraic Geometry

    Phillip Griffiths and Joseph Harris. Principles of Algebraic Geometry . New York: John Wiley & Sons, 1978

    1978

  27. [33]

    Griffiths and Darrell F

    David J. Griffiths and Darrell F. Schroeter. Introduction to Quantum Mechanics . 3rd ed. Cambridge University Press, 2018

    2018

  28. [34]

    Un complexe resolvant pour certain id´ eaux d´ et` erminentiels

    Tor H Gulliksen and OG Neg ˚ ard. “Un complexe resolvant pour certain id´ eaux d´ et` erminentiels.” In: Preprint series: Pure mathematics http://urn. nb. no/URN: NBN: no-8076 (1971)

    1971

  29. [35]

    Weyl points in ball-and-spring mechanical systems

    Zolt´ an Guba et al. “Weyl points in ball-and-spring mechanical systems”. In: Phys. Rev. B 111 (22 June 2025), 224109. doi: 10.1103/PhysRevB.111.224109. url: https://link.aps.org/ doi/10.1103/PhysRevB.111.224109

    2025 doi

  30. [36]

    Algebraic Geometry

    Robin Hartshorne. Algebraic Geometry. Springer New York, NY, 1977

    1977

  31. [37]

    On symmetric and skew-symmetric determinantal varieties

    Joe Harris and Loring W. Tu. “On symmetric and skew-symmetric determinantal varieties”. In: Topology 23.1 (1984), 71–84. issn: 0040-9383. doi: https://doi.org/10.1016/0040- 9383(84 ) 90026 - 0. url: https : / / www . sciencedirect . com / science / article / pii / 0040938384900260

    1984 doi

  32. [38]

    Geometry of symmetric determinantal loci

    Shinobu Hosono and Hiromichi Takagi. Geometry of symmetric determinantal loci. 2015. arXiv: 1508.01995 [math.AG]. url: https://arxiv.org/abs/1508.01995

    2015 arXiv

  33. [39]

    Gr¨ obner bases and multiplicity of determinantal and pfaffian ideals

    J¨ urgen Herzog and Ngˆ oViˆ et Trung. “Gr¨ obner bases and multiplicity of determinantal and pfaffian ideals”. In: Advances in Mathematics 96.1 (1992), 1–37. issn: 0001-8708. doi: https: //doi.org/10.1016/0001- 8708(92)90050- U . url: https://www.sciencedirect.com/ science/ar...

    1992

  34. [40]

    Weyl semimetals, Fermi arcs and chiral anomalies

    Shuang Jia, Su-Yang Xu, and M. Zahid Hasan. “Weyl semimetals, Fermi arcs and chiral anomalies”. In: Nature Materials 15.11 (2016), 1140–1144

    2016

  35. [41]

    A Weyl Fermion semimetal with surface Fermi arcs in the transition metal monopnictide TaAs class

    Shin-Ming Huang et al. “A Weyl Fermion semimetal with surface Fermi arcs in the transition metal monopnictide TaAs class”. In: Nature Communications 6 (June 2015), 7373. doi: 10. 1038/ncomms8373. url: https://www.nature.com/articles/ncomms8373

    2015

  36. [42]

    Perturbation theory for linear operators

    Tosio Kato. Perturbation theory for linear operators . Springer Berlin, Heidelberg, 2013. doi: 10.1007/978-3-642-66282-9

    2013 doi

  37. [43]

    Ideals generated by minors of a symmetric matrix

    Tadeusz J´ ozefiak. “Ideals generated by minors of a symmetric matrix”. In:Commentarii math- ematici Helvetici 53 (1978), 595–607

    1978

  38. [44]

    Characteristic classes

    John Willard Milnor and James D Stasheff. Characteristic classes . 76. Princeton university press, 1974

    1974

  39. [45]

    On minimality of determinantal varieties

    Khazhgali Kozhasov. “On minimality of determinantal varieties”. In: Linear Algebra and its Applications 626 (2021), 56–78. issn: 0024-3795. doi: https : / / doi . org / 10 . 1016 / j . laa . 2021 . 05 . 011. url: https : / / www . sciencedirect . com / science / article / pii ...

    2021

  40. [46]

    On contact between submanifolds

    James A. Montaldi. “On contact between submanifolds”. In: Michigan Math. J. 33(2) (1986), 195. doi: 10.1307/mmj/1029003348

    1986

  41. [47]

    Singularities of mappings: The local behaviour of smooth and complex analytic mappings

    David Mond and Juan J Nu˜ no-Ballesteros. Singularities of mappings: The local behaviour of smooth and complex analytic mappings . Vol. 357. Springer Nature, 2020

    2020

  42. [48]

    Stability of Weyl Node Merging Processes under Symmetry Con- straints

    Gabriele Naselli et al. “Stability of Weyl Node Merging Processes under Symmetry Con- straints”. In: Phys. Rev. Lett. 133 (19 2024), 196602. doi: 10.1103/PhysRevLett.133.196602. url: https://link.aps.org/doi/10.1103/PhysRevLett.133.196602

