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arxiv: 2606.08868 · v2 · pith:PZXZIIKInew · submitted 2026-06-07 · ❄️ cond-mat.mes-hall · hep-th· math-ph· math.MP

Topological invariant responsible for the integer QHE and non-commutative geometry

G. Kovyrshin , A. Mekrami , J. Miller , M.A. Zubkov , A. Zuevsky This is my paper

Pith reviewed 2026-06-27 17:41 UTC · model grok-4.3

classification ❄️ cond-mat.mes-hall hep-thmath-phmath.MP
keywords topological invariantinteger quantum Hall effecttight-binding modelsMatsubara Green functionK-theorycyclic cohomologyWigner transformationHall conductivity quantization
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0 comments X

The pith

The topological invariant N3 for the integer quantum Hall effect equals the pairing of a K^{-1} group element from the electron Green function with a class in cyclic cohomology HC^3.

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

The paper expresses the topological invariant N3 that controls quantization of Hall conductivity in two-dimensional tight-binding models as a specific pairing involving the Wigner-transformed two-point Matsubara Green function. This pairing is defined between an element of the K^{-1} group generated by the Green function and a fixed element of the cyclic cohomology group HC^3. Local index theorems then imply that N3 takes integer values for a limited subclass of these models, including non-homogeneous ones, thereby accounting for the observed integer quantization of the Hall conductivity.

Core claim

The topological invariant N3 responsible for the quantization of the Hall conductivity, for the specific case of the integer quantum Hall effect in 2D, is expressed through the Wigner transformation of the two-point electron Matsubara Green function. We express this invariant as a pairing of the element of the K^{-1} group (generated by the Green function) with the specific element of the cyclic cohomology group HC^3. According to a set of local index theorems the values of N3 can be shown to be integer for a limited class of tight-binding models.

What carries the argument

The pairing between the K^{-1} group element generated by the Green function and the specific HC^3 cyclic cohomology class, which defines the topological invariant N3.

If this is right

  • Hall conductivity remains quantized in integers for the considered non-homogeneous 2D tight-binding models whenever N3 is defined.
  • The invariant N3 can be evaluated directly from the Green function without requiring the full spectrum of the Hamiltonian.
  • The same construction applies to a wide class of 2D lattice models that may lack translational invariance.

Where Pith is reading between the lines

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

  • The Green-function formulation might extend the invariant to cases with weak interactions if the K-group element remains well-defined.
  • Numerical evaluation of the Wigner-transformed Green function in lattice models could provide a practical route to compute N3 for disordered systems.
  • The approach links the integer quantum Hall effect to non-commutative geometry in a way that could be compared with other K-theoretic invariants used in topological insulators.

Load-bearing premise

The tight-binding models must belong to the limited class for which local index theorems guarantee that the pairing N3 is an integer.

What would settle it

A direct computation of the pairing N3 for a concrete non-homogeneous tight-binding model that lies outside the limited class but still has a Green function generating the required K^{-1} element, yielding a non-integer value.

Figures

Figures reproduced from arXiv: 2606.08868 by A. Mekrami, A. Zuevsky, G. Kovyrshin, J. Miller, M.A. Zubkov.

Figure 1
Figure 1. Figure 1: FIG. 1: In this figure we represent schematically [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
read the original abstract

We consider a wide class of $2D$ tight - binding models of solid state physics. These models are, in the most general case, non - homogeneous. The topological invariant ${\cal N}_3$ responsible for the quantization of the Hall conductivity, for the specific case of the integer quantum Hall effect in $2D$, is expressed through the Wigner transformation of the two-point electron Matsubara Green function. We express this invariant as a pairing of the element of the $K^{-1}$ group (generated by the Green function) with the specific element of the cyclic cohomology group $HC^3$. According to a set of local index theorems the values of ${\cal N}_3$ can be shown to be integer for a limited class of tight - binding models.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The manuscript considers a wide class of 2D tight-binding models, possibly non-homogeneous, and expresses the topological invariant N3 responsible for integer quantum Hall conductivity quantization via the Wigner transform of the two-point Matsubara Green function. This invariant is formulated as the pairing between the K^{-1} class generated by the Green function and a specific element of the cyclic cohomology group HC^3. The abstract states that local index theorems imply N3 is integer-valued only for a limited subclass of such models.

