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arxiv: 2605.19036 · v1 · pith:5NAD2EEMnew · submitted 2026-05-18 · ❄️ cond-mat.str-el · cond-mat.supr-con· quant-ph

Hidden weak-pairing superconductivity of non-interacting anyons obeying frac{1}{3} statistics

Zheng-Duo Fan , Ashvin Vishwanath , Zijian Wang This is my paper

Pith reviewed 2026-05-20 07:48 UTC · model grok-4.3

classification ❄️ cond-mat.str-el cond-mat.supr-conquant-ph
keywords anyonssuperconductivityfractional Chern insulatorscomposite fermionsweak pairingchiral central chargeflux attachmentp-ip pairing
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The pith

Non-interacting anyons with one-third statistics superconduct through hidden weak pairing of composite fermions.

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

The paper establishes that a gas of charge-e/3 anyons obeying exchange statistics θ = -π/3 can form a superconductor even in the absence of interactions. This anyon gas emerges in doped fractional Chern insulators at fillings 1/3 or 2/3, where projective lattice translations create three degenerate pockets. A flux-attachment construction exploits this structure to cancel the average statistical flux, mapping the system to three species of composite fermions in zero effective field. Statistical gauge-field fluctuations then supply the pairing interaction that drives the fermions into a p-ip state, which translates to an f-if physical superconductor. The weak-pairing phase produces an edge with chiral central charge c_- = -1/2, differing from the integer value predicted by binding anyons into charge-2e/3 molecules.

Core claim

A non-interacting gas of charge-e/3 anyons with θ = -π/3 superconducts via a hidden weak-pairing mechanism. In doped fractional Chern insulators at 1/3 or 2/3 filling, the three-pocket structure from projective lattice translations allows a flux-attachment construction where average statistical flux vanishes, mapping to three species of composite fermions in zero effective field. The anyon statistics encoded in gauge field fluctuations supplies the pairing interaction, leading to a p-ip paired state of CFs equivalent to an f-if physical superconductor. The strong-pairing phase connects to Laughlin's anyon-binding picture, while the weak-pairing phase yields an edge with chiral central charge

What carries the argument

Flux-attachment construction that maps the anyon gas onto three species of composite fermions in zero effective field, with statistical gauge fluctuations acting as the pairing glue.

Load-bearing premise

The three-pocket structure arising from projective lattice translations in doped fractional Chern insulators at filling 1/3 or 2/3 enables a flux-attachment construction where the average statistical flux vanishes.

What would settle it

Measurement of the edge chiral central charge c_- = -1/2 (rather than an integer value) in the superconducting state adjacent to a fractional Chern insulator at 1/3 or 2/3 filling.

read the original abstract

We show that a non-interacting gas of charge-$e/3$ anyons with exchange statistics $\theta=-\pi/3$ can superconduct through a hidden weak-pairing mechanism. Such an anyon gas arises naturally in doped fractional Chern insulators at filling $1/3$ or $2/3$, where projective lattice translations enforce three degenerate anyon pockets. Exploiting this three-pocket structure, we develop a flux-attachment construction in which the average statistical flux vanishes, thereby mapping the problem to three species of composite fermions (CFs) in zero effective magnetic field. We show that the anyon statistics itself, encoded in statistical gauge field fluctuations, supplies the pairing glue and drives the CFs into a $p-\mathrm{i}p$ paired state, which corresponds to a $f-\mathrm{i}f$ physical superconductor. The CF strong-pairing phase is adiabatically connected to Laughlin's picture of anyon superconductivity, where charge-$e/3$ anyons bind into charge-$2e/3$ molecules, which then lead to superconductivity. By contrast, the more natural weak-pairing phase of CFs realizes a distinct superconducting phase - its edge is characterized by a chiral central charge $c_-=-1/2$, in contrast to the prediction of integer $c_-$ for the anyon superconductor based on Laughlin's picture, thereby resolving the discrepancy between previous theories and recent numerical results. Our theory provides a natural framework for understanding superconductivity near fractional Chern insulators, as observed in recent experiments. Finally, we discuss extensions of our theory that predict new chiral superconductors adjacent to FCIs at other fillings.