    2024 doi

  43. [49]

    Geometry, Topology and Physics

    Mikio Nakahara. Geometry, Topology and Physics . 2nd. Boca Raton, FL: CRC Press, 2003. isbn: 9780750306065

    2003

  44. [52]

    The geometry of the Hermitian matrix space and the Schrieffer–Wolff trans- formation

    G. Pint´ er et al. “The geometry of the Hermitian matrix space and the Schrieffer–Wolff trans- formation”. In: arXiv:2407.10478 (2024). url: https://arxiv.org/abs/2407.10478

    2024

  45. [53]

    Upper bound on the number of Weyl points born from a nongeneric degeneracy point

    Gerg˝ o Pint´ er et al. “Upper bound on the number of Weyl points born from a nongeneric degeneracy point”. In: Phys. Rev. B 110 (24 Dec. 2024), 245124. doi: 10.1103/PhysRevB. 110.245124. url: https://link.aps.org/doi/10.1103/PhysRevB.110.245124

    2024 doi

  46. [54]

    Multifold topological semimetals

    I˜ nigo Robredo et al. “Multifold topological semimetals”. In: Europhysics Letters 147.4 (Aug. 2024), 46001. doi: 10.1209/0295-5075/ad6bbc. url: https://dx.doi.org/10.1209/0295- 5075/ad6bbc

    2024 doi

  47. [55]

    Sakurai and Jim Napolitano

    J.J. Sakurai and Jim Napolitano. Modern Quantum Mechanics. 2nd. San Francisco: Addison- Wesley, 2011. isbn: 978-0-8053-8291-4

    2011

  48. [56]

    Observation of spin–orbit coupling induced Weyl points in a two- electron double quantum dot

    Zolt´ an Scher¨ ubl et al. “Observation of spin–orbit coupling induced Weyl points in a two- electron double quantum dot”. In: Communications Physics 2.1 (2019), 108. doi: 10.1038/ s42005-019-0200-2 . url: https://www.nature.com/articles/s42005-019-0200-2

    2019

  49. [57]

    Relation between the Anderson and Kondo Hamiltonians

    J. R. Schrieffer and P. A. Wolff. “Relation between the Anderson and Kondo Hamiltonians”. In: Phys. Rev. 149 (2 Sept. 1966), 491–492. doi: 10.1103/PhysRev.149.491

    1966 doi

  50. [58]

    G´ eom´ etrie alg´ ebrique et g´ eom´ etrie analytique

    Jean-Pierre Serre. “G´ eom´ etrie alg´ ebrique et g´ eom´ etrie analytique”. fr. In:Annales de l’Institut Fourier 6 (1956), 1–42. doi: 10.5802/aif.59 . url: https://www.numdam.org/articles/ 10.5802/aif.59/

    1956 doi

  51. [59]

    Shafarevich

    Igor R. Shafarevich. Basic Algebraic Geometry 1: Varieties in Projective Space. Springer Berlin, Heidelberg, 2013

    2013

  52. [60]

    Application of singularity theory to bifurcation of band structures in crystals

    Hiroshi Teramoto et al. “Application of singularity theory to bifurcation of band structures in crystals”. In: Journal of Singularities 21 (2020), 289–302. doi: 10.5427/jsing.2020.21p

    2020 doi

  53. [61]

    Classification of Hamiltonians in neighborhoods of band crossings in terms of the theory of singularities

    Hiroshi Teramoto et al. “Classification of Hamiltonians in neighborhoods of band crossings in terms of the theory of singularities”. In: J. Math. Phys. 58.7 (2017), 073502. doi: 10.1063/1. 4991662

    2017 doi

  54. [62]

    Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differ- entialgleichungen (mit einer Anwendung auf die Theorie der Hohlraumstrahlung)

    Hermann Weyl. “Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differ- entialgleichungen (mit einer Anwendung auf die Theorie der Hohlraumstrahlung)”. In: Math- ematische Annalen 71.4 (Dec. 1, 1912), 441–479. doi: 10 . 1007 / BF01456804. url: https : //d...

    1912 doi

  55. [63]

    Discovery of a Weyl fermion semimetal and topological Fermi arcs

    Su-Yang Xu et al. “Discovery of a Weyl fermion semimetal and topological Fermi arcs”. In: Science 349.6248 (2015), 613–617. doi: 10.1126/science.aaa9297 . eprint: https://www. science.org/doi/pdf/10.1126/science.aaa9297 . url: https://www.science.org/doi/ abs/10.1126/science.aaa9297

    2015 doi

  56. [64]

    Encyclopedia of emergent particles in three-dimensional crystals

    Zhi-Ming Yu et al. “Encyclopedia of emergent particles in three-dimensional crystals”. In: Science Bulletin 67.4 (2022), 375–380. issn: 2095-9273. doi: https://doi.org/10.1016/ j.scib.2021.10.023 . url: https://www.sciencedirect.com/science/article/pii/ S2095927321006927. 57

    2022

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.