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arxiv: 2604.22628 · v2 · submitted 2026-04-24 · ❄️ cond-mat.str-el · cond-mat.mes-hall

Relations between density-density correlators of states in the maximal spin multiplet

Ritajit Kundu , Ajit C. Balram This is my paper

Pith reviewed 2026-05-08 09:51 UTC · model grok-4.3

classification ❄️ cond-mat.str-el cond-mat.mes-hall
keywords density-density correlatorsmaximal spin multipletpair-correlation functionsstatic structure factorsfractional quantum Hall statesHalperin statesspin symmetry
0
0 comments X

The pith

Identities relate the pair-correlation functions and static structure factors of all states in a maximal spin multiplet to those of the highest-weight state.

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

The paper establishes identities that connect the density-density correlators between different states sharing the same maximal spin multiplet. These identities mean that the pair-correlation functions and static structure factors for every state in the multiplet can be derived directly from the corresponding functions of the highest-weight state alone. This approach simplifies the calculation of energies for fractional quantum Hall states that form such multiplets. The authors use it to find analytic energies for the Halperin (1,1,1) state varying with density imbalance and layer separation, and numerical values for other states.

Core claim

We present identities relating the pair-correlation functions and static structure factors of states in the maximal spin multiplet. This allows us to compute these density-density correlation functions of all members of the multiplet using just these correlation functions of the highest-weight state. We apply these relations to obtain energies for many fractional quantum Hall (FQH) states. In particular, we analytically compute the energies of the Halperin-(1,1,1) state as a function of density imbalance and layer separation, and numerically evaluate these energies for many other FQH states.

What carries the argument

The identities linking pair-correlation functions and static structure factors across states in the maximal spin multiplet.

If this is right

  • All density-density correlation functions within the multiplet follow from data on the single highest-weight state.
  • Energies of the Halperin-(1,1,1) state are obtained analytically as functions of density imbalance and layer separation.
  • Numerical energies for many other fractional quantum Hall states become accessible without separate correlator calculations for each member.
  • Computational cost drops for studying multi-component or partially polarized quantum Hall systems that realize spin multiplets.

Where Pith is reading between the lines

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

  • The same symmetry-based reduction could apply to other observables that transform under spin rotations, such as spin-density correlations.
  • The identities may extend to time-dependent or dynamical correlators if the underlying wave-function symmetries persist.
  • Similar relations could simplify calculations in lattice models or ultracold-atom systems that exhibit maximal spin multiplets.

Load-bearing premise

The wave functions of the states belong to a maximal spin multiplet and obey the symmetry properties required for the identities to hold exactly.

What would settle it

Direct computation of the pair-correlation function for a lower-weight state from its own wave function, followed by comparison to the value predicted by applying the identity to the highest-weight state's correlator.

Figures

Figures reproduced from arXiv: 2604.22628 by Ajit C. Balram, Ritajit Kundu.

Figure 1
Figure 1. Figure 1: FIG. 1. Coulomb energies in units of view at source ↗
read the original abstract

We present identities relating the pair-correlation functions and static structure factors of states in the maximal spin multiplet. This allows us to compute these density-density correlation functions of all members of the multiplet using just these correlation functions of the highest-weight state. We apply these relations to obtain energies for many fractional quantum Hall (FQH) states. In particular, we analytically compute the energies of the Halperin-$(1,1,1)$ state as a function of density imbalance and layer separation, and numerically evaluate these energies for many other FQH states.

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 derives identities relating the pair-correlation functions and static structure factors of states belonging to the same maximal spin multiplet. These relations follow from the total density operator being a scalar under SU(2) rotations, allowing all correlators within the multiplet to be obtained from those of the highest-weight state. The identities are applied to compute interaction energies for multiple fractional quantum Hall states, including an analytic expression for the Halperin (1,1,1) state as a function of density imbalance and layer separation, together with numerical evaluations for other states.