Significance. If the pairing construction is rigorously established and the models satisfy the hypotheses of the cited local index theorems, the work would provide a Green-function-based non-commutative geometry formulation of the integer QHE that extends to disordered systems. The explicit use of K-theory and cyclic cohomology pairings is a potential strength, but the restriction to a limited class and lack of verification for the broader models considered limits the immediate impact.

major comments (1)
  1. [Abstract] Abstract: The manuscript states that it considers a 'wide class' of models (including non-homogeneous cases) but asserts integrality of N3 only 'for a limited class of tight-binding models' via local index theorems. No explicit verification is supplied that the Wigner-transformed Green functions of the wide class obey the smoothness, decay, or spectral-gap conditions required by those theorems. This gap directly affects whether the integrality claim applies to the models under consideration.
minor comments (1)
  1. [Abstract] Abstract: Notation such as ${\cal N}_3$ and HC^3 is introduced without prior definition or reference to standard conventions in non-commutative geometry literature.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for highlighting the need for greater clarity on the scope of the integrality result. We address the single major comment below and will revise the manuscript to resolve the identified gap.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The manuscript states that it considers a 'wide class' of models (including non-homogeneous cases) but asserts integrality of N3 only 'for a limited class of tight-binding models' via local index theorems. No explicit verification is supplied that the Wigner-transformed Green functions of the wide class obey the smoothness, decay, or spectral-gap conditions required by those theorems. This gap directly affects whether the integrality claim applies to the models under consideration.

    Authors: The core contribution of the manuscript is the derivation of the topological invariant N3 as the pairing between the K^{-1} class generated by the Wigner-transformed Matsubara Green function and the indicated element of HC^3; this construction is carried out for the stated wide class of 2D tight-binding models, including non-homogeneous cases. The abstract and text explicitly restrict the integrality statement to the limited subclass for which the hypotheses of the cited local index theorems are known to hold. We agree that the manuscript does not supply an explicit check that the Green functions arising from the full wide class satisfy the requisite smoothness, decay, and spectral-gap conditions. In the revised version we will insert a clarifying paragraph (most naturally in the introduction or a dedicated subsection of the discussion) that (i) reiterates the distinction between the general pairing construction and the additional analytic hypotheses needed for integrality, (ii) states that verification of those hypotheses must be performed model-by-model, and (iii) notes that the wide-class formulation remains valid even when integrality is not guaranteed. This change improves precision without altering the main technical results. revision: yes

Circularity Check

0 steps flagged

No circularity; integrality cited from external local index theorems

full rationale

The paper constructs N3 explicitly as the pairing of a K^{-1} class (generated by the Wigner-transformed Matsubara Green function) with an HC^3 cocycle. Integrality is not derived internally but is attributed to 'a set of local index theorems' that apply only to a limited subclass of models. The abstract distinguishes the considered 'wide class' from that limited subclass, so the integrality step is presented as an external assumption rather than a self-derived or self-cited result. No equations, fitted parameters, or self-referential reductions appear in the provided text. The central pairing construction stands independently of the integrality claim.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Abstract-only review yields no explicit free parameters or invented entities. The claim rests on two domain assumptions drawn directly from the abstract.

axioms (2)
  • domain assumption The two-point Matsubara Green function generates an element of the K^{-1} group
    Stated as the generator of the pairing in the abstract.
  • domain assumption Local index theorems apply and guarantee integrality for the limited class of models considered
    Invoked in the final sentence of the abstract to conclude N3 is integer.

