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Superconductivity survives deep inside the quantum Hall regime of rhombohedral hexalayer graphene, coexisting with multi-Landau-level charge density wave order.

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2026-07-11 09:42 UTC pith:26KBROTM

load-bearing objection First clear transport evidence of field-stabilized superconductivity coexisting with multi-LL CDW deep in the quantum-Hall regime of R6G; data density is high and the central interpretation holds. the 2 major comments →

arxiv 2607.05039 v1 pith:26KBROTM submitted 2026-07-06 cond-mat.mes-hall

Coexisting Charge Density Wave and Superconducting Order in Quantizing Magnetic Fields

Ron Q. Nguyen , Peiyu Qin , Hai-Tian Wu , Sparsh Mishra , Tobias Wolf , Joseph Roll , Erin Morissette , Naiyuan J. Zhang
show 7 more authors
Sarah Alkidim Kenji Watanabe Takashi Taniguchi Aaron W. Hui Dima E. Feldman Allan MacDonald J.I.A. Li
This is my paper
classification cond-mat.mes-hall PACS 73.43.-f74.25.F-73.21.-b71.45.Lr
keywords rhombohedral hexalayer graphenecharge density wavequantum Hall superconductivityLandau level mixingre-entrant integer quantum Hallflat band edgeBKT transition
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper reports that near the flat-band edge of rhombohedral hexalayer graphene, a large displacement field and perpendicular magnetic field create a manifold of nearly degenerate Landau levels. Strong Coulomb mixing of those levels produces charge-density-wave order that reorganizes many levels at once, yielding re-entrant integer quantum Hall states whose Hall conductance numbers lie far below the nearby filling factors. The same mixed-Landau-level manifold also hosts a superconducting phase that is stabilized, rather than destroyed, by the perpendicular field and remains zero-resistance deep inside the quantum Hall regime. Both orders melt with first-order thermal hysteresis on comparable temperature scales of roughly half a kelvin. The result shows that superconductivity and charge order can emerge together from strongly mixed Landau levels, offering a concrete experimental window into their interplay even at zero field.

Core claim

In rhombohedral hexalayer graphene under large displacement field, Landau quantization of the flat-band edge produces a dense set of nearly degenerate levels. Charge-density-wave order that mixes many of these levels generates re-entrant integer quantum Hall plateaus whose Hall conductance quantum numbers deviate strongly from nearby integer fillings, while a superconducting phase is stabilized by the same perpendicular field and coexists with the charge order deep inside the quantum Hall regime.

What carries the argument

Multi-Landau-level charge-density-wave reconstruction: Coulomb mixing of a manifold of nearly degenerate levels redistributes Chern numbers so that some occupied levels become localized (Chern zero) while others remain topologically nontrivial, producing Hall quantization well below the total filling and a continuously tunable density of localized electrons n_CDW.

Load-bearing premise

That simultaneous vanishing of both longitudinal and Hall resistance inside the experimental noise floor of a few ohms, under a large perpendicular field, can be produced only by bulk superconductivity that shorts the Hall voltage.

What would settle it

A measurement that shows finite bulk Hall conductivity or non-superconducting dissipationless transport (for example from edge modes or a different incompressible state) inside the claimed zero-resistance pockets while both resistances remain at the noise floor.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Superconductivity can be compatible with Landau quantization when low-energy collective modes of a multi-level charge-density wave supply retarded pairing interactions that overcome broken time-reversal symmetry.
  • Re-entrant Hall plateaus whose Chern numbers lie far from the filling factor become a spectroscopic signature of multi-level charge-density-wave reconstruction rather than conventional single-level bubble or Wigner phases.
  • The same flat-band-edge manifold that hosts these phases at finite field is the parent state of the zero-field superconducting and resistive phases of rhombohedral hexalayer graphene.
  • Continuous tunability of the charge-density-wave period with density and field implies interaction-driven, not lattice-commensurate, order that can self-dope reconstructed sub-bands.

Where Pith is reading between the lines

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

  • If the charge-density wave supplies the pairing glue, deliberately engineering similar multi-level mixing in other flat-band graphene systems or moiré materials could stabilize field-resilient superconductivity.
  • The first-order melting of the charge order that bounds the superconducting dome suggests that thermal or current-driven destruction of the crystal should abruptly suppress pairing, a prediction that can be tested by simultaneous noise or compressibility measurements.
  • The inverted Landau-level hierarchy near the flat-band edge may be a general design rule for materials in which charge order and superconductivity compete on equal footing.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. The manuscript reports magneto-transport experiments on rhombohedral hexalayer graphene near a displacement-field-tuned flat-band edge. In a dense manifold of nearly degenerate Landau levels, the authors identify charge-density-wave order through re-entrant integer quantum Hall plateaus whose Hall conductance quantum numbers lie well below the nearby filling factors, continuous evolution of an extracted CDW density n_CDW, first-order thermal hysteresis, and current-driven breakdown with negative differential resistance. Adjacent to these states, and for 1 < n/n_CDW < 2, they observe a zero-resistance phase that is stabilized by perpendicular field, persists deep into the quantum-Hall regime, exhibits simultaneous vanishing of R_∥ and R_⊥ within the noise floor, a BKT-like power-law evolution of I–V characteristics, and thermal hysteresis that tracks the CDW melting line. The two orders are argued to emerge from the same strongly mixed Landau-level manifold, offering insight into zero-field CDW–superconductivity interplay in R6G.