Significance. If the identities hold exactly, the work supplies a symmetry-based shortcut for evaluating density-density correlators and energies in spinful or multilayer quantum Hall systems without separate calculations for each multiplet member. The analytic treatment of the Halperin (1,1,1) state is a concrete strength, as it enables direct exploration of parameter dependence. The derivation is parameter-free and rests on standard representation theory of SU(2).

major comments (2)
  1. [§2] §2 (derivation of the identities): while the SU(2) argument is standard, the manuscript does not include an explicit algebraic verification or a small-system numerical check (e.g., two or three electrons in the lowest Landau level) confirming that the relations survive Landau-level projection and the specific form of the wave functions; such a check is load-bearing for the central claim that the identities apply directly to the FQH states considered later.
  2. [§4] §4 (Halperin (1,1,1) energies): the analytic energy expression is stated as a function of imbalance and separation, but the manuscript does not display the final closed-form result or compare it against the known balanced, zero-separation limit; without this, it is difficult to confirm that the application of the identities reproduces established results.
minor comments (2)
  1. [§2] Notation for the component-resolved pair-correlation functions g_{σσ'}(r) should be defined once at the beginning of §2 and used consistently; the current text occasionally switches between g and S without explicit cross-reference.
  2. Figure captions for the numerical energy plots should state the system size, number of Monte Carlo samples, and statistical error bars; these details are needed to assess the precision of the reported energies.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript and the recommendation for minor revision. We address each major comment below.

read point-by-point responses

  1. Referee: [§2] §2 (derivation of the identities): while the SU(2) argument is standard, the manuscript does not include an explicit algebraic verification or a small-system numerical check (e.g., two or three electrons in the lowest Landau level) confirming that the relations survive Landau-level projection and the specific form of the wave functions; such a check is load-bearing for the central claim that the identities apply directly to the FQH states considered later.

    Authors: We agree that an explicit verification strengthens the presentation. Although the identities follow from the total density operator being an SU(2) scalar (which commutes with Landau-level projection), we will add in the revised manuscript a small-system algebraic verification for two electrons in the lowest Landau level, confirming that the relations hold for the projected wave functions. revision: yes

  2. Referee: [§4] §4 (Halperin (1,1,1) energies): the analytic energy expression is stated as a function of imbalance and separation, but the manuscript does not display the final closed-form result or compare it against the known balanced, zero-separation limit; without this, it is difficult to confirm that the application of the identities reproduces established results.

    Authors: We thank the referee for this observation. In the revised manuscript we will display the explicit closed-form analytic expression for the Halperin (1,1,1) energy as a function of density imbalance and layer separation, and we will include a direct comparison to the known balanced, zero-separation limit to confirm consistency with established results. revision: yes

Circularity Check

0 steps flagged

No significant circularity; identities follow from SU(2) representation theory

full rationale

The central derivation consists of algebraic identities relating pair-correlation functions and static structure factors within a maximal spin multiplet. These follow directly from the total density operator being an SU(2) scalar and the states forming an irreducible representation, with lower-weight correlators obtained by repeated application of the spin-lowering operator to the highest-weight wave function. No parameters are fitted to data and then relabeled as predictions, no self-citation supplies a load-bearing uniqueness theorem, and no ansatz is smuggled in. The subsequent use of the resulting correlators to evaluate energies via standard interaction Hamiltonians for FQH states is a downstream application that does not close any loop back to the input definitions. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the algebraic properties of spin multiplets and the assumption that the FQH wave functions transform appropriately under total-spin rotations; no free parameters or new entities are introduced in the abstract.

axioms (1)
  • domain assumption The states under study form a maximal spin multiplet whose members are related by total-spin lowering operators.
    Invoked when the identities are stated to hold for all members once the highest-weight correlators are known.

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Works this paper leans on

75 extracted references · 2 canonical work pages

  1. [1]

    Kubo, Statistical-Mechanical Theory of Irreversible Processes

    R. Kubo, Statistical-Mechanical Theory of Irreversible Processes. I. General Theory and Simple Applications to Magnetic and Conduction Problems, Journal of the Physical Society of Japan12, 570–586 (1957)

    1957

  2. [2]

    Hansen and I

    J.-P. Hansen and I. R. McDonald,Theory of simple liq- uids: with applications to soft matter(Academic press, 2013)

    2013

  3. [3]

    Friedrich, P

    W. Friedrich, P. Knipping, and M. Laue, Interferenz- erscheinungen bei R¨ ontgenstrahlen, Annalen der Physik 346, 971 (1913)

    1913

  4. [4]

    R. E. Franklin and R. G. Gosling, Molecular Configu- ration in Sodium Thymonucleate, Nature171, 740–741 (1953)

    1953

  5. [5]

    N. W. Ashcroft and J. Lekner, Structure and Resistivity of Liquid Metals, Physical Review145, 83–90 (1966)

    1966

  6. [6]

