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arxiv: 2606.12104 · v1 · pith:VHRUN77N · submitted 2026-06-10 · cond-mat.supr-con

Proximity-induced unconventional superconductivity and chiral topological phases in twisted graphene/NbSe₂ van der Waals heterostructure

Adam Hlo\v{z}n\'y , Marko Milivojevi\'c This is my paper

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classification cond-mat.supr-con
keywords proximity-induced superconductivitytwisted grapheneNbSe2 heterostructurechiral topological superconductivityChern numbersC3 symmetryBogoliubov-de Gennesvan der Waals materials
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The pith

Twisted graphene on NbSe2 can realize chiral topological superconductivity with Chern numbers in {-4,-2,2,4} via proximity-induced pairing under C3 symmetry.

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

The paper uses the Bogoliubov-de Gennes formalism on parameters taken from ab initio calculations at a 23.4 degree twist to model how superconductivity from NbSe2 enters the graphene layer. With the common symmetry reduced to the C3 point group, the authors enumerate all allowed combinations of singlet and triplet gap functions and track how the topological invariants change with the relative strength of each channel. The resulting phase diagram contains extended regions of chiral topological superconductivity distinguished by nonzero Chern numbers. If realized, these phases would supply a van der Waals platform in which topological superconductivity is induced rather than engineered from scratch.

Core claim

Using symmetry-allowed gap functions classified by the irreducible representations of the C3 group, the calculation shows that mixtures of singlet and triplet pairing channels produce a phase diagram containing chiral topological superconducting states with Chern numbers C belonging to the set {-4,-2,2,4}.

What carries the argument

Symmetry-allowed superconducting gap functions under the C3 point group that mix singlet and triplet channels, whose topological character is diagnosed by the Chern number.

If this is right

  • The heterostructure supplies a concrete platform for proximity-induced chiral topological superconductivity.
  • The topological phases remain detectable by quasiparticle interference imaging and transport measurements.
  • Symmetry reduction at the interface can stabilize chiral pairing components that are not dominant in bulk NbSe2.
  • Nonzero Chern numbers imply protected boundary modes whose presence follows directly from the bulk invariants.

Where Pith is reading between the lines

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

  • Different twist angles that preserve C3 symmetry could be scanned to enlarge or shrink the topological regions in the phase diagram.
  • If chiral pairing is confirmed, defects or edges in the graphene layer should host zero-energy modes whose statistics could be tested separately.
  • The same symmetry-classification approach could be applied to other transition-metal dichalcogenide substrates to generate different sets of Chern numbers.
  • Control of the relative strength of singlet versus triplet channels might be achieved by gating or strain, offering a route to switch between topological phases.

Load-bearing premise

The van der Waals interface and resulting symmetry lowering can change which pairing channel is most stable and thereby favor a chiral component that gets induced into the graphene layer.

What would settle it

If the computed Chern number remains zero for every combination of mixing parameters between the allowed gap functions, or if quasiparticle interference and transport experiments detect no signatures consistent with nonzero Chern numbers, the claimed topological phases would be ruled out.

Figures

Figures reproduced from arXiv: 2606.12104 by Adam Hlo\v{z}n\'y, Marko Milivojevi\'c.

Figure 1
Figure 1. Figure 1: FIG. 1. Top (a) and side (b) view of the graphene/NbSe [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Chern number [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Local Chern number contributions in small regions around the [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Comparison of the electronic band structure of graphene on NbSe [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. WFC flow, computed via the Wilson loop method, [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. WFC flow computed via the Wilson loop method [PITH_FULL_IMAGE:figures/full_fig_p012_6.png] view at source ↗
read the original abstract

We study proximity-induced unconventional superconductivity in a twisted graphene/NbSe$_2$ van der Waals heterostructure using the Bogoliubov-de Gennes formalism. The normal-state parameters of proximitized graphene are extracted from ab initio calculations at a twist angle of $23.4^\circ$, which reduces the common symmetry of the heterostructure to $\mathbf{C}_3$. We construct symmetry-allowed superconducting gap functions of the graphene layer according to the irreducible representations of the $\mathbf{C}_3$ group, containing singlet and triplet pairing channels and their mixtures. Computing the topological invariants as a function of the mixing parameters, we find a rich phase diagram of chiral topological superconducting phases, characterized by nonzero Chern numbers $C\in\{-4,-2,2,4\}$. While the nature of the superconducting order parameter of NbSe$_2$ remains debated, the formation of the van der Waals heterostructure and the related symmetry reduction can alter the relative stability of competing pairing channels, potentially stabilizing a chiral component that is proximity-induced into graphene and triggers the topological phases identified here, making the twisted graphene/NbSe$_2$ heterostructure a promising platform for chiral topological superconductivity detectable via quasiparticle interference imaging and transport measurements.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 2 minor