Significance. If the multi-LL CDW interpretation of the Hall-index mismatch and the identification of field-stabilized bulk superconductivity hold, the work supplies the first clear experimental realization of superconductivity coexisting with quantum-Hall physics in a Landau-quantized 2D system. The continuous tunability of the CDW period, the Středa-slope deviations, and the hierarchical relation between CDW, RIQH and SC phases constitute a new regime of interaction-driven reconstruction that is of broad interest to both quantum-Hall and unconventional-superconductivity communities. The data set is extensive (n–D–B maps, conductivity tensors, I–V, thermal hysteresis, BKT analysis) and the continuum LL spectra are calculated from established SWMc parameters without free-parameter fitting into the central claims.

major comments (2)
  1. The assignment of the zero-resistance pocket (Figs. 3b,d and M8) to bulk superconductivity rests on simultaneous vanishing of R_∥ and R_⊥ within a ~3 Ω noise floor, together with BKT power-law I–V and critical-current peaks. While these signatures are collectively persuasive, the manuscript should more explicitly address and exclude alternative dissipationless or edge-dominated states that could short the Hall voltage in a large perpendicular field (e.g., chiral edge modes of a reconstructed topological CDW or a field-induced quantum anomalous Hall crystal). A short discussion of contact geometry, current-path checks, or noise-floor limits would strengthen the claim without altering the data.
  2. The extraction n_CDW = n − σ_xy B / ((e²/h) Φ_0) (text around Fig. 2e and Methods) assumes that the reduction in Hall conductance is caused solely by localization of a density n_CDW into C = 0 bands. Given the strong multi-LL mixing emphasized throughout, residual Chern-number redistribution among itinerant bands could systematically shift the extracted n_CDW. The paper should quantify the robustness of this formula under the level-mixing scenarios of Fig. M2b, or show that the extrapolated n_CDW → n limit remains consistent across independent Chern sequences.
minor comments (4)
  1. Fig. 1c and M1: the inverted LL hierarchy is central; a brief quantitative estimate of n* (~100/B[T]) already appears in Methods but would help the main-text reader if restated near Fig. 1.
  2. Notation for Hall conductance quantum number versus filling factor is occasionally loose (e.g., “effective LL filling of approximately ν ~ 8”); consistent use of ν and C would improve clarity.
  3. Several supplementary figures (S3, S5, M7) are essential for the BKT and onset-temperature claims; cross-references in the main text could be more explicit.
  4. Typographical: “bag Taniguchi” in the author list of Ref. [53] and a few missing spaces around units (e.g., “2T”) should be cleaned.

Circularity Check

0 steps flagged

No significant circularity: experimental transport signatures and standard continuum LL spectra; nCDW is extracted from measured Hall data via a definitional relation that is not fed back as a prediction.

full rationale

The paper is almost entirely experimental. The central claims (RIQH states whose Hall quantum numbers deviate strongly from filling, field-stabilized zero-resistance phase coexisting with CDW) rest on measured σxy/σxx plateaus, Streda-slope deviations, first-order thermal hysteresis, current-driven breakdown with negative dV/dI, BKT power-law I–V evolution, and simultaneous R∥ = R⊥ = 0 within the noise floor. nCDW is defined from the measured Hall conductivity by the standard relation nCDW = n - σ xy B / ((e2/h)Φ0); this is a diagnostic extraction, not a fitted parameter that is later re-used as a prediction. Continuum LL spectra (Fig. M1) are computed from established SWMc parameters and serve only as qualitative backdrop for strong multi-LL mixing; they are not fitted to the transport data and do not enter any quantitative claim. No uniqueness theorem, self-citation chain, or ansatz is load-bearing for the reported phases. The derivation chain therefore contains no circular step.

Axiom & Free-Parameter Ledger

0 free parameters · 4 axioms · 0 invented entities

Experimental transport study. The load-bearing content is measured resistances, conductivities, and their temperature/current/field dependence. Theoretical scaffolding (continuum LL spectra, Chern-number redistribution under CDW) is used only for interpretation and rests on standard domain assumptions; no new free parameters are fitted to produce the central claims.

axioms (4)
  • domain assumption Vanishing of both R∥ and R⊥ (within ~3 Ω noise) in finite B⊥ implies bulk superconductivity that shorts the Hall voltage.
    Stated in main text and Methods (Fig. M8); alternative dissipationless states are not exhaustively excluded.
  • domain assumption Hall conductivity reduced below the filling factor implies a multi-LL CDW that converts some occupied LLs into C = 0 localized bands while conserving total Chern number.
    Core interpretive step (Fig. M2 and surrounding text); rests on standard topological band theory plus the observed Hall-index mismatch.
  • domain assumption Continuum SWMc model (with or without trigonal warping) correctly captures the inverted LL hierarchy near the flat-band edge.
    Used in Methods and Fig. M1 to motivate strong LL mixing; parameters taken from prior DFT literature.
  • domain assumption Thermal hysteresis and negative-dV/dI peaks are diagnostic of first-order melting of a crystalline CDW (Wigner-solid-like) state.
    Standard interpretation drawn from semiconductor 2DEG literature and applied throughout regimes I–III.