    A. J. Greenfield, J. Wellendorf, and N. Wiser, X-Ray Determination of the Static Structure Factor of Liquid Na and K, Physical Review A4, 1607–1616 (1971)

    1971

  7. [7]

    J. L. Yarnell, M. J. Katz, R. G. Wenzel, and S. H. Koenig, Structure Factor and Radial Distribution Function for Liquid Argon at 85 ◦K, Physical Review A7, 2130–2144 (1973)

    1973

  8. [8]

    K. S. Vahvaselk¨ a, X-Ray Diffraction Analysis of Liquid Hg, Sn, Zn, Al and Cu, Physica Scripta18, 266–274 (1978)

    1978

  9. [9]

    E. C. Svensson, V. F. Sears, A. D. B. Woods, and P. Mar- tel, Neutron-diffraction study of the static structure fac- tor and pair correlations in liquid 4He, Phys. Rev. B21, 3638 (1980)

    1980

  10. [10]

    Furrer, J

    A. Furrer, J. F. Mesot, and T. Str¨ assle,Neutron scatter- ing in condensed matter physics, Vol. 4 (World Scientific Publishing Company, 2009)

    2009

  11. [11]

    Giuliani and G

    G. Giuliani and G. Vignale,Quantum Theory of the Elec- tron Liquid(Cambridge University Press, The Edinburgh Building, Cambridge CB2 2RU, UK, 2008)

    2008

  12. [12]

    G. D. Mahan,Many-particle physics(Springer Science & Business Media, 2013)

    2013

  13. [13]

    Pines,Theory of quantum liquids: normal Fermi liq- uids(CRC Press, 2018)

    D. Pines,Theory of quantum liquids: normal Fermi liq- uids(CRC Press, 2018)

    2018

  14. [14]

    Yoshioka, B

    D. Yoshioka, B. I. Halperin, and P. A. Lee, Ground state of two-dimensional electrons in strong magnetic fields and 1 3 quantized Hall effect, Phys. Rev. Lett.50, 1219 (1983). 8

    1983

  15. [15]

    Lin, Exact diagonalization of quantum-spin models, Physical Review B42, 6561 (1990)

    H. Lin, Exact diagonalization of quantum-spin models, Physical Review B42, 6561 (1990)

    1990

  16. [16]

    Pereira, J

    R. Pereira, J. Sirker, J. Caux, R. Hagemans, J. M. Mail- let, S. White, and I. Affleck, Dynamical structure factor at small q for the xxz spin-1/2 chain, Journal of Statis- tical Mechanics: Theory and Experiment2007, P08022 (2007)

    2007

  17. [17]

    Fehske, R

    H. Fehske, R. Schneider, and A. Weisse,Computational many-particle physics(Springer, 2007)

    2007

  18. [18]

    Zhang and R

    J. Zhang and R. Dong, Exact diagonalization: the Bose– Hubbard model as an example, European Journal of Physics31, 591 (2010)

    2010

  19. [19]

    Binder and D

    K. Binder and D. Heermann,Monte Carlo Simulation in Statistical Physics(Springer-Verlag Berlin Heidelberg, 2010)

    2010

  20. [20]

    R. K. Dora and A. C. Balram, Dispersion of collective modes in spinful fractional quantum Hall states on the sphere, Phys. Rev. B113, 115420 (2026)

    2026

  21. [21]

    Eisenstein, Exciton condensation in bilayer quantum Hall systems, Annu

    J. Eisenstein, Exciton condensation in bilayer quantum Hall systems, Annu. Rev. Condens. Matter Phys.5, 159 (2014)

    2014

  22. [22]

    B. I. Halperin, Theory of the quantized Hall conductance, Helvetica Physica Acta56, 75 (1983)

    1983

  23. [23]

    R. B. Laughlin, Anomalous quantum Hall effect: An in- compressible quantum fluid with fractionally charged ex- citations, Phys. Rev. Lett.50, 1395 (1983)

    1983

  24. [24]

    S. M. Girvin, A. H. MacDonald, and P. M. Platzman, Magneto-roton theory of collective excitations in the frac- tional quantum Hall effect, Phys. Rev. B33, 2481 (1986)

    1986

  25. [25]

    R. K. Dora and A. C. Balram, Static structure factor and the dispersion of the Girvin-MacDonald-Platzman density mode for fractional quantum Hall fluids on the Haldane sphere, Phys. Rev. B111, 115132 (2025)

    2025

  26. [26]

