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arxiv: 2512.10944 · v2 · pith:F47LNZVGnew · submitted 2025-12-11 · ❄️ cond-mat.mes-hall · cond-mat.str-el· cond-mat.supr-con

Pair density wave in quarter metals from a repulsive fermionic interaction in graphene heterostructures: A renormalization group study

Sk Asrap Murshed , Bitan Roy This is my paper

Pith reviewed 2026-05-25 07:32 UTC · model grok-4.3

classification ❄️ cond-mat.mes-hall cond-mat.str-elcond-mat.supr-con
keywords pair density wavequarter metalgraphene heterostructuresrenormalization grouprepulsive interactionvalley polarizationchiral superconductivityodd-parity pairing
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The pith

Repulsive density-density interactions among polarized fermions in quarter metals of graphene heterostructures can drive a chiral odd-parity pair density wave at low temperatures via renormalization group flow.

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

In chirally stacked multilayer graphene heterostructures under perpendicular electric displacement fields, low doping produces a non-degenerate quarter metal in which quasiparticles are fully polarized around one spontaneously chosen valley. The paper shows that repulsive density-density interactions between these polarized excitations can generate an instability toward a superconducting pair density wave. A leading-order renormalization group analysis establishes that this paired state develops at low temperatures and is chiral with odd parity. The mechanism is offered as an explanation for superconductivity observed experimentally near quarter-metal regimes in several members of the graphene heterostructure family.

Core claim

From a leading order renormalization group analysis, repulsive density-density interaction among the polarized fermionic excitations in the quarter metal can foster the pair density wave phase that is chiral and odd parity in nature at low temperatures.

What carries the argument

Leading order renormalization group analysis of repulsive density-density interactions among valley-polarized quasiparticles in the quarter metal.

If this is right

  • The pair density wave constitutes the unique local superconducting ground state permitted by the non-degenerate quarter metal.
  • The paired state is chiral and odd-parity.
  • The mechanism connects directly to superconducting states observed experimentally near the quarter metal in several graphene heterostructures.
  • Analogous paired states can be pursued in optical honeycomb lattices with repulsive interactions.

Where Pith is reading between the lines

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

  • If the RG flow is robust, similar pairing instabilities may appear in other spontaneously valley-polarized two-dimensional systems under purely repulsive interactions.
  • Controlled realization in optical lattices would allow direct tuning of interaction strength without substrate disorder, providing a clean test of the predicted temperature scale.
  • Spontaneous valley selection implies possible domain walls between regions of opposite valley polarization that could host gapless or fractionalized excitations.

Load-bearing premise

The quarter metal realized around one spontaneously chosen valley can sustain a single local superconducting ground state that is chiral and odd parity.

What would settle it

Detection of even-parity pairing symmetry or complete absence of superconductivity inside the quarter-metal doping window under perpendicular displacement field would falsify the predicted instability.

Figures

Figures reproduced from arXiv: 2512.10944 by Bitan Roy, Sk Asrap Murshed.

Figure 2
Figure 2. Figure 2: FIG. 2. Feynman diagrams representing (a) the bare vertex [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 1
Figure 1. Figure 1: FIG. 1. Feynman diagrams for (a) the bare four-fermion [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Cuts of the global phase diagram for the quarter-metal in chirally-stacked [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Same as Fig [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗
read the original abstract

Electronic bands in chirally stacked $n$ layer carbon-based honeycomb heterostructures, encompassing rhombohedral or ABC ($n \geq 3$), Bernal or AB bilayer ($n=2$), and monolayer ($n=1$) graphene, possess four-fold valley and spin degeneracy. Such systems with $n \geq 2$, when subject to external perpendicular electric displacement fields, feature a fully degenerate metal at high doping, a spin polarized but valley degenerate half-metal at moderate doping, and a non-degenerate quarter metal at low doping. Due to the fully polarized nature of the quasiparticles in the quarter metal, realized around one particular valley otherwise chosen spontaneously, it can sustain a single local superconducting ground state, representing a pair density wave that is chiral and odd parity in nature. From a leading order renormalization group analysis, here we show that repulsive density-density interaction among such polarized fermionic excitations can foster the pair density wave phase at low temperatures. Connections with experimentally observed superconducting states in the close vicinity of the quarter metal in some members of such graphene heterostructures family are discussed and possible routes to realize such a paired state in optical honeycomb lattices are highlighted.