pith-pipeline@v1.1.0-grok45 · 39527 in / 2466 out tokens · 25610 ms · 2026-07-11T09:42:50.487831+00:00 · methodology

0 comments
read the original abstract

Charge density wave (CDW) and superconductivity are both common in strongly interacting electron systems. While CDW order is ubiquitous in both quantum Hall systems and unconventional superconductors, superconductivity is generally suppressed by the strong magnetic fields required for Landau quantization. Here we investigate the intertwined CDW and superconducting phases of rhombohedral hexalayer graphene (R6G) in a large displacement field, which generates tunable flat band edges, and a strong magnetic field, which generates a manifold of nearly degenerate Landau levels. CDW order is accompanied by pronounced thermal hysteresis as expected for first-order melting transitions. Surprisingly, we find a series of strong integer quantum Hall effects at magnetic fields above ~2T with Hall conductance quantum numbers that deviate strongly from nearby integer filling factors, an observation that can be explained only by CDW order that mixes many Landau levels. We also find a nearby superconducting phase that is stabilized by perpendicular magnetic fields and persists deep within the quantum Hall regime. The CDW and superconducting phases develop on comparable temperature scales and emerge from the same manifold of strongly mixed Landau levels. These observations provide new insight into the interplay between superconductivity and CDW order in R6G at zero magnetic field.

Figures

Figures reproduced from arXiv: 2607.05039 by Aaron W. Hui, Allan MacDonald, Dima E. Feldman, Erin Morissette, Hai-Tian Wu, J.I.A. Li, Joseph Roll, Kenji Watanabe, Naiyuan J. Zhang, Peiyu Qin, Ron Q. Nguyen, Sarah Alkidim, Sparsh Mishra, Takashi Taniguchi, Tobias Wolf.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗

discussion (0)

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Reference graph

Works this paper leans on

106 extracted references · 3 linked inside Pith

  1. [1]

    On the interaction of electrons in metals,

    E. Wigner, “On the interaction of electrons in metals,” Phys. Rev.46, 1002–1011 (1934)

    1934

  2. [2]

    The dynamics of charge-density waves,

    G. Gr¨ uner, “The dynamics of charge-density waves,” Rev. Mod. Phys.60, 1129–1181 (1988)

    1988

  3. [3]

    Rudolf Ernst Peierls,Quantum theory of solids(Oxford university press, 1955)

    1955

  4. [4]

    25 (Elsevier, 2012)

    Lev Petrovich Gor’kov and George Gr¨ uner,Charge den- sity waves in solids, Vol. 25 (Elsevier, 2012)

    2012

  5. [5]

    Strong-coupling theory of charge-density-wave transitions,

    C. M. Varma and A. L. Simons, “Strong-coupling theory of charge-density-wave transitions,” Phys. Rev. Lett.51, 138–141 (1983)

    1983

  6. [6]

    Pseu- dogap and charge density waves in two dimensions,

    S. V. Borisenko, A. A. Kordyuk, A. N. Yaresko, V. B. Zabolotnyy, D. S. Inosov, R. Schuster, B. B¨ uchner, R. Weber, R. Follath, L. Patthey, and H. Berger, “Pseu- dogap and charge density waves in two dimensions,” Phys. Rev. Lett.100, 196402 (2008). 8

    2008

  7. [7]

    Classification of charge den- sity waves based on their nature,

    Xuetao Zhu, Yanwei Cao, Jiandi Zhang, EW Plum- mer, and Jiandong Guo, “Classification of charge den- sity waves based on their nature,” Proceedings of the National Academy of Sciences112, 2367–2371 (2015)

    2015

  8. [8]

    Crystallization of a two- dimensional electron gas in a magnetic field,

    Yu E Lozovik and VI Yudson, “Crystallization of a two- dimensional electron gas in a magnetic field,” ZhETF Pisma Redaktsiiu22, 26 (1975)

    1975

  9. [9]

    Observation of a magnetically induced wigner solid,

    E. Y. Andrei, G. Deville, D. C. Glattli, F. I. B. Williams, E. Paris, and B. Etienne, “Observation of a magnetically induced wigner solid,” Phys. Rev. Lett.60, 2765–2768 (1988)

    1988

  10. [10]

    Conduction threshold and pinning frequency of magnetically induced wigner solid,

    F. I. B. Williams, P. A. Wright, R. G. Clark, E. Y. Andrei, G. Deville, D. C. Glattli, O. Probst, B. Etienne, C. Dorin, C. T. Foxon, and J. J. Harris, “Conduction threshold and pinning frequency of magnetically induced wigner solid,” Phys. Rev. Lett.66, 3285–3288 (1991)