    A. C. Balram, C. T˝ oke, and J. K. Jain, Luttinger theo- rem for the strongly correlated Fermi liquid of composite fermions, Phys. Rev. Lett.115, 186805 (2015)

    2015

  27. [27]

    D. X. Nguyen, T. Can, and A. Gromov, Particle-hole duality in the lowest Landau level, Phys. Rev. Lett.118, 206602 (2017)

    2017

  28. [28]

    J. K. Jain, Composite-fermion approach for the fractional quantum Hall effect, Phys. Rev. Lett.63, 199 (1989)

    1989

  29. [29]

    Moore and N

    G. Moore and N. Read, Nonabelions in the fractional quantum Hall effect, Nucl. Phys. B360, 362 (1991)

    1991

  30. [30]

    Fulsebakke, M

    J. Fulsebakke, M. Fremling, N. Moran, and J. K. Slinger- land, Parametrization and thermodynamic scaling of pair correlation functions for the fractional quantum Hall ef- fect, SciPost Phys.14, 149 (2023)

    2023

  31. [31]

    B. I. Halperin, P. A. Lee, and N. Read, Theory of the half-filled Landau level, Phys. Rev. B47, 7312 (1993)

    1993

  32. [32]

    Rezayi and N

    E. Rezayi and N. Read, Fermi-liquid-like state in a half- filled Landau level, Phys. Rev. Lett.72, 900 (1994)

    1994

  33. [33]

    A. C. Balram, C. T˝ oke, A. W´ ojs, and J. K. Jain, Sponta- neous polarization of composite fermions in then= 1 Landau level of graphene, Phys. Rev. B92, 205120 (2015)

    2015

  34. [34]

    A. C. Balram and J. K. Jain, Fermi wave vector for the partially spin-polarized composite-fermion Fermi sea, Phys. Rev. B96, 235102 (2017)

    2017

  35. [35]

    Ciftja and C

    O. Ciftja and C. Wexler, Monte Carlo simulation method for Laughlin-like states in a disk geometry, Phys. Rev. B 67, 075304 (2003)

    2003

  36. [36]

    R. K. Dora and A. C. Balram, Competition between frac- tional quantum Hall liquid and electron solid phases in the Landau levels of multilayer graphene, Phys. Rev. B 108, 235153 (2023)

    2023

  37. [37]

    J. K. Jain and R. K. Kamilla, Quantitative study of large composite-fermion systems, Phys. Rev. B55, R4895 (1997)

    1997

  38. [38]

    A. C. Balram, C. T¨ oke, A. W´ ojs, and J. K. Jain, Phase diagram of fractional quantum Hall effect of composite fermions in multicomponent systems, Phys. Rev. B91, 045109 (2015)

    2015

  39. [39]

    J. K. Jain and R. K. Kamilla, Composite fermions in the Hilbert space of the lowest electronic Landau level, Int. J. Mod. Phys. B11, 2621 (1997)

    1997

  40. [40]

    J. K. Jain,Composite Fermions(Cambridge University Press, New York, US, 2007)

    2007

  41. [41]

    A. C. Balram, C. T¨ oke, A. W´ ojs, and J. K. Jain, Frac- tional quantum Hall effect in graphene: Quantitative comparison between theory and experiment, Phys. Rev. B92, 075410 (2015)

    2015

  42. [42]

    A. C. Balram and A. W´ ojs, Fractional quantum Hall ef- fect atν= 2+4/9, Phys. Rev. Research2, 032035 (2020)

    2020

  43. [43]

    K. Moon, H. Mori, K. Yang, S. M. Girvin, A. H. Mac- Donald, L. Zheng, D. Yoshioka, and S.-C. Zhang, Sponta- neous interlayer coherence in double-layer quantum Hall systems: Charged vortices and kosterlitz-thouless phase transitions, Phys. Rev. B95, 5138 (1995)

    1995

  44. [44]

    V. W. Scarola and J. K. Jain, Phase diagram of bi- layer composite fermion states, Phys. Rev. B64, 085313 (2001)

    2001

  45. [45]

    M¨ oller, S

    G. M¨ oller, S. H. Simon, and E. H. Rezayi, Paired compos- ite fermion phase of quantum Hall bilayers atν= 1 2 + 1 2, Phys. Rev. Lett.101, 176803 (2008)

    2008

  46. [46]