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Reference graph

Works this paper leans on

131 extracted references · 1 canonical work pages

  1. [1]

    In this approach, the algebra of smooth functions on a given topological space is replaced by an abstract (and in general, non-commutative) algebra A

    basic definitions Noncommutative geometry is essentially a generalization of the geometry of topological spaces, whereby the algebra of functions defined over a given space is replaced by a noncommutative algebra. In this approach, the algebra of smooth functions on a given topological space is replaced by an abstract (and in general, non-commutative) alg...

  2. [2]

    the space spanned by bra-vectors, whileHis spanned by ket-vectors

    Basic algebraAand its derivatives 19 ByH ∗ =Hom(H,C)we denote the space conjugate toH, i.e. the space spanned by bra-vectors, whileHis spanned by ket-vectors. As mentioned above we can takeA:=H ∗ ⊗ H. But this choice is not unique. We should impose various conditions on the operators thus coming to differentC∗ algebras. The transition to Weyl symbols is a...

  3. [3]

    In this section, for simplicity we disregard the internal finite dimensional space

    GroupsK 0,K −1 for various choices ofA. In this section, for simplicity we disregard the internal finite dimensional space. Its effect on theKgroups is obvious and can be postponed until later in the discussion

  4. [4]

    Even more let us discretize the direction of imaginary time

    Let us consider the case of finite spatial lattice size,N. Even more let us discretize the direction of imaginary time. In this case A= End(H),H=C ZN ×C |O|,O=Z N ×Z N . We deal with matrix algebra A= End(H) ∼= MN 3(C), and, by Morita invariance, obtain K 0(A) =K 0(MN 3(C)) =K 0(C) ∼= Z, K 1(A) =K 1(MN 3(C)) =K 1(C) ∼= 0

  5. [5]

    In this case A= End(H),H=C R ×C |O|,O=Z N ×Z N

    Once again let us consider finite spatial lattice size,N, but continuous values of imaginary time. In this case A= End(H),H=C R ×C |O|,O=Z N ×Z N . SpaceC |O| is finite-dimensional, thusdimC |O| =N 2,. We interpretC R as space of functions of a continuous variable (time or frequency). We can tryH∼= L2(R)⊗C N 2 . Using standardK-theory properties we can se...

  6. [6]

    Let us assume now that the direction of Matsubara frequencies is compactified (i.e

    CompactificationR→S 1 Above we have seen that the groupK1 is trivial for the most straightforward definition of algebraAin case of finite lattice and continuous imaginary time. Let us assume now that the direction of Matsubara frequencies is compactified (i.e. direction of imaginary time is discretized), and in addition we suppose that the operators are d...

  7. [7]

    When we pass to Weyl symbols we deal with A=C ∞(R× O ′ × M′), whereRis the domain of Matsubara frequencies

    Transition to Weyl symbols at finiteN. When we pass to Weyl symbols we deal with A=C ∞(R× O ′ × M′), whereRis the domain of Matsubara frequencies. Calculation ofK- groups in this case is given in Appendix F1. We obtain K 0(A) =Z (2N) 4 , K 1(A) = 0 Again, if we replaceRof Matsubara frequency byS1, we obtain K 0(A) =Z (2N) 4 , K 1(A) =Z (2N) 4

  8. [8]

    In this case instead of the operators ˆGwe consider Weyl symbolsG W defined on the spatial latticeO ′ and momentum spaceR× M ′

    Computation of groupsKi for the case when the limit of infiniteNis taken partially. In this case instead of the operators ˆGwe consider Weyl symbolsG W defined on the spatial latticeO ′ and momentum spaceR× M ′. The latter space is in the limitN→ ∞replaced byR×T 2. Calculation of theK - groups in this case is presented in Appendix F2. The result is K 0(A)...