    1991

  11. [11]

    Low-frequency noise in the reentrant insulating phase around the 1/5 fractional quantum hall liquid,

    Yuan P. Li, T. Sajoto, L. W. Engel, D. C. Tsui, and M. Shayegan, “Low-frequency noise in the reentrant insulating phase around the 1/5 fractional quantum hall liquid,” Phys. Rev. Lett.67, 1630–1633 (1991)

    1991

  12. [12]

    Wigner crystallization and metal- insulator transition of two-dimensional holes in gaas at B= 0,

    Jongsoo Yoon, C. C. Li, D. Shahar, D. C. Tsui, and M. Shayegan, “Wigner crystallization and metal- insulator transition of two-dimensional holes in gaas at B= 0,” Phys. Rev. Lett.82, 1744–1747 (1999)

    1999

  13. [13]

    Microwave resonance of the 2d wigner crystal around integer landau fillings,

    Yong Chen, R. M. Lewis, L. W. Engel, D. C. Tsui, P. D. Ye, L. N. Pfeiffer, and K. W. West, “Microwave resonance of the 2d wigner crystal around integer landau fillings,” Phys. Rev. Lett.91, 016801 (2003)

    2003

  14. [14]

    Sharp tun- nelling resonance from the vibrations of an electronic wigner crystal,

    Joonho Jang, Benjamin M Hunt, Loren N Pfeiffer, Ken- neth W West, and Raymond C Ashoori, “Sharp tun- nelling resonance from the vibrations of an electronic wigner crystal,” Nat. Phys.13, 340–344 (2017)

    2017

  15. [15]

    Direct observation of a magnetic-field-induced wigner crystal,

    Yen-Chen Tsui, Minhao He, Yuwen Hu, Ethan Lake, Taige Wang, Kenji Watanabe, Takashi Taniguchi, Michael P Zaletel, and Ali Yazdani, “Direct observation of a magnetic-field-induced wigner crystal,” Nature628, 287–292 (2024)

    2024

  16. [16]

    Astability and negative differential resistance of the wigner solid,

    G. A. Cs´ athy, D. C. Tsui, L. N. Pfeiffer, and K. W. West, “Astability and negative differential resistance of the wigner solid,” Phys. Rev. Lett.98, 066805 (2007)

    2007

  17. [17]

    Evidence for two-dimentional quantum wigner crystal,

    V. J. Goldman, M Santos, M Shayegan, and J. E. Cunningham, “Evidence for two-dimentional quantum wigner crystal,” Phys. Rev. Lett.65, 2189–2192 (1990)

    1990

  18. [18]

    Insulating and fractional quantum Hall states in the first excited Landau level,

    J. P. Eisenstein, K. B. Cooper, L. N. Pfeiffer, and K. W. West, “Insulating and fractional quantum Hall states in the first excited Landau level,” Phys. Rev. Lett.88, 076801 (2002)

    2002

  19. [19]

    Stripe and bubble phases in quantum Hall systems,

    Michael M Fogler, “Stripe and bubble phases in quantum Hall systems,” inHigh Magnetic Fields: Applications in Condensed Matter Physics and Spectroscopy(Springer,

  20. [20]

    Mi- croscopic theory of the reentrant integer quantum Hall effect in the first and second excited Landau levels,

    M. O. Goerbig, P. Lederer, and C. Morais Smith, “Mi- croscopic theory of the reentrant integer quantum Hall effect in the first and second excited Landau levels,” Phys. Rev. B68, 241302 (2003)

    2003

  21. [21]

    Competition between quantum-liquid and electron-solid phases in intermediate Landau levels,

    M. O. Goerbig, P. Lederer, and C. Morais Smith, “Competition between quantum-liquid and electron-solid phases in intermediate Landau levels,” Phys. Rev. B69, 115327 (2004)

    2004

  22. [22]

    Electron correlation in the second Landau level: A competition between many nearly degenerate quantum phases,

    J. S. Xia, W. Pan, C. L. Vicente, E. D. Adams, N. S. Sullivan, H. L. Stormer, D. C. Tsui, L. N. Pfeiffer, K. W. Baldwin, and K. W. West, “Electron correlation in the second Landau level: A competition between many nearly degenerate quantum phases,” Phys. Rev. Lett. 93, 176809 (2004)

    2004

  23. [23]

    Collective nature of the reen- trant integer quantum Hall states in the second Landau level,

    N. Deng, A. Kumar, M. J. Manfra, L. N. Pfeiffer, K. W. West, and G. A. Cs´ athy, “Collective nature of the reen- trant integer quantum Hall states in the second Landau level,” Phys. Rev. Lett.108, 086803 (2012)

    2012

  24. [24]

    Competition between a fractional quantum Hall liquid and bubble and wigner crystal phases in the third Landau level,

    G. Gervais, L. W. Engel, H. L. Stormer, D. C. Tsui, K. W. Baldwin, K. W. West, and L. N. Pfeiffer, “Competition between a fractional quantum Hall liquid and bubble and wigner crystal phases in the third Landau level,” Phys. Rev. Lett.93, 266804 (2004)