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 / 0 minor

Summary. The manuscript claims that in chirally stacked multilayer graphene heterostructures (n≥2) under perpendicular displacement fields, the quarter-metal regime features spontaneously valley-polarized fermions that, under repulsive density-density interactions, develop a chiral odd-parity pair density wave (PDW) superconducting instability at low temperatures, as obtained from a leading-order renormalization group analysis. Connections to nearby experimental superconducting states and possible optical-lattice realizations are discussed.

Significance. If the RG result holds, the work supplies a controlled perturbative mechanism for PDW order driven purely by repulsion in a fully polarized quarter metal, offering a possible explanation for superconductivity observed near quarter-metal fillings in rhombohedral and Bernal graphene devices and a concrete proposal for optical lattices.

major comments (1)
  1. [Abstract] Abstract: the central claim that a leading-order RG analysis produces a PDW instability is asserted without any flow equations, beta functions, cutoff scheme, or numerical/analytic results, so the support for the instability cannot be evaluated from the provided text.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their comments. We address the single major comment below.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claim that a leading-order RG analysis produces a PDW instability is asserted without any flow equations, beta functions, cutoff scheme, or numerical/analytic results, so the support for the instability cannot be evaluated from the provided text.

    Authors: The abstract is a concise summary and therefore omits the explicit flow equations, beta functions, cutoff scheme, and numerical/analytic results. These are derived and presented in full in Sections III and IV of the manuscript, together with the associated figures. We will revise the abstract to include a brief statement that the PDW instability follows from the leading-order RG flow. revision: yes

Circularity Check

0 steps flagged

No significant circularity in RG derivation of PDW instability

full rationale

The paper derives the pair-density-wave instability from a leading-order renormalization-group flow applied to repulsive density-density interactions among valley-polarized fermions. The quarter-metal polarization and single-valley selection are stated as model inputs (spontaneously chosen), not outputs of the RG equations themselves. No fitted parameters are renamed as predictions, no self-citations are invoked as uniqueness theorems, and the abstract presents the PDW phase as an emergent result of the flow rather than a self-definitional or ansatz-smuggled quantity. This is the expected non-circular outcome for a standard perturbative RG analysis.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The analysis rests on standard renormalization-group techniques for interacting fermions; no new free parameters, ad-hoc axioms, or invented entities are introduced in the abstract.

axioms (1)
  • standard math Leading-order renormalization-group flow equations for density-density interactions among polarized fermions are applicable to the quarter-metal regime.
    The abstract states that the result follows from a leading-order RG analysis.

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Reference graph

Works this paper leans on

76 extracted references · 76 canonical work pages · 1 internal anchor

  1. [1]

    S. A. Murshed, B. Roy, Nodal pair density waves from a quarter-metal in crystalline graphene multilayers, Phys. Rev. B 112, 085121 (2025)

    work page 2025

  2. [2]

    Roy and I

    B. Roy and I. F. Herbut, Unconventional superconduc- tivity on honeycomb lattice: Theory of Kekule order pa- rameter, Phys. Rev. B 82, 035429 (2010)

    work page 2010

  3. [4]

    T. Han, Z. Lu, Z. Hadjri, L. Shi, Z. Wu, W. Xu, Y. Yao, A. A. Cotten, O. S. 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, Signatures of Chiral Superconductivity in Rhom- bohedral Graphene, Nature 643, 654 (2025)

    work page 2025

  4. [5]

    Y. Choi, Y. Choi, M. Valentini, C. L. Patterson, L. F. W. Holleis, O. I. Sheekey, H. Stoyanov, X. Cheng, T. Taniguchi, K. Watanabe and A. F. Young, Superconduc- tivity and quantized anomalous Hall effect in rhombohe- dral graphene, Nature 639, 342 (2025)

    work page 2025

  5. [6]

    Y.Z. Chou, J. Zhu, and S. Das Sarma, Intravalley spin- polarized superconductivity in rhombohedral tetralayer graphene, Phys. Rev. B 111, 174523 (2025)

    work page 2025

  6. [7]

    A. S. Patri and M. Franz, Family of multilayer graphene superconductors with tunable chirality: Momentum- space vortices nucleated by a ring of Berry curvature, Phys. Rev. B 112, 214505 (2025)

    work page 2025

  7. [8]