  9. [9]

    We have two representations of the model: coordinate representation and momentum representation

    LimitN→ ∞for operator algebra. We have two representations of the model: coordinate representation and momentum representation. In mo- mentum representation we can try A= End(C ∞(R×T 2)) but then bothK-groups vanish. Let us replace it by algebra of pseudodifferential operators onR×T 2. This results in K 0(A) =K −1(A) ∼= Z4 (56)

  10. [10]

    We can try A= End(C ∞(R× O)) but this choice leads to trivialK-groups

    The limitN→ ∞for operator algebra in coordinate representation. We can try A= End(C ∞(R× O)) but this choice leads to trivialK-groups. There exist ways to reduce the size of the algebra thus giving nontrivial Kgroups, but we do not discuss these further, having in mind those versions described above in the previous items. B. Cyclic cohomology LetMbe anA-b...

  11. [11]

    In this case A= End(H) ∼= MN 3(C), 24 and we obtain HC 3(A) ∼= 0

    FiniteNand discretized direction of imaginary time with finite sizeN. In this case A= End(H) ∼= MN 3(C), 24 and we obtain HC 3(A) ∼= 0

  12. [12]

    Operator algebra

    The case of finiteN. Operator algebra. In this case A= End(H) ∼= K(L2(R))b⊗MN 2(C), we obtain HC 3(A) = 0

  13. [13]

    CompactificationR→S 1 A= End(H) ∼= K(L2(S1))b⊗MN 2(C), we obtain HC 3(A) = 0

  14. [14]

    A=C ∞(R× O ′ × M′), whereRis the axis of Matsubara frequencies

    Transition to Weyl symbols at finiteN. A=C ∞(R× O ′ × M′), whereRis the axis of Matsubara frequencies. The calculation ofHC 3 is given in Appendix G1. With this setup we obtain HC 3(A) = 0 However, if we compactify the Matsubara frequencies (for example, if we discretize the axis of imaginary time), then A=C ∞(S1 × O′ × M′), and HC 3(A) =C (2N) 4 , 25

  15. [15]

    We consider Weyl symbolsGW defined on the spatial latticeO′ and momentum spaceR×T 2

    The case when the limit of infiniteNis taken partially. We consider Weyl symbolsGW defined on the spatial latticeO′ and momentum spaceR×T 2. Therefore, A=C ∞(O′ ×R×T 2) The calculation ofHC 3 groups is presented in Appendix G2. The results is HC 3(A) = C2 4N 2 (65) If we replace hereRbyS 1, we obtain instead: HC 3(A) = C4 4N 2 (66) For the limitN→ ∞, the ...

  16. [16]

    We have two representations of the model: the coordinate representation and the momentum representation

    The limitN→ ∞for operator algebra. We have two representations of the model: the coordinate representation and the momentum representation. In the momentum representation we can try A= End(C ∞(R×T 2)) but thenHC 3 as well as bothK-groups vanish. Therefore, we can replace it by the algebra of pseudodifferential operators onR×T 2 . This results in HC 3(A) ∼...

  17. [17]

    As for theKgroups, if we take A= End(C ∞(R× O)) then the groupHC 3 is trivial

    The limitN→ ∞for the operator algebra in coordinate representation. As for theKgroups, if we take A= End(C ∞(R× O)) then the groupHC 3 is trivial. 26 IV. TOPOLOGICAL INVARIANTN 3 ON THE LANGUAGE OF CYCLIC COHOMOLOGIES. A.N 3 in terms of cyclic cohomology First of all, for any reasonable choice ofAwe can think of the Green functionGas an element ofGL(1,A),...

  18. [18]

    We will consider the two dimensional Weyl transform

    Weyl symbols of the simplest operators Below we will be interested in the Weyl symbols of the operators of the form ofQ(ω,p−A(i∂ p)). We will consider the two dimensional Weyl transform. The direction of imaginary time remains continuous. Let us start our consideration from the Weyl symbol for a general function ofˆp f( ˆp)W (p, q) = X ni=0,1;u∈M′ ⟨p+πn/N...