    2004

  25. [25]

    Observation of reen- trant integer quantum Hall states in the lowest Landau level,

    Yang Liu, C. G. Pappas, M. Shayegan, L. N. Pfeiffer, K. W. West, and K. W. Baldwin, “Observation of reen- trant integer quantum Hall states in the lowest Landau level,” Phys. Rev. Lett.109, 036801 (2012)

    2012

  26. [26]

    Bertrand I Halperin and Jainendra K Jain,Fractional quantum Hall effects: New developments(World Scien- tific, 2020)

    2020

  27. [27]

    Competing fractional quantum Hall and electron solid phases in graphene,

    Shaowen Chen, Rebeca Ribeiro-Palau, Kang Yang, Kenji Watanabe, Takashi Taniguchi, James Hone, Mark O. Goerbig, and Cory R. Dean, “Competing fractional quantum Hall and electron solid phases in graphene,” Phys. Rev. Lett.122, 026802 (2019)

    2019

  28. [28]

    Current flow in the bubble and stripe phases,

    B. Friess, V. Umansky, K. von Klitzing, and J. H. Smet, “Current flow in the bubble and stripe phases,” Phys. Rev. Lett.120, 137603 (2018)

    2018

  29. [29]

    Ground state of a two-dimensional electron liquid in a weak magnetic field,

    M. M. Fogler, A. A. Koulakov, and B. I. Shklovskii, “Ground state of a two-dimensional electron liquid in a weak magnetic field,” Phys. Rev. B54, 1853–1871 (1996)

    1996

  30. [30]

    Charge density wave in two-dimensional electron liquid in weak magnetic field,

    A. A. Koulakov, M. M. Fogler, and B. I. Shklovskii, “Charge density wave in two-dimensional electron liquid in weak magnetic field,” Phys. Rev. Lett.76, 499–502 (1996)

    1996

  31. [31]

    Evidence for an anisotropic state of two-dimensional electrons in high landau levels,

    M. P. Lilly, K. B. Cooper, J. P. Eisenstein, L. N. Pfeiffer, and K. W. West, “Evidence for an anisotropic state of two-dimensional electrons in high landau levels,” Phys. Rev. Lett.82, 394–397 (1999)

    1999

  32. [32]

    Anisotropic states of two-dimensional electron systems in high Landau levels: Effect of an in-plane magnetic field,

    M. P. Lilly, K. B. Cooper, J. P. Eisenstein, L. N. Pfeiffer, and K. W. West, “Anisotropic states of two-dimensional electron systems in high Landau levels: Effect of an in-plane magnetic field,” Phys. Rev. Lett.83, 824–827 (1999)

    1999

  33. [33]

    Strongly anisotropic trans- port in higher two-dimensional Landau levels,

    R.R. Du, D.C. Tsui, H.L. Stormer, L.N. Pfeiffer, K.W. Baldwin, and K.W. West, “Strongly anisotropic trans- port in higher two-dimensional Landau levels,”109, 389–394 (1999)

    1999

  34. [34]

    Strongly anisotropic electronic transport at Landau level filling factor ν = 9 /2 and ν = 5 /2 under a tilted magnetic field,

    W. Pan, R. R. Du, H. L. Stormer, D. C. Tsui, L. N. Pfeiffer, K. W. Baldwin, and K. W. West, “Strongly anisotropic electronic transport at Landau level filling factor ν = 9 /2 and ν = 5 /2 under a tilted magnetic field,” Phys. Rev. Lett.83, 820–823 (1999)

    1999

  35. [35]

    Liquid-crystal phases of quantum hall systems,

    Eduardo Fradkin and Steven A. Kivelson, “Liquid-crystal phases of quantum hall systems,” Phys. Rev. B59, 8065– 8072 (1999)

    1999

  36. [36]

    Quantum theory of quantum hall smectics,

    A. H. MacDonald and Matthew P. A. Fisher, “Quantum theory of quantum hall smectics,” Phys. Rev. B61, 5724–5733 (2000)

    2000

  37. [37]

    Resis- tivity anisotropy of quantum hall stripe phases,

    M. Sammon, X. Fu, Yi Huang, M. A. Zudov, B. I. Shklovskii, G. C. Gardner, J. D. Watson, M. J. Manfra, K. W. Baldwin, L. N. Pfeiffer, and K. W. West, “Resis- tivity anisotropy of quantum hall stripe phases,” Phys. Rev. B100, 241303(R) (2019)

    2019

  38. [38]

    Nematic 9 fermi fluids in condensed matter physics,

    Eduardo Fradkin, Steven A Kivelson, Michael J Lawler, James P Eisenstein, and Andrew P Mackenzie, “Nematic 9 fermi fluids in condensed matter physics,” The Annual Review of Condensed Matter Physics is1, 153–78 (2010)

    2010

  39. [39]