    Christos, P

    M. Christos, P. M. Bonetti, M. S. Scheurer, Finite- momentum pairing and superlattice superconduc- tivity in valley-imbalanced rhombohedral graphene, arXiv:2503.15471 (2025)

    work page arXiv 2025

  8. [9]

    Z. Dong, P. A. Lee, A controlled expansion for pairing in a polarized band with strong repulsion, arXiv:2503.11079 (2025)

    work page arXiv 2025

  9. [10]

    Parra-Mart´ ınez, A

    G. Parra-Mart´ ınez, A. Jimeno-Pozo, V. T. Phong, H. Sainz-Cruz, D. Kaplan, P. Emanuel, Y. Oreg, P. A. Pan- tale´ on, J.´A. Silva-Guill´ en, and F. Guinea, Band Renor- malization, Quarter Metals, and Chiral Superconductiv- ity in Rhombohedral Tetralayer Graphene, Phys. Rev. Lett. 135, 136503 (2025)

    work page 2025

  10. [11]

    Z. Han, J. Herzog-Arbeitman, Q. Gao, and E. Kha- laf, Exact models of chiral flat-band superconductors, arXiv:2508.21127 (2025)

    work page arXiv 2025

  11. [12]

    Barlas, F

    Y. Barlas, F. Zhang, and E. Rossi, Quantum Geometry Induced Kekul´ e Superconductivity in Haldane phases, arXiv:2508.2179 (2025)

    work page arXiv 2025

  12. [13]

    Yang and Y-H

    H. Yang and Y-H. Zhang, Topological incommen- surate Fulde-Ferrell-Larkin-Ovchinnikov superconductor and Bogoliubov Fermi surface in rhombohedral tetralayer graphene, Phys. Rev. B 112, L020506 (2025)

    work page 2025

  13. [14]

    Geier, M

    M. Geier, M. Davydova, and L. Fu, Chiral and topo- logical superconductivity in isospin polarized multilayer graphene, arXiv:2409.13829 (2024)

    work page arXiv 2024

  14. [15]

    H. Zhou, L. Holleis, Y. Saito, L. Cohen, W. Huynh, C. L. Patterson, F. Yang, T. Taniguchi, K. Watanabe, and A. F. Young, Isospin magnetism and spin-polarized su- perconductivity in Bernal bilayer graphene, Science 375, 774 (2022)

    work page 2022

  15. [16]

    S. C. de la Barrera, S. Aronson, Z. Zheng, K. Watanabe, T. Taniguchi, Q. Ma, P. Jarillo-Herrero, and R. Ashoori, Cascade of isospin phase transitions in Bernal-stacked bilayer graphene at zero magnetic field, infrared spec- troscopy, Nat. Phys. 18, 771 (2022)

    work page 2022

  16. [17]

    A. M. Seiler, F. R. Geisenhof, F. Winterer, K. Watanabe, T. Taniguchi, T. Xu, F. Zhang, and R. T. Weitz, Quan- tum cascade of correlated phases in trigonally warped bilayer graphene, Nature (London) 608, 298 (2022)

    work page 2022

  17. [18]

    C. Li, F. Xu, B. Li, J. Li, G. Li, K. Watanabe, T. Taniguchi, B. Tong, J. Shen, L. Lu, J. Jia, F. Wu, X. Liu and T. Li, Tunable superconductivity in electron- and hole-doped Bernal bilayer graphene, Nature631, 300 (2024)

    work page 2024

  18. [19]

    H. Zhou, T. Xie, A. Ghazaryan, T. Holder, J. R. Ehrets, E. M. Spanton, T. Taniguchi, K. Watanabe, E. Berg, M. Serbyn, and A. F. Young, Half and quarter metals in rhombohedral trilayer graphene, Nature (London) 598, 429 (2021)

    work page 2021

  19. [20]

    H. Zhou, T. Xie, T. Taniguchi, K. Watanabe, and A. F. Young, Superconductivity in rhombohedral trilayer graphene, Nature (London) 598, 434 (2021)

    work page 2021

  20. [21]