    Papi´ c, G

    Z. Papi´ c, G. M¨ oller, M. V. Milovanovi´ c, N. Regnault, and M. O. Goerbig, Fractional quantum Hall state atν= 1 4 in a wide quantum well, Phys. Rev. B79, 245325 (2009)

    2009

  47. [47]

    M. R. Peterson, Y.-L. Wu, M. Cheng, M. Barkeshli, Z. Wang, and S. Das Sarma, Abelian and non-Abelian states inν= 2/3 bilayer fractional quantum Hall sys- tems, Phys. Rev. B92, 035103 (2015)

    2015

  48. [48]

    Isobe and L

    H. Isobe and L. Fu, Interlayer pairing symmetry of com- posite fermions in quantum Hall bilayers, Phys. Rev. Lett.118, 166401 (2017)

    2017

  49. [49]

    W. Zhu, Z. Liu, F. D. M. Haldane, and D. N. Sheng, Frac- tional quantum Hall bilayers at half filling: Tunneling- driven non-abelian phase, Phys. Rev. B94, 245147 (2016)

    2016

  50. [50]

    Z. Zhu, L. Fu, and D. N. Sheng, Numerical study of quan- tum Hall bilayers at total fillingν T = 1: A new phase at intermediate layer distances, Phys. Rev. Lett.119, 177601 (2017)

    2017

  51. [51]

    Lian and S.-C

    B. Lian and S.-C. Zhang, Wave function and emergent su(2) symmetry in theν T = 1 quantum Hall bilayer, Phys. Rev. Lett.120, 077601 (2018)

    2018

  52. [52]

    W. N. Faugno, A. C. Balram, A. W´ ojs, and J. K. Jain, Theoretical phase diagram of two-component composite fermions in double-layer graphene, Phys. Rev. B101, 085412 (2020)

    2020

  53. [53]

    Wagner, D

    G. Wagner, D. X. Nguyen, S. H. Simon, and B. I. Halperin,s-wave paired electron and hole composite fermion trial state for quantum Hall bilayers withν= 1, Phys. Rev. Lett.127, 246803 (2021)

    2021

  54. [54]

    Sharma, A

    A. Sharma, A. C. Balram, and J. K. Jain, Composite- fermion pairing at half-filled and quarter-filled lowest Landau level, Phys. Rev. B109, 035306 (2024)

    2024

  55. [55]

    X. Liu, Z. Hao, K. Watanabe, T. Taniguchi, B. I. Halperin, and P. Kim, Interlayer fractional quantum Hall 9 effect in a coupled graphene double layer, Nature Physics 15, 893 (2019)

    2019

  56. [56]

    J. I. A. Li, Q. Shi, Y. Zeng, K. Watanabe, T. Taniguchi, J. Hone, and C. R. Dean, Pairing states of composite fermions in double-layer graphene, Nature Physics15, 898 (2019)

    2019

  57. [57]

    Shi, E.-M

    Q. Shi, E.-M. Shih, D. Rhodes, B. Kim, K. Barmak, K. Watanabe, T. Taniguchi, Z. Papi´ c, D. A. Abanin, J. Hone, and C. R. Dean, Bilayer WSe2 as a natural platform for interlayer exciton condensates in the strong coupling limit, Nature Nanotechnology17, 577 (2022)

    2022

  58. [58]

    N. J. Zhang, R. Q. Nguyen, N. Batra, X. Liu, K. Watan- abe, T. Taniguchi, D. E. Feldman, and J. I. A. Li, Exci- tons in the fractional quantum Hall effect, Nature637, 327 (2025)

    2025

  59. [59]

    R. Q. Nguyen, N. J. Zhang, N. Khurana-Batra, S. Alkidim, X. Liu, K. Watanabe, T. Taniguchi, D. Feld- man, and J. Li, Bilayer excitons in the laughlin fractional quantum Hall state, arXiv preprint arXiv:2410.24208 (2024)

    work page arXiv 2024

  60. [60]

    Lebreuilly, Physical Review A96, 10.1103/Phys- RevA.96.033828 (2017)

    M. Christos, S. Sachdev, and M. S. Scheurer, Correlated insulators, semimetals, and superconductivity in twisted trilayer graphene, Physical Review X12, 10.1103/phys- revx.12.021018 (2022)

    work page doi:10.1103/phys- 2022

  61. [61]