    Theory of the striped superconductor,

    Erez Berg, Eduardo Fradkin, and Steven A. Kivelson, “Theory of the striped superconductor,” Phys. Rev. B 79, 064515 (2009)

    2009

  40. [40]

    Charge-4e superconductivity from pair-density-wave or- der in certain high-temperature superconductors,

    Erez Berg, Eduardo Fradkin, and Steven A. Kivelson, “Charge-4e superconductivity from pair-density-wave or- der in certain high-temperature superconductors,” Nat. Phys.5, 830–833 (2009)

    2009

  41. [41]

    How to detect fluctuating stripes in the high-temperature su- perconductors,

    S. A. Kivelson, I. P. Bindloss, E. Fradkin, V. Oganesyan, J. M. Tranquada, A. Kapitulnik, and C. Howald, “How to detect fluctuating stripes in the high-temperature su- perconductors,” Rev. Mod. Phys.75, 1201–1241 (2003)

    2003

  42. [42]

    Lattice symmetry breaking in cuprate superconductors: stripes, nematics, and superconductiv- ity,

    Matthias Vojta, “Lattice symmetry breaking in cuprate superconductors: stripes, nematics, and superconductiv- ity,” Advances in Physics58, 699–820 (2009)

    2009

  43. [43]

    High-temperature su- perconductivity,

    Xingjiang Zhou, Wei-Sheng Lee, Masatoshi Imada, Nan- dini Trivedi, Philip Phillips, Hae-Young Kee, P¨ aivi T¨ orm¨ a, and Mikhail Eremets, “High-temperature su- perconductivity,” Nature Reviews Physics3, 462–465 (2021)

    2021

  44. [44]

    Band structure of abc-stacked graphene trilayers,

    Fan Zhang, Bhagawan Sahu, Hongki Min, and A. H. MacDonald, “Band structure of abc-stacked graphene trilayers,” Phys. Rev. B82, 035409 (2010)

    2010

  45. [45]

    Trigonal warping and berry’s phase nπ in abc-stacked multilayer graphene,

    Mikito Koshino and Edward McCann, “Trigonal warping and berry’s phase nπ in abc-stacked multilayer graphene,” Phys. Rev. B80, 165409 (2009)

    2009

  46. [46]

    Effect of orbital quan- tization on the critical field of type-ii superconductors,

    L. W. Gruenberg and L. Gunther, “Effect of orbital quan- tization on the critical field of type-ii superconductors,” Phys. Rev.176, 606–613 (1968)

    1968

  47. [47]

    Theoretical aspects of superconductivity in very high magnetic fields,

    Mark Rasolt and Zlatko Tes˘anovi´ c, “Theoretical aspects of superconductivity in very high magnetic fields,” Rev. Mod. Phys.64, 709–754 (1992)

    1992

  48. [48]

    Lan- dau quantization and particle-particle ladder sums in a magnetic field,

    A. H. MacDonald, H. Akera, and M. R. Norman, “Lan- dau quantization and particle-particle ladder sums in a magnetic field,” Phys. Rev. B45, 10147(R)–10150(R) (1992)

    1992

  49. [49]

    Quantum mechanics and superconductivity in a magnetic field,

    A. H. MacDonald, Hiroshi Akera, and M. R. Norman, “Quantum mechanics and superconductivity in a magnetic field,” International Journal of Modern Physics B7, 333– 349 (1993)

    1993

  50. [50]

    Strong landau- quantization effects in high-magnetic-field superconduc- tivity of a two-dimensional multiple-band metal near the lifshitz transition,

    Kok Wee Song and Alexei E. Koshelev, “Strong landau- quantization effects in high-magnetic-field superconduc- tivity of a two-dimensional multiple-band metal near the lifshitz transition,” Phys. Rev. B95, 174503 (2017)

    2017

  51. [51]

    Quantum hall superconductivity from moir´ e lan- dau levels,

    Gaurav Chaudhary, A. H. MacDonald, and M. R. Nor- man, “Quantum hall superconductivity from moir´ e lan- dau levels,” Phys. Rev. Res.3, 033260 (2021)

    2021

  52. [52]

    Half-and quarter-metals in rhombohedral trilayer graphene,

    Haoxin Zhou, Tian Xie, Areg Ghazaryan, Tobias Holder, James R Ehrets, Eric M Spanton, Takashi Taniguchi, Kenji Watanabe, Erez Berg, Maksym Serbyn, et al., “Half-and quarter-metals in rhombohedral trilayer graphene,” Nature598, 429–433 (2021)

    2021

  53. [53]

    Superconductivity in rhombohedral trilayer graphene,

    Haoxin Zhou, Tian Xie, Takashi bag Taniguchi, Kenji Watanabe, and Andrea F Young, “Superconductivity in rhombohedral trilayer graphene,” Nature598, 434–438 (2021)

    2021

  54. [54]

    Isospin magnetism and spin-polarized super- conductivity in Bernal bilayer graphene,

    Haoxin Zhou, Ludwig Holleis, Yu Saito, Liam Cohen, William Huynh, Caitlin L Patterson, Fangyuan Yang, Takashi Taniguchi, Kenji Watanabe, and Andrea F Young, “Isospin magnetism and spin-polarized super- conductivity in Bernal bilayer graphene,” Science375, 774–778 (2022)