  19. [19]

    Weyl symbol ofexp i(ˆpk −A ke−ωωω∂p ) For a functionϕ(x)≡ ⟨x|ϕ⟩defined on the latticeOwe can define the Fourier transform ⟨p|ϕ⟩ ≡ϕ(p) = 1 N D/2 X x∈O e−ipxϕ(x)(E3) 39 Correspondingly, we can define the operatorˆx≡i∂p that acts as ⟨p|ˆx|ϕ⟩ ≡i∂ pϕ(p) = 1 N D/2 X x∈O e−ipxxϕ(x)(E4) Obviously, its action onϕ(x)is simplyxϕ(x). Now let us calculate the commutat...

  20. [20]

    The Weyl symbol ofF= exp i(ˆpµ −PN J=1 AJ µekJ i∂p ) The next step is the consideration of the functionF= exp i(ˆpµ −PN J=1 AJµ ekJ i∂p) , which appears in the series expansion for arbitrary function (Fourier or Laplace series). As above, we will be using the following relation: iAJµ ekJ i∂p iˆpµ =i(ˆpµ +ik Jµ)iAJµ ekJ i∂p (E13) 41 It gives [iˆpµ, iAJµ ek...

  21. [21]

    We should representAin the form of the Laplace transformation: Aµ(x) = Z ˜Aµ(k)ekx +c.c

    The Weyl symbol ofF= exp i(ˆpµ −A µ(i∂p)) For the case of arbitrary functionAµ(i∂p)the Weyl symbol ofF= exp i(ˆpµ −A µ(i∂p)) may be calculated using the above obtained expressions. We should representAin the form of the Laplace transformation: Aµ(x) = Z ˜Aµ(k)ekx +c.c. dk(E28) that is Aµ(i∂p) = Z ˜Aµ(k)eki∂p +c.c. dk(E29) In turn, this integral may be dis...

  22. [22]

    The Weyl symbol of Dirac operator ˆQ(p−A(i∂ p)) The homogeneous tight-binding model is defined here as the one, in which the Dirac operator is expressed through the momentum operator as a sum over exponents of the form F µ n = exp in(ˆpµ −A µ(i∂p)) (E35) with integern. One can check that for each term of this type we have (F µ n )W = exp in(pµ − A(n) µ (x...

  23. [23]

    When we pass to Weyl symbols we deal with A=C ∞(R× O ′ × M′), whereRis the range of Matsubara frequencies

    K - groups for the algebra of Weyl symbols at finiteN. When we pass to Weyl symbols we deal with A=C ∞(R× O ′ × M′), whereRis the range of Matsubara frequencies. Typical expression forQW isQ W =iω−H(p, x), whereω∈R. Its inverse will be(iω−H(p, x)) −1 ⋆ . Our space of functions should contain both these forms Let us decompose the algebra. SinceO′ andM ′ ar...

  24. [24]

    Computation of groupsK i for the case when the limit of infiniteNis taken partially. a. The case when the range of Matsubara frequencies isR In this case, instead of the operators ˆGwe consider Weyl symbolsG W defined on the spatial latticeO ′ and momentum spaceR× M ′. The latter space is in the limitN→ ∞replaced byR×T 2. Let us decompose the algebra. Sin...

  25. [25]

    A=C ∞(R× O ′ × M′), whereRis the axis of Matsubara frequencies

    Algebra of Weyl symbols at finiteN. A=C ∞(R× O ′ × M′), whereRis the axis of Matsubara frequencies. SinceO ′ andM ′ are finite discrete sets we have the algebraic decomposition A=C ∞(R)⊗C (2N) 2 ⊗C (2N) 2 =C ∞(R)⊗C (2N) 4 = (2N) 4 M i=1 C ∞(R). Let us computeHC3(C ∞(R)). Start with de Rham cohomology ofR. SinceRis contractible its de Rham cohomology is tr...

  26. [26]

    We consider Weyl symbolsGW defined on the spatial latticeO′ and momentum spaceR×T 2

    The case when the limit of infiniteNis taken partially. We consider Weyl symbolsGW defined on the spatial latticeO′ and momentum spaceR×T 2. Therefore, A=C ∞(O′ ×R×T 2) SinceO ′ is a finite discrete set with(2N)2 points, functions on it reduce to tuples which givesA=C∞(R×T 2)⊗ C(2N) 2 =L(2N) 2 i=1 C ∞(R×T 2). Recall de Rham cohomology ofR×T 2: SinceRis co...