Summary. The manuscript studies proximity-induced unconventional superconductivity in a twisted graphene/NbSe2 van der Waals heterostructure via the Bogoliubov-de Gennes formalism. Normal-state parameters are taken from ab initio calculations at 23.4° twist angle (reducing symmetry to C3). Symmetry-allowed gap functions are constructed from the irreducible representations of C3, incorporating singlet and triplet channels and their mixtures. Topological invariants are evaluated over the Brillouin zone as a function of the mixing parameters, producing a phase diagram containing chiral topological superconducting phases with Chern numbers C ∈ {-4, -2, 2, 4}. The work proposes the heterostructure as a platform for realizing and detecting such phases.

Significance. If the phase diagram is robust, the result would identify a concrete 2D van der Waals platform hosting multiple chiral topological superconducting phases with sizable Chern numbers. The combination of ab initio normal-state parameters with C3 symmetry classification of the gap functions is a methodological strength that grounds the calculation in material-specific inputs.

major comments (2)
  1. [Results (phase-diagram computation)] The central claim of a rich phase diagram with C = ±2, ±4 rests on scanning the singlet-triplet mixing parameters and evaluating the Chern number, yet no numerical details (Brillouin-zone discretization, convergence checks, or validation against the pure-singlet or pure-triplet limits where C must vanish) are supplied. This absence is load-bearing for the reported phase boundaries.
  2. [Discussion / abstract] The assertion that heterostructure formation and C3 symmetry reduction can stabilize a chiral component (abstract, final paragraph) is presented as a possibility but is not supported by any energetic comparison or self-consistent calculation of the mixing parameters; the parameters are scanned rather than determined from the microscopic model.
minor comments (2)
  1. [Methods] Define the precise parametrization of the singlet-triplet mixing (e.g., the range and normalization of the coefficients) in the gap-function construction.
  2. [Methods] Specify the ab initio method, k-point sampling, and relaxation protocol used to extract the normal-state parameters at 23.4° twist.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive report. Below we provide point-by-point responses to the major comments.

read point-by-point responses

  1. Referee: [Results (phase-diagram computation)] The central claim of a rich phase diagram with C = ±2, ±4 rests on scanning the singlet-triplet mixing parameters and evaluating the Chern number, yet no numerical details (Brillouin-zone discretization, convergence checks, or validation against the pure-singlet or pure-triplet limits where C must vanish) are supplied. This absence is load-bearing for the reported phase boundaries.

    Authors: We agree with the referee that additional numerical details are necessary to substantiate the phase diagram. In the revised version, we will include information on the Brillouin zone discretization (e.g., the number of k-points used), convergence tests with respect to grid density, and explicit checks confirming that the Chern number is zero in the pure-singlet and pure-triplet limits. These additions will be placed in the methods section or a new appendix. revision: yes

  2. Referee: [Discussion / abstract] The assertion that heterostructure formation and C3 symmetry reduction can stabilize a chiral component (abstract, final paragraph) is presented as a possibility but is not supported by any energetic comparison or self-consistent calculation of the mixing parameters; the parameters are scanned rather than determined from the microscopic model.