    2022

  55. [55]

    Extreme anisotropy in the metallic and superconducting phases of rhombohedral hexalayer graphene,

    Peiyu Qin, Hai-Tian Wu, Ron Q. Nguyen, Erin Moris- sette, Naiyuan J. Zhang, K. Watanabe, T. Taniguchi, and Jia Li, “Extreme anisotropy in the metallic and superconducting phases of rhombohedral hexalayer graphene,” arXiv preprint arXiv:2504.05129 (2025)

    Pith/arXiv arXiv 2025

  56. [56]

    Signatures of chiral superconductivity in rhombohedral graphene,

    Tonghang Han, Zhengguang Lu, Zach Hadjri, Lihan Shi, Zhenghan Wu, Wei Xu, Yuxuan Yao, Armel A. Cotten, Omid Sharifi Sedeh, Henok Weldeyesus, Jixiang Yang, Junseok Seo, Shenyong Ye, Muyang Zhou, Haoyang ... Liu,et al., “Signatures of chiral superconductivity in rhombohedral graphene,” Nature643, 654–661 (2025)

    2025

  57. [57]

    Superconductivity and quantized anomalous Hall effect in rhombohedral graphene,

    Youngjoon Choi, Ysun Choi, Marco Valentini, Caitlin L. Patterson, Ludwig F. W. Holleis, Owen I. Sheekey, Hari Stoyanov, Xiang Cheng, Takashi Taniguchi, Kenji Watanabe, Andrea F. Young,et al., “Superconductivity and quantized anomalous Hall effect in rhombohedral graphene,” Nature639, 342–347 (2025)

    2025

  58. [58]

    Ex- tended quantum anomalous hall states in graphene/hbn moir´ e superlattices,

    Zhengguang Lu, Tonghang Han, Yuxuan Yao, Zach Had- jri, Jixiang Yang, Junseok Seo, Lihan Shi, Shenyong Ye, Kenji Watanabe, Takashi Taniguchi, and Long Ju, “Ex- tended quantum anomalous hall states in graphene/hbn moir´ e superlattices,” Nature637, 1090–1095 (2025)

    2025

  59. [59]

    Charge den- sity wave state of two-dimensional electrons in strong magnetic fields,

    Daijiro Yoshioka and Hidetoshi Fukuyama, “Charge den- sity wave state of two-dimensional electrons in strong magnetic fields,” Journal of the Physical Society of Japan 47, 394–402 (1979)

    1979

  60. [60]

    Ground-state energy of a two-dimensional charge-density-wave state in a strong magnetic field,

    D. Yoshioka and P. A. Lee, “Ground-state energy of a two-dimensional charge-density-wave state in a strong magnetic field,” Phys. Rev. B27, 4986–4996 (1983)

    1983

  61. [61]

    Influence of landau-level mixing on the charge-density-wave state of a two-dimensional electron gas in a strong magnetic field,

    A. H. MacDonald, “Influence of landau-level mixing on the charge-density-wave state of a two-dimensional electron gas in a strong magnetic field,” Phys. Rev. B 30, 4392–4398 (1984)

    1984

  62. [62]

    Landau level mix- ing and the fractional quantum hall effect,

    I. Sodemann and A. H. MacDonald, “Landau level mix- ing and the fractional quantum hall effect,” Phys. Rev. B87, 245425 (2013)

    2013

  63. [63]

    Charge-ordered states in twisted mote 2,

    Sparsh Mishra, Tobias M. R. Wolf, and Allan H. Mac- Donald, “Charge-ordered states in twisted mote 2,” (2026), arXiv:2603.05714 [cond-mat.str-el]

    arXiv 2026

  64. [64]

    Berry trashcan with short range attrac- tion: Exact px + ipy superconductivity in rhombohedral graphene,

    Ming-Rui Li, Yves H. Kwan, Hong Yao, and B. An- drei Bernevig, “Berry trashcan with short range attrac- tion: Exact px + ipy superconductivity in rhombohedral graphene,” arXiv preprint arXiv:2509.16312 (2025)

    arXiv 2025

  65. [65]

    “Berry Trash- can

    B. Andrei Bernevig and Yves H. Kwan, ““Berry Trash- can” model of interacting electrons in rhombohedral graphene,” arXiv preprint arXiv:2503.09692 (2025)

    Pith/arXiv arXiv 2025

  66. [66]

    Quantum cascade of correlated phases in trigonally warped bilayer graphene,

    Anna M. Seiler, Fabian R. Geisenhof, Felix Winterer, Kenji Watanabe, Takashi Taniguchi, Tianyi Xu, Fan Zhang, and R. Thomas Weitz, “Quantum cascade of correlated phases in trigonally warped bilayer graphene,” Nature608, 298–302 (2022)

    2022

  67. [67]