    C. L. Patterson, O. I. Sheekey, T. B. Arp, L. F. W. Holleis, J. Koh, Y. Choi, T. Xie, S. Xu, Y. Guo, H. Stoy- anov, E. Redekop, C. Zhang, G. Babikyan, D. Gong, H. Zhou, X. Cheng, T. Taniguchi, K. Watanabe, M. E. Hu- ber, C. Jin, ´E. L.Hurtubise, J. Alicea and A. F. Young, 6 Superconductivity and spin canting in spin–orbit-coupled trilayer graphene, Nature...

    work page 2025

  21. [22]

    T. Arp, O. Sheekey, H. Zhou, C. L. Tschirhart, C. L. Pat- terson, H. M. Yoo, L. Holleis, E. Redekop, G. Babikyan, T. Xie, J. Xiao, Y. Vituri, T. Holder, T. Taniguchi, K. Watanabe, M. E. Huber, E. Berg and A. F. Young, In- tervalley coherence and intrinsic spin–orbit coupling in rhombohedral trilayer graphene, Nat. Phys. 20, 1413 (2024)

    work page 2024

  22. [23]

    Y. Liu, A. Gupta, Y. Choi, Y. Vituri, H. Stoyanov, J. Xiao, Y. Wang, H. Zhou, B. Barick, T. Taniguchi, K. Watanabe, B. Yan, E. Berg, A. F. Young, H. Beidenkopf, N. Avraham, Visualizing incommensurate inter-valley co- herent states in rhombohedral trilayer graphene, arXiv: 2411.11163 (2024)

    work page arXiv 2024

  23. [24]

    K. Liu, J. Zheng, Y. Sha, B. Lyu, F. Li, Y. Park, Y. Ren, K. Watanabe, T. Taniguchi, J. Jia, W. Luo, Z. Shi, J. Jung and G. Chen, Spontaneous broken-symmetry in- sulator and metals in tetralayer rhombohedral graphene, Nat. Nanotechnol. 19, 188 (2024)

    work page 2024

  24. [25]

    Y. Sha, J. Zheng, K. Liu, H. Du, K. Watanabe, T. Taniguchi, J. Jia, Z. Shi, R. Zhong, and G. Chen, Ob- servation of a Chern insulator in crystalline ABCA- tetralayer graphene with spin-orbit coupling, Science 384, 414 (2024)

    work page 2024

  25. [26]

    T. Han, Z. Lu, G. Scuri, J. Sung, J. Wang, T. Han, K. Watanabe, T. Taniguchi, H. Park, and L. Ju, Cor- related insulator and Chern insulators in pentalayer rhombohedral-stacked graphene, Nat. Nanotechnol. 19, 181 (2024)

    work page 2024

  26. [27]

    Z. Lu, T. Han, Y. Yao, A. P. Reddy, J. Yang, J. Seo, K. Watanabe, T. Taniguchi, L. Fu and L. Ju, Fractional quantum anomalous Hall effect in multilayer graphene, Nature 626, 759 (2024)

    work page 2024

  27. [28]

    T. Han, Z. Lu, G. Scuri, J. Sung, J. Wang, T. Han, K. Watanabe, T. Taniguchi, L. Fu, H. Park and L. Ju, Orbital multiferroicity in pentalayer rhombohedral graphene, Nature 623, 41 (2023)

    work page 2023

  28. [29]

    Morissette, P

    E. Morissette, P. Qin, H.-T. Wu, N. J. Zhang, R. Q. Nguyen, K. Watanabe, T. Taniguchi, and J. I. A. Li, Striped Superconductor in Rhombohedral Hexalayer Graphene, arXiv:2504.05129 (2025)

    work page arXiv 2025

  29. [30]

    Morissette, P

    E. Morissette, P. Qin, K. Watanabe, T. Taniguchi, and J. I. A. Li, Coulomb-driven Momentum Space Condensation in Rhombohedral Hexalayer Graphene, arXiv:2503.09954 (2025)

    work page arXiv 2025

  30. [31]

    J. Deng, J. Xie, H. Li, T. Taniguchi, K. Watanabe, J. Shan, K. F. Mak, and X. Liu, Superconductivity and Ferroelectric Orbital Magnetism in Semimetallic Rhom- bohedral Hexalayer Graphene, arXiv:2508.15909 (2025)

    work page arXiv 2025

  31. [32]