  27. [27]

    von Klitzing, G

    K. von Klitzing, G. Dorda, and M. Pepper, New method for high-accuracy determination of the fine-structure constant based on quantized hall resistance, Phys. Rev. Lett.45, 494 (1980)

    1980

  28. [28]

    R. B. Laughlin, Quantized hall conductivity in two dimensions, Phys. Rev. B23, 5632 (1981)

    1981

  29. [29]

    D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs, Quantized Hall Conductance in a Two-Dimensional Periodic Potential, Phys. Rev. Lett.49, 405 (1982)

    1982

  30. [30]

    Kohmoto, Topological invariant and the quantization of the hall conductance, Ann

    M. Kohmoto, Topological invariant and the quantization of the hall conductance, Ann. Phys. (N.Y.)160, 343 (1985). 51

    1985

  31. [31]

    M. V. Berry, Quantal phase factors accompanying adiabatic changes, Proc. R. Soc. Lond. A392, 45 (1984)

    1984

  32. [32]

    Simon, Holonomy, the quantum adiabatic theorem, and berry’s phase, Phys

    B. Simon, Holonomy, the quantum adiabatic theorem, and berry’s phase, Phys. Rev. Lett.51, 2167 (1983)

    1983

  33. [33]

    Q. Niu, D. J. Thouless, and Y.-S. Wu, Quantized hall conductance as a topological invariant, Phys. Rev. B31, 3372 (1985)

    1985

  34. [34]

    J. E. Avron, R. Seiler, and B. Simon, Homotopy and quantization in condensed matter physics, Phys. Rev. Lett.51, 51 (1983)

    1983

  35. [35]

    B. I. Halperin, Quantized hall conductance, current-carrying edge states, and the existence of extended states in a two- dimensional disordered potential, Phys. Rev. B25, 2185 (1982)

    1982

  36. [36]

    Strěda, Theory of quantised hall conductivity in two dimensions, J

    P. Strěda, Theory of quantised hall conductivity in two dimensions, J. Phys. C: Solid State Phys.15, L717 (1982)

    1982

  37. [37]

    Hatsugai, Chern number and edge states in the integer quantum hall effect, Phys

    Y. Hatsugai, Chern number and edge states in the integer quantum hall effect, Phys. Rev. Lett.71, 3697 (1993)

    1993

  38. [38]

    Bellissard, A

    J. Bellissard, A. van Elst, and H. Schulz-Baldes, The noncommutative geometry of the quantum hall effect, J. Math. Phys.35, 5373 (1994)

    1994

  39. [39]

    R. E. Prange and S. M. Girvin, eds.,The Quantum Hall Effect, 2nd ed. (Springer-Verlag, New York, 1990)

    1990

  40. [40]

    Xiao, M.-C

    D. Xiao, M.-C. Chang, and Q. Niu, Berry phase effects on electronic properties, Rev. Mod. Phys.82, 1959 (2010)

    1959

  41. [41]

    M. Z. Hasan and C. L. Kane, Colloquium: Topological insulators, Rev. Mod. Phys.82, 3045 (2010)

    2010

  42. [42]

    Qi and S.-C

    X.-L. Qi and S.-C. Zhang, Topological insulators and superconductors, Rev. Mod. Phys.83, 1057 (2011)

    2011

  43. [43]

    Chang, C.-X

    C.-Z. Chang, C.-X. Liu, and A. H. MacDonald, Colloquium: Quantum anomalous hall effect, Rev. Mod. Phys.95, 011002 (2023)

    2023

  44. [44]

    Chi and J

    H. Chi and J. S. Moodera, Progress and prospects in the quantum anomalous hall effect, APL Mater.10, 090903 (2022)

    2022

  45. [45]