    Parent berry curva- ture and the ideal anomalous hall crystal,

    Tixuan Tan and Trithep Devakul, “Parent berry curva- ture and the ideal anomalous hall crystal,” Phys. Rev. X14, 041040 (2024)

    2024

  68. [68]

    Anomalous hall crystals in rhombo- hedral multilayer graphene. i. interaction-driven chern bands and fractional quantum hall states at zero mag- netic field,

    Junkai Dong, Taige Wang, Tianle Wang, Tomohiro Soejima, Michael P. Zaletel, Ashvin Vishwanath, and Daniel E. Parker, “Anomalous hall crystals in rhombo- hedral multilayer graphene. i. interaction-driven chern bands and fractional quantum hall states at zero mag- netic field,” Phys. Rev. Lett.133, 206503 (2024)

    2024

  69. [69]

    Anomalous hall crystals in rhombo- hedral multilayer graphene. ii. general mechanism and a minimal model,

    Tomohiro Soejima, Junkai Dong, Taige Wang, Tianle Wang, Michael P. Zaletel, Ashvin Vishwanath, and Daniel E. Parker, “Anomalous hall crystals in rhombo- hedral multilayer graphene. ii. general mechanism and a minimal model,” Phys. Rev. B110, 205124 (2024). 10

    2024

  70. [70]

    Stabil- ity of anomalous Hall crystals in multilayer rhombohedral graphene,

    Zhihuan Dong, Adarsh S. Patri, and T. Senthil, “Stabil- ity of anomalous Hall crystals in multilayer rhombohedral graphene,” Phys. Rev. B110, 205130 (2024)

    2024

  71. [71]

    Ex- tended quantum anomalous Hall effect in moir´ e struc- tures: Phase transitions and transport,

    Adarsh S. Patri, Zhihuan Dong, and T. Senthil, “Ex- tended quantum anomalous Hall effect in moir´ e struc- tures: Phase transitions and transport,” Phys. Rev. B 110, 245115 (2024)

    2024

  72. [72]

    The- ory of quantum anomalous Hall phases in pentalayer rhombohedral graphene moir´ e structures,

    Zhihuan Dong, Adarsh S. Patri, and T. Senthil, “The- ory of quantum anomalous Hall phases in pentalayer rhombohedral graphene moir´ e structures,” Phys. Rev. Lett.133, 206502 (2024)

    2024

  73. [73]

    George Gruner,Density waves in solids(CRC press, 2018)

    2018

  74. [74]

    Electronic crystals: an experimental overview,

    Pierre Monceau, “Electronic crystals: an experimental overview,” Advances in Physics61, 325–581 (2012)

    2012

  75. [75]

    Magneto-oscillatory effects in two-dimensional systems with spin-orbit interaction,

    P. Stˇ reda and L. Smrˇ cka, “Magneto-oscillatory effects in two-dimensional systems with spin-orbit interaction,” Journal of Physics C: Solid State Physics16, L895–L900 (1983)

    1983

  76. [76]

    Apparent inconsistency between streda formula and hall conductivity in reentrant integer quantum anomalous hall effect in twisted mote2,

    Yi Huang, Seth Musser, Jihang Zhu, Yang-Zhi Chou, and Sankar Das Sarma, “Apparent inconsistency between streda formula and hall conductivity in reentrant integer quantum anomalous hall effect in twisted mote2,” Phys. Rev. B112, 195136 (2025)

    2025

  77. [77]

    Quantum melting a wigner crystal into hall liquids,

    Aidan P. Reddy and Liang Fu, “Quantum melting a wigner crystal into hall liquids,” Phys. Rev. B113, L161403 (2026)

    2026

  78. [78]

    Melting of a 2d quantum electron solid in high magnetic field,

    Yong P. Chen, G. Sambandamurthy, Z. H. Wang, R. M. Lewis, L. W. Engel, D. C. Tsui, P. D. Ye, L. N. Pfeiffer, and K. W. West, “Melting of a 2d quantum electron solid in high magnetic field,” Nat. Phys.2, 452–455 (2006)

    2006

  79. [79]

    Imag- ing quantum melting in a disordered 2d wigner solid,

    Ziyu Xiang, Hongyuan Li, Jianghan Xiao, Mit H. Naik, Zhehao Ge, Zehao He, Sudi Chen, Jiahui Nie, Shiyu Li, Yifan Jiang, Renee Sailus, Rounak Banerjee, Takashi Taniguchi, Kenji Watanabe, Sefaattin Tongay, Steven G. Louie, Michael F. Crommie, and Feng Wang, “Imag- ing quantum melting in a disordered 2d wigner solid,” Science388, 736–740 (2025)

    2025

  80. [80]

    Thermal and quantum melting phase diagrams for a magnetic-field-induced wigner solid,

    Meng K. Ma, K. A. Villegas Rosales, H. Deng, Y. J. Chung, L. N. Pfeiffer, K. W. West, K. W. Baldwin, R. Winkler, and M. Shayegan, “Thermal and quantum melting phase diagrams for a magnetic-field-induced wigner solid,” Phys. Rev. Lett.125, 036601 (2020)

    2020

Showing first 80 references.