    Kr¨ otzsch, K

    K. Kr¨ otzsch, K. Watanabe, T. Taniguchi, A. Ma- grez, M. Banerjee, Magnetoelectric Switching of Com- peting Magnetic Orders in Rhombohedral Graphene, arXiv:2509.24672 (2025)

    work page arXiv 2025

  32. [33]

    R. Q. Nguyen, H.-T. Wu, E. Morissette, N. J. Zhang, P. Qin, K. Watanabe, T. Taniguchi, A. W. Hui, D. E. Feld- man and J. I. A. Li, A Hierarchy of Superconductivity and Topological Charge Density Wave States in Rhom- bohedral Graphene, arXiv:2507.22026 (2025)

    work page arXiv 2025

  33. [34]

    Y.-Z. Chou, F. Wu, J. D. Sau, and S. Das Sarma, Acoustic-phonon-mediated superconductivity in rhombo- hedral trilayer graphene, Phys. Rev. Lett. 127, 187001 (2021)

    work page 2021

  34. [35]

    Chatterjee, T

    S. Chatterjee, T. Wang, E. Berg, and M. P. Zaletel, Inter- valley coherent order and isospin fluctuation mediated superconductivity in rhombohedral trilayer graphene, Nat. Commun. 13, 6013 (2022)

    work page 2022

  35. [36]

    Ghazaryan, T

    A. Ghazaryan, T. Holder, M. Serbyn, and E. Berg, Un- conventional superconductivity in systems with annu- lar fermi surfaces: Application to rhombohedral trilayer graphene, Phys. Rev. Lett. 127, 247001 (2021)

    work page 2021

  36. [37]

    Vituri, J

    Y. Vituri, J. Xiao1, K. Pareek, T. Holder, and E. Berg, Incommensurate intervalley coherent states in ABC graphene: Collective modes and superconductivity, Phys. Rev. B 111, 075103 (2025)

    work page 2025

  37. [38]

    Zhumagulov, D

    Y. Zhumagulov, D. Kochan, and J. Fabian, Emergent Correlated Phases in Rhombohedral Trilayer Graphene Induced by Proximity Spin-Orbit and Exchange Cou- pling, Phys. Rev. Lett. 132, 186401 (2024)

    work page 2024

  38. [39]

    W. Qin, C. Huang, T. Wolf, N. Wei, I. Blinov, and A. H. MacDonald, Functional Renormalization Group Study of Superconductivity in Rhombohedral Trilayer Graphene, Phys. Rev. Lett. 130, 146001 (2023)

    work page 2023

  39. [40]

    D.-C. Lu, T. Wang, S. Chatterjee, and Y. -Z. You, Correlated metals and unconventional superconductivity in rhombohedral trilayer graphene: A renormalization group analysis, Phys. Rev. B 106, 155115 (2022)

    work page 2022

  40. [41]

    You and Ashvin Vishwanath, Kohn-Luttinger super- conductivity and intervalley coherence in rhombohedral trilayer graphene, Phys

    Y. You and Ashvin Vishwanath, Kohn-Luttinger super- conductivity and intervalley coherence in rhombohedral trilayer graphene, Phys. Rev. B 105, 134524 (2022)

    work page 2022

  41. [42]

    A. L. Szab´ o and B. Roy, Metals, fractional metals, and superconductivity in rhombohedral trilayer graphene, Phys. Rev. B 105, L081407 (2022)

    work page 2022

  42. [43]

    T. Cea, P. A. Pantale´ on, V. T. Phong, and F. Guinea, Superconductivity from repulsive interactions in rhombo- hedral trilayer graphene: A Kohn-Luttinger-like mecha- nism, Phys. Rev. B 105, 075432 (2022)

    work page 2022

  43. [44]

    Huang, T

    C. Huang, T. M. R. Wolf, W. Qin, N. Wei, I. V. Blinov, and A. H. MacDonald, Spin and orbital metallic mag- netism in rhombohedral trilayer graphene, Phys. Rev. B 107, L121405 (2023)

    work page 2023

  44. [45]

    Juriˇ ci´ c, E

    V. Juriˇ ci´ c, E. Mu˜ noz, and R. Soto-Garrido, Optical con- ductivity as a probe of the interaction-driven metal in rhombohedral trilayer graphene, Nanomaterials 12, 3727 (2022)

    work page 2022

  45. [46]