    Weber, M

    B. Weber, M. S. Fuhrer, X.-L. Sheng, S. A. Yang, R. Thomale,et al., 2024 roadmap on 2d topological insulators, J. Phys. Mater.7, 022501 (2024)

    2024

  46. [46]

    Y. Bai, Y. Li, J. Luan, R. Liu, W. Song, Y. Chen, P.-F. Ji, Q. Zhang, F. Meng, B. Tong, L. Li, Y. Jiang, Z. Gao, L. Gu, J. Zhang, Y. Wang, Q.-K. Xue, K. He, Y. Feng, and X. Feng, Quantized anomalous hall resistivity achieved in molecular beam epitaxy-grown MnBi2Te4 thin films, Natl. Sci. Rev.11, nwad189 (2024)

    2024

  47. [47]

    Bosnar, A

    M. Bosnar, A. Y. Vyazovskaya, E. K. Petrov, E. V. Chulkov, and M. M. Otrokov, High chern number van der waals magnetic topological multilayers MnBi2Te4/hbn, npj 2D Mater. Appl.7, 33 (2023)

    2023

  48. [48]

    Baù and A

    N. Baù and A. Marrazzo, Local chern marker for periodic systems, Phys. Rev. B109, 014206 (2024)

    2024

  49. [49]

    H. Park, J. Cai, E. Anderson,et al., Observation of fractionally quantized anomalous hall effect, Nature622, 74 (2023)

    2023

  50. [50]

    J. Cai, E. Anderson, C. Wang,et al., Signatures of fractional quantum anomalous hall states in twisted MoTe2, Nature 622, 63 (2023)

    2023

  51. [51]

    Z. Lu, T. Han, Y. Yao,et al., Fractional quantum anomalous hall effect in multilayer graphene, Nature626, 759 (2024)

    2024

  52. [52]

    L. Ju, T. Cao, L. Fu, D. Xiao, and X. Xu, The fractional quantum anomalous hall effect, Nat. Rev. Mater.9, 455 (2024)

    2024

  53. [53]

    R. M. Kaufmann, D. Li, and B. Wehefritz-Kaufmann, Notes on topological insulators, Rev. Math. Phys.28, 1630003 (2016), arXiv:1501.02874 [math-ph]

    Pith/arXiv arXiv 2016

  54. [54]

    Fradkin,Field Theories of Condensed Matter Physics (Addison Wesley Publishing Company, 1991)

    E. Fradkin,Field Theories of Condensed Matter Physics (Addison Wesley Publishing Company, 1991)

    1991

  55. [55]

    Tong, Lectures on the quantum hall effect (2016), arXiv:1606.06687 [hep-th]

    D. Tong, Lectures on the quantum hall effect (2016), arXiv:1606.06687 [hep-th]

    Pith/arXiv arXiv 2016

  56. [56]

    Hatsugai, Topological aspects of the quantum Hall effect, J

    Y. Hatsugai, Topological aspects of the quantum Hall effect, J. Phys. Condens. Matter9, 2507 (1997)

    1997

  57. [57]

    X.-L. Qi, T. L. Hughes, and S.-C. Zhang, Topological field theory of time-reversal invariant insulators, Phys. Rev. B78, 195424 (2008)

    2008

  58. [58]

    Ishikawa and T

    K. Ishikawa and T. Matsuyama, Magnetic field induced multi component QED in three-dimensions and quantum Hall effect, Z. Phys. C33, 41 (1986)

    1986

  59. [59]

    G. E. Volovik, An analog of the quantum Hall effect in a superfluid 3He film, JETP67, 9 (1988), zhETF, Vol. 94, No. 3(9), 123. 52

    1988

  60. [60]

    G. E. Volovik,The Universe in a Helium Droplet (Clarendon Press, Oxford, 2003)

    2003

  61. [61]

    Coleman and B

    S. Coleman and B. Hill, Phys. Lett. B159, 184 (1985)

    1985

  62. [62]