    Authors: The manuscript frames the stabilization of the chiral component as a potential outcome of the symmetry reduction in the heterostructure, consistent with the phrasing 'potentially stabilizing' in the abstract. We do not claim to have performed an energetic comparison or self-consistent determination of the mixing parameters, as that would require a detailed microscopic model of the interface, which lies outside the scope of this work focused on the topological consequences. To clarify this, we will revise the discussion to explicitly state that the mixing parameters are varied to map the phase diagram, and note that future work could address the microscopic determination of these parameters. revision: partial

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The central computation constructs symmetry-allowed gap functions from the irreps of the C3 point group, inserts ab-initio-derived normal-state parameters into the BdG Hamiltonian, and evaluates Chern numbers over the Brillouin zone while scanning the singlet-triplet mixing parameters. These steps follow directly from group theory and standard topological band theory without any fitted parameter being renamed as a prediction, without load-bearing self-citations, and without any self-definitional reduction of the target invariants to the inputs. The phase diagram is an output of the explicit diagonalization and invariant calculation rather than an input.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The claim rests on the applicability of the Bogoliubov-de Gennes formalism to the proximitized layer, the accuracy of the ab-initio normal-state parameters at the chosen twist angle, and the assumption that the C3 symmetry classification fully captures the allowed gap functions.

free parameters (1)
  • mixing parameters between singlet and triplet channels
    These parameters are varied continuously to trace the phase diagram; their specific values are not fixed by external data in the abstract.
axioms (2)
  • domain assumption Bogoliubov-de Gennes formalism is adequate for describing the proximity-induced state
    Standard mean-field treatment invoked without further justification in the abstract.
  • domain assumption The twist angle of 23.4° reduces the symmetry exactly to the C3 group
    Stated as extracted from ab initio calculations.

pith-pipeline@v0.9.1-grok · 5767 in / 1416 out tokens · 21629 ms · 2026-06-27T08:09:36.849601+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

73 extracted references · 1 canonical work pages

  1. [1]

    Singletd 0 gap functions In the nearest-neighbor approximation, the ∆ AB k,Am(s) block in the singlet case can be constructed as ∆AB k,Am(s) = X g∈C3 Γ∗ Am(g)eik·(D(g−1)rAB′ )d0,(11) wherem= 0,±1, whiler nn =a/ √ 3(0,1,0) represents the distance between the representative nearest-neghbor atoms on sublattice A and B (see FIG. 1c), whileDcorre- sponds to th...

  2. [2]

    Tripletd z gap functions In the case of thed z triplet gap functions, the same construction as in the singlet case, i.e. using the formulaP g∈C3 Γ∗ Am(g)eik·(D(g−1)rAB′ )D(g−1)dz =f m(k)dz, does not give us gap function that have the proper behavior with respect to the TRS, Θ ˆ∆nn A0,(z)Θ−1 = ˆ∆nn A0,(z) and Θ ˆ∆nn A1,(z)Θ−1 = ˆ∆nn A−1,(z). However, a pro...

  3. [3]

    (18) have the correct TRS behavior, since ˆ∆nn A0,(z) preserves TRS, while ˆ∆nn A1,(z) and ˆ∆nn A−1,(z) are time-reversal partners

    Thus, ∆ nn Am,(z)(k) is equal to ∆nn Am,(z)(k) = 0 0 0g m(k) 0 0g m(k) 0 0−g m(−k) 0 0 −gm(−k) 0 0 0 ! ,(17) or in the second quantized form, ˆ∆nn Am,(z) = X k h gm(k)(a† k↑b† −k↓ +a † k↓b† −k↑) −gm(−k)(b† k↑a† −k↓ +b † k↓a† −k↑) i .(18) The gap functions constructed using the Eq. (18) have the correct TRS behavior, since ˆ∆nn A0,(z) preserves TRS, while ...

  4. [4]

    Θ ˆ∆nn A0,(xy)Θ−1 = ˆ∆nn A0,(xy) and Θ ˆ∆nn A1,(xy)Θ−1 = ˆ∆nn A−1,(xy)

    Triplet(d x, dy)gap functions Finally, we construct the gap functions in the (d x, dy) space, and demand, similarly as in the previous two cases, the proper TRS behavior of the constructed gap functions, i.e. Θ ˆ∆nn A0,(xy)Θ−1 = ˆ∆nn A0,(xy) and Θ ˆ∆nn A1,(xy)Θ−1 = ˆ∆nn A−1,(xy). Such a behavior can be ob- tained by defining a gap function in the followin...

  5. [5]

    Singlet-triplet mixing The classification of the superconducting gap functions in terms of the IRs of theC 3 group allows us to mix all gap functions belonging to the same IR. This is because gap functions within the same IR share identical transfor- mation properties under all symmetry operations of the group, making them indistinguishable from a symmetr...