    H. Kim, Y. Choi, E. Lantagne-Hurtubise, C. Lewandowski, A. Thomson, L. Kong, H. Zhou, E. Baum, Y. Zhang, L. Holleis, K. Watanabe, T. Taniguchi, A. F. Young, J. Alicea, and S. Nadj-Perge, Imaging inter-valley coherent order in magic-angle twisted trilayer graphene, Nature623, 942–948 (2023)

    2023

  62. [62]

    Mukherjee, S

    A. Mukherjee, S. Layek, S. Sinha, R. Kundu, A. H. Marchawala, M. Hingankar, J. Sarkar, L. D. V. San- gani, H. Agarwal, S. Ghosh, A. B. Tazi, K. Watanabe, T. Taniguchi, A. N. Pasupathy, A. Kundu, and M. M. Deshmukh, Superconducting magic-angle twisted trilayer graphene with competing magnetic order and moir´ e in- homogeneities, Nature Materials24, 1400–14...

    2025

  63. [63]

    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, Nature626, 759 (2024)

    2024

  64. [64]

    Xie, C.-P

    Y.-M. Xie, C.-P. Zhang, J.-X. Hu, K. F. Mak, and K. T. Law, Valley-polarized quantum anomalous hall state in moir´ e mote2/wse2 heterobilayers, Phys. Rev. Lett.128, 026402 (2022)

    2022

  65. [65]

    Z. Lian, Y. Meng, L. Ma, I. Maity, L. Yan, Q. Wu, X. Huang, D. Chen, X. Chen, X. Chen, M. Blei, T. Taniguchi, K. Watanabe, S. Tongay, J. Lischner, Y.-T. Cui, and S.-F. Shi, Valley-polarized excitonic Mott insu- lator in WS2/WSe2 moir´ e superlattice, Nature Physics 20, 34–39 (2023)

    2023

  66. [66]

    J. Cai, E. Anderson, C. Wang, X. Zhang, X. Liu, W. Holtzmann, Y. Zhang, F. Fan, T. Taniguchi, K. Watanabe, Y. Ran, T. Cao, L. Fu, D. Xiao, W. Yao, and X. Xu, Signatures of fractional quantum anomalous Hall states in twisted MoTe2, Nature622, 63 (2023)

    2023

  67. [67]

    H. Park, J. Cai, E. Anderson, Y. Zhang, J. Zhu, X. Liu, C. Wang, W. Holtzmann, C. Hu, Z. Liu, T. Taniguchi, K. Watanabe, J.-H. Chu, T. Cao, L. Fu, W. Yao, C.- Z. Chang, D. Cobden, D. Xiao, and X. Xu, Observation of fractionally quantized anomalous Hall effect, Nature 622, 74 (2023)

    2023

  68. [68]

    Y. Zeng, Z. Xia, K. Kang, J. Zhu, P. Kn¨ uppel, C. Vaswani, K. Watanabe, T. Taniguchi, K. F. Mak, and J. Shan, Thermodynamic evidence of fractional Chern insulator in moir´ e MoTe2, Nature622, 69 (2023)

    2023

  69. [69]

    F. Xu, Z. Sun, T. Jia, C. Liu, C. Xu, C. Li, Y. Gu, K. Watanabe, T. Taniguchi, B. Tong, J. Jia, Z. Shi, S. Jiang, Y. Zhang, X. Liu, and T. Li, Observation of integer and fractional quantum anomalous Hall effects in twisted bilayer mote2, Phys. Rev. X13, 031037 (2023)

    2023

  70. [70]

    Github repository: Exact relations between the density- density correlators of states in a spin multiplet (2026)

    2026

  71. [71]

    DiagHam,https://www.nick-ux.org/diagham

  72. [72]

    F. D. M. Haldane, Fractional quantization of the Hall ef- fect: A hierarchy of incompressible quantum fluid states, Phys. Rev. Lett.51, 605 (1983)

    1983

  73. [73]

    S. H. Simon and B. I. Halperin, Response function of the fractional quantized Hall state on a sphere. i. fermion Chern-Simons theory, Phys. Rev. B50, 1807 (1994)

    1994

  74. [74]

    S. He, S. H. Simon, and B. I. Halperin, Response function of the fractional quantized Hall state on a sphere. ii. exact diagonalization, Phys. Rev. B50, 1823 (1994)

    1994

  75. [75]

    S. M. Girvin, A. H. MacDonald, and P. M. Platzman, Collective-excitation gap in the fractional quantum Hall effect, Phys. Rev. Lett.54, 581 (1985)

    1985