    A. S. Patri and T. Senthil, Strong correlations in ABC- stacked trilayer graphene: Moir´ e is important, Phys. Rev. B 107, 165122 (2023)

    work page 2023

  46. [47]

    H. Dai, R. Ma, X. Zhang, T. Guo, and T. Ma, Quantum Monte Carlo study of superconductivity in rhombohedral trilayer graphene under an electric field, Phys. Rev. B 107, 245106 (2023)

    work page 2023

  47. [48]

    C. W. Chau, S. A. Chen, and K. T. Law, Origin of Superconductivity in Rhombohedral Trilayer Graphene: Quasiparticle Pairing within the Inter-Valley Coherent Phase, arXiv:2404.19237 (2024)

    work page internal anchor Pith review Pith/arXiv arXiv 2024

  48. [49]

    Zhang and J

    Q.C. Zhang and J. Wang, Fermion-fermion interac- tion driven phase transitions in rhombohedral trilayer graphene, Phys. Rev. B 111, 014111 (2025)

    work page 2025

  49. [50]

    Jimeno-Pozo, H

    A. Jimeno-Pozo, H. Sainz-Cruz, T. Cea, P. A. Pantale´ on, and F. Guinea, Superconductivity from electronic inter- actions and spin-orbit enhancement in bilayer and tri- layer graphene, Phys. Rev. B 107, L161106 (2023)

    work page 2023

  50. [51]

    Y.-Z. Chou, F. Wu, and S. Das Sarma, Enhanced super- conductivity through virtual tunneling in Bernal bilayer graphene coupled to WSe 2, Phys. Rev. B 106, L180502 (2022)

    work page 2022

  51. [52]

    A. L. Szab´ o and B. Roy, Competing orders and cascade 7 of degeneracy lifting in doped Bernal bilayer graphene, Phys. Rev. B 105, L201107 (2022)

    work page 2022

  52. [53]

    Wagner, Y

    G. Wagner, Y. H. Kwan, N. Bultinck, S. H. Simon, and S. A. Parameswaran, Superconductivity from repulsive in- teractions in Bernal-stacked bilayer graphene, Phys. Rev. B 110, 214517 (2024)

    work page 2024

  53. [54]

    J. B. Curtis, N. R. Poniatowski, Y. Xie, A. Yacoby, E. Demler, and P. Narang, Stabilizing fluctuating spin- triplet superconductivity in graphene via induced spin- orbit coupling, Phys. Rev. Lett. 130, 196001 (2023)

    work page 2023

  54. [55]

    Y.-Z. Chou, F. Wu, J. D. Sau, and S. Das Sarma, Acoustic-phonon-mediated superconductivity in Bernal bilayer graphene, Phys. Rev. B 105, L100503 (2022)

    work page 2022

  55. [56]

    Zhumagulov, D

    Y. Zhumagulov, D. Kochan, and J. Fabian, Swapping exchange and spin-orbit induced correlated phases in proximitized Bernal bilayer graphene, Phys. Rev. B 110, 045427 (2024)

    work page 2024

  56. [57]

    Fischer, L

    A. Fischer, L. Klebl, D. M. Kennes, and T. O. Wehling, Supercell Wannier functions and a faithful low-energy model for Bernal bilayer graphene, Phys. Rev. B 110, L201113 (2024)

    work page 2024

  57. [58]

    Y. Jang, Y. Park, J. Jung, and H. Min, Chirality and correlations in the spontaneous spin-valley polarization of rhombohedral multilayer graphene, Phys. Rev. B 108, L041101 (2023)

    work page 2023

  58. [59]

    Cea, Superconductivity induced by the intervalley Coulomb scattering in a few layers of graphene, Phys

    T. Cea, Superconductivity induced by the intervalley Coulomb scattering in a few layers of graphene, Phys. Rev. B 107, L041111 (2023)

    work page 2023

  59. [60]

    Cr´ epieux, E

    A. Cr´ epieux, E. Pangburn, L. Haurie, O. A. Awoga, A. M. Black-Schaffer, N. Sedlmayr, C. P´ epin, and C. Bena, Superconductivity in monolayer and few-layer graphene. II. Topological edge states and Chern numbers, Phys. Rev. B 108, 134515 (2023)

    work page 2023

  60. [61]