  6. [6]

    ,4 labels the occupied BdG bands

    Overlap matrix and Wannier center construction We analyze the BdG HamiltonianH BdG(k) (2) with Bloch eigenstates|Ψ nk⟩, wheren= 1, . . . ,4 labels the occupied BdG bands. For a fixed value of the crystal momentumk 1 alongb 1 = 4π/3 (3/2, √ 3/2,0), a closed string ofN 2 + 1 discretised points is constructed along the second reciprocal directionb 2 = 4π/ √ ...

  7. [7]

    Therefore, it is enough to compute the Wilson loop over half the BZ to determine theZ 2 invariant [43]

    Time-reversal symmetric case:Z 2 invariant When TRS is present, the WFC spectrum over the sec- ond half of the BZ is the mirror image of the first half. Therefore, it is enough to compute the Wilson loop over half the BZ to determine theZ 2 invariant [43]. TheZ 2 invariant is extracted by counting the number of cross- ings of the WFC flow with an arbitrar...

  8. [8]

    The odd FIG

    over the half of the BZ. The odd FIG. 5. WFC flow, computed via the Wilson loop method, for the singlet gap function ∆ s A0(π/4, π/3). The main panel is obtained using the 3000×6000k-point mesh alongb 1 ×b 2, and suggest at first sight an odd number of crossings with an arbitrary horizontal line, indicatingZ 2 = 1. The insets, computed on a refined 60×300...

  9. [9]

    The Chern numberCis ex- tracted by counting the crossings of the WFC flow with an arbitrary horizontal reference lineθref ∈(− 1 2 , 1

    Chern number When TRS is broken, the relevant topological invari- ant is the Chern numberC. The Chern numberCis ex- tracted by counting the crossings of the WFC flow with an arbitrary horizontal reference lineθref ∈(− 1 2 , 1

  10. [10]

    However, the crossings are slope dependent

    in the full BZ. However, the crossings are slope dependent. Pre- cisely, the WFC flow crossingθ ref with a positive slope counts as +1, and with a negative slope as−1. After calculating the intersection of all WFC branchesθ i (B5) withθ ref, the Chern number is determined. Whereas C̸= 0 signals topologically nontrivial phase,C= 0 cor- responds to a trivia...

  11. [11]

    ˇZuti´ c, J

    I. ˇZuti´ c, J. Fabian, and S. Das Sarma, Review of Modern Physics76, 323 (2004)

    2004

  12. [12]

    Fabian, A

    J. Fabian, A. Matos-Abiague, C. Ertler, P. Stano, and 13 I. ˇZuti´ c, Acta Phys. Slovaca57, 565 (2007)

    2007

  13. [13]

    Offidani, M

    M. Offidani, M. Milletar` ı, R. Raimondi, and A. Ferreira, Phys. Rev. Lett.119, 196801 (2017)

    2017

  14. [14]

    T. S. Ghiasi, A. A. Kaverzin, P. J. Blah, and B. J. van Wees, Nano Lett.19, 5959 (2019)

    2019

  15. [15]

    Sinova, S

    J. Sinova, S. O. Valenzuela, J. Wunderlich, C. H. Back, and T. Jungwirth, Rev. Mod. Phys.87, 1213 (2015)

    2015

  16. [16]

    Tombros, C

    N. Tombros, C. Jozsa, M. Popinciuc, H. T. Jonkman, and B. J. van Wees, Nature448, 571–574 (2007)

    2007

  17. [17]

    K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, Science306, 666 (2004)

    2004

  18. [18]

    E. W. Hill, A. K. Geim, K. Novoselov, F. Schedin, and P. Blake, IEEE Trans. Magn.42, 2694 (2006)

    2006

  19. [19]

    Ohishi, M

    M. Ohishi, M. Shiraishi, R. Nouchi, T. Nozaki, T. Shinjo, and Y. Suzuki, Jpn. J. Appl. Phys.46, L605 (2007)

    2007

  20. [20]

    Gurram, S

    M. Gurram, S. Omar, and B. J. van Wees, Nat. Com- mun.8, 248 (2017)

    2017

  21. [21]

    Gurram, S

    M. Gurram, S. Omar, and B. J. van Wees, 2D Mater.5, 032004 (2018)

    2018

  22. [22]