    Lee, Phys

    T. Lee, Phys. Lett. B171, 247 (1986)

    1986

  63. [63]

    M. A. Zubkov, Momentum space topology of QCD, Annals Phys393, 264 (2018), arXiv:1610.08041

    Pith/arXiv arXiv 2018

  64. [64]

    C. X. Zhang and M. A. Zubkov, Influence of interactions on the anomalous quantum Hall effect (2019), arXiv:arXiv:1902.06545 [cond-mat.mes-hall]

    arXiv 2019

  65. [65]

    Zubkov, Momentum space topology of qcd, Annals of Physics393, 264 (2018)

    M. Zubkov, Momentum space topology of qcd, Annals of Physics393, 264 (2018)

    2018

  66. [66]

    Zubkov and G

    M. Zubkov and G. Volovik, Momentum space topological invariants for the 4d relativistic vacua with mass gap, Nuclear Physics B860, 295 (2012)

    2012

  67. [67]

    Volovik and M

    G. Volovik and M. Zubkov, Standard model as the topological material, New Journal of Physics19, 015009 (2017)

    2017

  68. [68]

    M. A. Zubkov and Z. V. Khaidukov, Topology of the momentum space, wigner transformations, and a chiral anomaly in lattice models, JETP Letters106, 172 (2017)

    2017

  69. [69]

    G. E. Volovik and M. Zubkov, Nambu sum rule in the njl models: from superfluidity to top quark condensation, JETP letters97, 301 (2013)

    2013

  70. [70]

    Katsnelson, G

    M. Katsnelson, G. Volovik, and M. Zubkov, Euler–heisenberg effective action and magnetoelectric effect in multilayer graphene, Annals of Physics331, 160 (2013)

    2013

  71. [71]

    Volovik and M

    G. Volovik and M. Zubkov, Scalar excitation with leggett frequency in he 3-b and the 125 gev higgs particle in top quark condensation models as pseudo-goldstone bosons, Physical Review D92, 055004 (2015)

    2015

  72. [72]

    Selch, M

    M. Selch, M. Zubkov, S. Pramanik, and M. Lewkowicz, Non-renormalization of the fractional quantum hall conductivity by interactions, Annals of Physics , 170202 (2025)

    2025

  73. [73]

    Bakker, A

    B. Bakker, A. Veselov, and M. Zubkov, Central dominance and the confinement mechanism in gluodynamics, Physics Letters B471, 214 (1999)

    1999

  74. [74]

    Bakker, A

    B. Bakker, A. Veselov, and M. Zubkov, Standard model with the additional z6 symmetry on the lattice, Physics Letters B620, 156 (2005)

    2005

  75. [75]

    M. A. Zubkov and X. Wu, Topological invariant in terms of the Green functions for the Quantum Hall Effect in the presence of varying magnetic field (2019), arXiv:arXiv:1901.06661 [cond-mat.mes-hall]

    arXiv 2019

  76. [76]

    Zhang and M

    C. Zhang and M. Zubkov, Influence of interactions on integer quantum hall effect, Annals of Physics444, 169016 (2022)

    2022

  77. [77]

    I. V. Fialkovsky and M. A. Zubkov, Elastic deformations and Wigner-Weyl formalism in graphene, Symmetry12, 317 (2020), arXiv:1905.11097

    arXiv 2020

  78. [78]

    M. N. Chernodub and M. Zubkov, Scale magnetic effect in quantum electrodynamics and the wigner-weyl formalism, Physical Review D96, 056006 (2017)

    2017

  79. [79]

    Zhang and M

    C. Zhang and M. Zubkov, Influence of interactions on the anomalous quantum hall effect, Journal of Physics A: Mathe- matical and Theoretical53, 195002 (2020)

    2020

  80. [80]

    Suleymanov and M

    M. Suleymanov and M. Zubkov, Wigner–weyl formalism and the propagator of wilson fermions in the presence of varying external electromagnetic field, Nuclear Physics B938, 171 (2019)

    2019

Showing first 80 references.