    E. V. Bostr¨ om, A. Fischer, J. B. Profe, J. Zhang, D. M. Kennes, and A. Rubio, Phonon-mediated unconventional s- and f-wave pairing superconductivity in rhombohedral stacked multilayer graphene, arXiv:2311.02494 (2023)

    work page arXiv 2023

  61. [62]

    Herasymchuk, S

    A. Herasymchuk, S. G. Sharapov, O. V. Yazyev, Y. Zhumagulov, Correlated phases in rhombohedral N-layer graphene, arXiv:2508.14630 (2025)

    work page arXiv 2025

  62. [63]

    Y. Chen, M. S. Scheurer, C. Schrade, Intrinsic supercon- ducting diode effect and nonreciprocal superconductivity in rhombohedral graphene multilayers, Phys. Rev. B112, L060505 (2025)

    work page 2025

  63. [64]

    S. A. Murshed and B. Roy, Charge-density waves and stripes in quarter metals of graphene heterostructures, arXiv:2510.20816 (2025)

    work page arXiv 2025

  64. [65]

    A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, The electronic properties of graphene, Rev. Mod. Phys. 81, 109 (2009)

    work page 2009

  65. [66]

    G. W. Semenoff, Condensed-Matter simulation of a three-dimensional anomaly, Phys. Rev. Lett. 53, 2449 (1984)

    work page 1984

  66. [67]

    Fierz, Zur Fermischen Theorie des β-Zerfalls, Z

    M. Fierz, Zur Fermischen Theorie des β-Zerfalls, Z. Physik 104, 553 (1937)

    work page 1937

  67. [68]

    B. Roy, P. Goswami, and V. Juriˇ ci´ c, Interacting Weyl fermions: Phases, phase transitions, and global phase diagram, Phys. Rev. B 95, 201102 (2017)

    work page 2017

  68. [69]

    Roy and M

    B. Roy and M. S. Foster, Quantum Multicriticality near the Dirac-Semimetal to Band-Insulator Critical Point in Two Dimensions: A Controlled Ascent from One Dimen- sion, Phys. Rev. X 8, 011049 (2018)

    work page 2018

  69. [70]

    Zinn-Justin, Quantum Field Theory and Critical Phe- nomena (Oxford Science, Oxford, 2002)

    J. Zinn-Justin, Quantum Field Theory and Critical Phe- nomena (Oxford Science, Oxford, 2002)

    work page 2002

  70. [71]

    A. L. Szab´ o and B. Roy, Extended Hubbard model in undoped and doped monolayer and bilayer graphene: Se- lection rules and organizing principle among competing orders, Phys. Rev. B 103, 205135 (2021)

    work page 2021

  71. [72]

    S. A. Murshed, S. K. Das, and B. Roy, Superconductiv- ity in doped planar Dirac insulators: A renormalization group study, Phys. Rev. B 111, 245153 (2025)

    work page 2025

  72. [73]

    J. M. Kosterlitz and D. J. Thouless, Ordering, metasta- bility and phase transitions in two-dimensional systems, J. Phys. C: Solid State Phys. 6, 1181 (1973)

    work page 1973

  73. [74]

    Uehlinger, G

    T. Uehlinger, G. Jotzu, M. Messer, D. Greif, W. Hofstet- ter, U. Bissbort, and T. Esslinger, Artificial graphene with tunable interactions, Phys. Rev. Lett. 111, 185307 (2013)

    work page 2013

  74. [75]

    Messer, R

    M. Messer, R. Desbuquois, T. Uehlinger, G. Jotzu, S. Huber, D. Greif, and T. Esslinger, Exploring Competing Density Order in the Ionic Hubbard Model with Ultra- cold Fermions, Phys. Rev. Lett. 115, 115303 (2015)

    work page 2015

  75. [76]

    Parity Anomaly

    F. D. M. Haldane, Model for a Quantum Hall Effect with- out Landau Levels: Condensed-Matter Realization of the “Parity Anomaly”, Phys. Rev. Lett. 61, 2015 (1988)

    work page 2015

  76. [77]

    Jotzu, M

    G. Jotzu, M. Messer, R. Desbuquois, M. Lebrat, T. Uehlinger, D. Greif, and T. Esslinger, Experimental real- ization of the topological Haldane model with ultracold fermions, Nature (London) 515, 237 (2014)

    work page 2014