    Popinciuc, C

    M. Popinciuc, C. J´ ozsa, P. J. Zomer, N. Tombros, A. Veligura, H. T. Jonkman, and B. J. van Wees, Phys. Rev. B80, 214427 (2009)

    2009

  23. [23]

    Ringer, M

    S. Ringer, M. Rosenauer, T. V¨ olkl, M. Kadur, F. Hop- perdietzel, D. Weiss, and J. Eroms, Appl. Phys. Lett. 113, 132403 (2018)

    2018

  24. [24]

    Avsar, J

    A. Avsar, J. Y. Tan, T. Taychatanapat, J. Balakrishnan, G. K. W. Koon, Y. Yeo, J. Lahiri, A. Carvalho, A. S. Rodin, E. C. T. O’Farrell, G. Eda, A. H. Castro Neto, and B. ¨Ozyilmaz, Nat. Commun.5, 4875 (2014)

    2014

  25. [25]

    Gmitra and J

    M. Gmitra and J. Fabian, Phys. Rev. B92, 155403 (2015)

    2015

  26. [26]

    W. Yan, O. Txoperena, R. Llopis, H. Dery, L. E. Hueso, and F. Casanova, Nat. Commun.7, 13372 (2016)

    2016

  27. [27]

    C. K. Safeer, F. Herling, W. Y. Choi, N. Ontoso, J. Ingla- Ayn´ es, L. E. Hueso, and F. Casanova, 2D Mater.9, 015024 (2022)

    2022

  28. [28]

    S. Lee, D. J. P. de Sousa, Y.-K. Kwon, F. de Juan, Z. Chi, F. Casanova, and T. Low, Phys. Rev. B106, 165420 (2022)

    2022

  29. [29]

    Ontoso, C

    N. Ontoso, C. K. Safeer, F. Herling, J. Ingla-Ayn´ es, H. Yang, Z. Chi, B. Martin-Garcia, I. Robredo, M. G. Vergniory, F. de Juan, M. R. Calvo, L. E. Hueso, and F. Casanova, Phys. Rev. Applied19, 014053 (2023)

    2023

  30. [30]

    Z. Chi, S. Lee, H. Yang, E. Dolan, C. K. Safeer, J. Ingla-Ayn´ es, F. Herling, N. Ontoso, B. Mart´ ın-Garc´ ıa, M. Gobbi, T. Low, L. E. Hueso, and F. Casanova, Adv. Mater.36, 2310768 (2024)

    2024

  31. [31]

    Z. Chi, E. Dolan, H. Yang, B. Mart´ ın-Garc´ ıa, M. Gobbi, L. E. Hueso, and F. Casanova, Phys. Rev. Applied24, 064076 (2025)

    2025

  32. [32]

    Milivojevi´ c, J

    M. Milivojevi´ c, J. Mnich, P. Jureczko, M. Kurpas, and M. Gmitra, Mater. Futures5, 015201 (2026)

    2026

  33. [33]

    Amundsen, J

    M. Amundsen, J. Linder, J. W. A. Robinson, I. ˇZuti´ c, and N. Banerjee, Rev. Mod. Phys.96, 021003 (2024)

    2024

  34. [34]

    Mart´ ınez, P

    I. Mart´ ınez, P. H¨ ogl, C. Gonz´ alez-Ruano, J. P. Cascales, C. Tiusan, Y. Lu, M. Hehn, A. Matos-Abiague, J. Fabian, I. ˇZuti´ c, and F. G. Aliev, Phys. Rev. Applied13, 014030 (2020)

    2020

  35. [35]

    Zhang, R

    Y. Zhang, R. Polski, A. Thomson, E. Lantagne- Hurtubise, C. Lewandowski, H. Zhou, K. Watanabe, T. Taniguchi, J. Alicea, and S. Nadj-Perge, Nature613, 268 (2023)

    2023

  36. [36]

    Zhang, G

    Y. Zhang, G. Shavit, H. Ma, Y. Han, C. W. Siu, A. Mukherjee, K. Watanabe, T. Taniguchi, D. Hsieh, C. Lewandowski, F. von Oppen, Y. Oreg, and S. Nadj- Perge, Nature641, 625 (2025)

    2025

  37. [37]

    Sanchez, A

    D. Sanchez, A. Junod, J. Muller, H. Berger, and F. Levy, Physica B204, 167 (1995)

    1995

  38. [38]

    X. Xi, Z. Wang, W. Zhao, J.-H. Park, K. T. Law, H. Berger, L. Forr´ o, J. Shan, and K. F. Mak, Nat. Phys. 12, 139 (2016)

    2016

  39. [39]

    Khestanova, J

    E. Khestanova, J. Birkbeck, M. Zhu, Y. Cao, G. Yu, D. Ghazaryan, J. Yin, H. Berger, L. Forro, T. Taniguchi, K. Watanabe, K. S. Novoselov, A. Mishchenko, R. Gor- bachev, A. K. Geim, and I. V. Grigorieva, Nano Lett. 18, 2623 (2018)

    2018

  40. [40]

    H¨ ogl, T

    P. H¨ ogl, T. Frank, D. Kochan, M. Gmitra, and J. Fabian, Phys. Rev. B101, 245441 (2020)

    2020

  41. [41]

    Naimer, M

    T. Naimer, M. Gmitra, and J. Fabian, Phys. Rev. B109, 205109 (2024)

    2024

  42. [42]

    Naritsuka, T

    M. Naritsuka, T. Machida, S. Asano, Y. Yanase, and T. Hanaguri, Nat. Phys.21, 746–753 (2025)

    2025

  43. [43]

    Trivini, T

    S. Trivini, T. Kokkeler, J. Ortuzar, E. Cort´ es-del R´ ıo, B. Vi˜ na-Baus´ a, P. Mallet, J.-Y. Veuillen, J. C. Cuevas, I. Brihuega, F. S. Bergeret, and J. I. Pascual, Nano Lett. 25, 16323–16329 (2025)

    2025

  44. [44]

    Altland and M

    A. Altland and M. R. Zirnbauer, Phys. Rev. B55, 1142 (1997)

    1997

  45. [45]

    David, P

    A. David, P. Rakyta, A. Korm´ anyos, and G. Burkard, Phys. Rev. B100, 085412 (2019)

    2019

  46. [46]

    Kochan, S

    D. Kochan, S. Irmer, and J. Fabian, Phys. Rev. B95, 165415 (2017)

    2017

  47. [47]

    Pangburn, L

    E. Pangburn, L. Haurie, A. Cr´ epieux, O. A. Awoga, A. M. Black-Schaffer, C. P´ epin, and C. Bena, Phys. Rev. B108, 134514 (2023)

    2023

  48. [48]

    W. Dai, A. Richardella, R. Du, W. Zhao, X. Liu, C. X. Liu, S.-H. Huang, R. Sankar, F. Chou, N. Samarth, and Q. Li, Sci. Rep.7, 7631 (2017)

    2017

  49. [49]

    J. K. Asb´ oth, L. Oroszl´ any, and A. P´ alyi, A Short Course on Topological Insulators, Lecture Notes in Physics, Vol. 909 (Springer International Publishing, 2016)

    2016

  50. [50]

    Maciejko, X.-L

    J. Maciejko, X.-L. Qi, H. D. Drew, and S.-C. Zhang, Phys. Rev. Lett.105, 166803 (2010)

    2010

  51. [51]

    B. J. Wieder, B. Bradlyn, J. Cano, Z. Wang, M. G. Vergniory, L. Elcoro, A. A. Soluyanov, C. Felser, T. Ne- upert, N. Regnault, and B. Andrei Bernevig, Nat. Rev. Mater.7, 196–216 (2021)

    2021

  52. [52]

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

    2010

  53. [53]

    R. Yu, X. L. Qi, A. Bernevig, Z. Fang, and X. Dai, Phys. Rev. B84, 075119 (2011)

    2011

  54. [54]

    Fu and C

    L. Fu and C. L. Kane, Phys. Rev. B74, 195312 (2006)

    2006

  55. [55]

    Fukui, Y

    T. Fukui, Y. Hatsugai, and H. Suzuki, J. Phys. Soc. Jpn. 74, 1674 (2005)

    2005

  56. [56]

    Gauge theory,

    D. Tong, “Gauge theory,” Lecture notes, University of Cambridge (2018)

    2018

  57. [57]

    Hloˇ zn´ y, J

    A. Hloˇ zn´ y, J. Haniˇ s, M. Gmitra, and M. Milivojevi´ c, arXiv preprint (2026), arXiv:2602.19791 [cond-mat]

    work page arXiv 2026

  58. [58]

    Haniˇ s, M

    J. Haniˇ s, M. Milivojevi´ c, and M. Gmitra, Phys. Rev. B 110, 104502 (2024)

    2024

  59. [59]

    Engstr¨ om, L

    L. Engstr¨ om, L. Zullo, T. Cren, A. Mesaros, and P. Si- mon, Phys. Rev. Lett.135, 236004 (2025)

    2025

  60. [60]

    Siegl, A

    J. Siegl, A. Bleibaum, W. Wan, M. Kurpas, J. Schlie- mann, M. M. Ugeda, M. Marganska, and M. Grifoni, Nat. Commun.16, 8228 (2025)

    2025

  61. [61]

    C.-W. Cho, J. Lyu, L. An, T. Han, K. T. Lo, C. Y. Ng, 14 J. Hu, Y. Gao, G. Li, M. Huang, N. Wang, J. Schmalian, and R. Lortz, Phys. Rev. Lett.129, 087002 (2022)

    2022

  62. [62]

    X. Wu, X. Hao, Z. Chen, Y. Cai, M. Wu, C. Chen, K. Wang, F. Ming, S. Johnston, R.-X. Zhang, and H. H. Weitering, Phys. Rev. X16, 011026 (2026)

    2026

  63. [63]

    T. Han, Z. Lu, Z. Hadjri, L. Shi, Z. Wu, W. Xu, Y. Yao, A. A. Cotten, O. Sharifi Sedeh, H. Weldeyesus, J. Yang, J. Seo, S. Ye, M. Zhou, H. Liu, G. Shi, Z. Hua, K. Watan- abe, T. Taniguchi, P. Xiong, D. M. Zumb¨ uhl, L. Fu, and L. Ju, Nature643, 654 (2025)

    2025

  64. [64]

    Y. Ding, Y. Wang, J. Ni, L. Shi, S. Shi, and W. Tang, Phys. B: Condens. Matter406, 2254 (2011)

    2011

  65. [65]

    Giannozzi, S

    P. Giannozzi, S. Baroni, N. Bonini, M. Calandra, R. Car, C. Cavazzoni, D. Ceresoli, G. L. Chiarotti, M. Cococ- cioni, I. Dabo, A. D. Corso, S. de Gironcoli, S. Fabris, G. Fratesi, R. Gebauer, U. Gerstmann, C. Gougoussis, A. Kokalj, M. Lazzeri, L. Martin-Samos, N. Marzari, F. Mauri, R. Mazzarello, S. Paolini, A. Pasquarello, L. Paulatto, C. Sbraccia, S. Sc...

    2009

  66. [66]

    Giannozzi, O

    P. Giannozzi, O. Andreussi, T. Brumme, O. Bunau, M. B. Nardelli, M. Calandra, R. Car, C. Cavazzoni, D. Ceresoli, M. Cococcioni, N. Colonna, I. Carnimeo, A. D. Corso, S. de Gironcoli, P. Delugas, R. A. DiStasio, A. Ferretti, A. Floris, G. Fratesi, G. Fugallo, R. Gebauer, U. Gerstmann, F. Giustino, T. Gorni, J. Jia, M. Kawa- mura, H.-Y. Ko, A. Kokalj, E. K¨...

    2017

  67. [67]

    J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett.77, 3865 (1996)

    1996

  68. [68]

    P. E. Bl¨ ochl, Phys. Rev. B50, 17953 (1994)

    1994

  69. [69]

    Methfessel and A

    M. Methfessel and A. T. Paxton, Phys. Rev. B40, 3616 (1989)

    1989

  70. [70]

    Grimme, J

    S. Grimme, J. Comput. Chem.27, 1787 (2006)

    2006

  71. [71]

    Barone, M

    V. Barone, M. Casarin, D. Forrer, M. Pavone, M. Sambi, and A. Vittadini, J. Comput. Chem.30, 934 (2008)

    2008

  72. [72]

    Bengtsson, Phys

    L. Bengtsson, Phys. Rev. B59, 12301 (1999)

    1999

  73. [73]

    N. J. Higham, Functions of Matrices: Theory and Computation (SIAM, Philadelphia, 2008)

    2008