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 / 2 minor

Summary. The manuscript claims that a non-interacting gas of charge-e/3 anyons with exchange statistics θ=-π/3 superconducts via a hidden weak-pairing mechanism. Such anyons arise in doped fractional Chern insulators at filling 1/3 or 2/3, where projective lattice translations produce three degenerate pockets. A flux-attachment construction is introduced in which the average statistical flux vanishes, mapping the system to three species of composite fermions at zero effective magnetic field. Statistical gauge fluctuations supply the pairing glue, driving the CFs into a p-ip state that corresponds to an f-if physical superconductor. The weak-pairing phase has edge chiral central charge c_-=-1/2, in contrast to the integer c_- of Laughlin's strong-pairing anyon superconductor, thereby resolving a discrepancy with numerical results and providing a framework for superconductivity near FCIs. Extensions to other fillings are discussed.

Significance. If the mapping and pairing mechanism are valid, the work is significant for identifying a statistics-driven weak-pairing superconducting phase in anyonic systems without additional interactions. It distinguishes this phase from the strong-pairing phase through a concrete topological invariant (c_-=-1/2), offers a parameter-free construction, and aligns theory with recent numerical and experimental observations of superconductivity adjacent to fractional Chern insulators. The falsifiable prediction of the edge central charge is a clear strength.

major comments (1)
  1. [flux-attachment construction] The flux-attachment construction (main text, around the discussion of three-pocket structure): the central claim that projective lattice translations at filling 1/3 or 2/3 produce three degenerate anyon pockets allowing an exact cancellation of average statistical flux (mapping to CFs at B_eff=0) is asserted from symmetry. No controlled microscopic calculation is provided to confirm that this cancellation survives lattice commensurability, residual Berry curvature, or filling-dependent corrections. If the effective field is only approximately zero, the pairing instability changes and the claimed weak-pairing phase with c_-=-1/2 is no longer guaranteed. This step is load-bearing for the distinction from Laughlin's picture and the resolution of the numerical discrepancy.
minor comments (2)
  1. [introduction and abstract] The notation for the chiral central charge c_- is introduced without an explicit definition or reference to its relation to the edge theory of the f-if superconductor upon first use.
  2. [results section] The distinction between the anyon statistics θ=-π/3 and the resulting physical f-if pairing could be clarified with a short table or diagram relating the CF pairing channel to the physical charge and statistics.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading, positive assessment of the work's significance, and constructive feedback on the flux-attachment construction. We address the major comment point by point below.

read point-by-point responses

  1. Referee: The flux-attachment construction (main text, around the discussion of three-pocket structure): the central claim that projective lattice translations at filling 1/3 or 2/3 produce three degenerate anyon pockets allowing an exact cancellation of average statistical flux (mapping to CFs at B_eff=0) is asserted from symmetry. No controlled microscopic calculation is provided to confirm that this cancellation survives lattice commensurability, residual Berry curvature, or filling-dependent corrections. If the effective field is only approximately zero, the pairing instability changes and the claimed weak-pairing phase with c_-=-1/2 is no longer guaranteed. This step is load-bearing for the distinction from Laughlin's picture and the resolution of the numerical discrepancy.

    Authors: We appreciate the referee highlighting this load-bearing step. The three-pocket structure follows directly from the projective representation of lattice translations acting on the anyons in a doped FCI at filling 1/3 (or 2/3), as established in the FCI literature; this symmetry enforces three degenerate, equally populated pockets with identical dispersions. Our flux-attachment construction attaches statistical flux to each anyon species such that the mean statistical magnetic field vanishes exactly for the composite fermions by construction: the total attached flux is chosen to cancel the effective field at the mean density per pocket, consistent with the anyon charge e/3 and statistics θ=-π/3. This cancellation is exact within the effective long-wavelength theory and protected by the underlying symmetry; it is not an approximation. Lattice commensurability, residual Berry curvature, and filling-dependent corrections enter as modifications to the dispersion or higher-order terms but do not generate a net nonzero B_eff due to the equal pocket populations and symmetry. While a full microscopic lattice diagonalization would be valuable, it lies outside the controlled effective-theory framework of the manuscript. In the revised version we will add an expanded paragraph clarifying the symmetry protection, the mean-field exactness of B_eff=0, and the stability of the resulting weak-pairing phase (c_-=-1/2) against these corrections in the long-wavelength limit. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation relies on external symmetry inputs and standard mappings

full rationale

The paper's central chain begins from the symmetry-enforced three-pocket structure of anyons in doped FCIs (asserted via projective lattice translations at 1/3 or 2/3 filling), proceeds to a flux-attachment choice that sets average statistical flux to zero (a deliberate construction mapping to zero-field CFs), and then invokes statistical gauge fluctuations as the pairing mechanism for the p-ip state. None of these steps reduce by definition or construction to the target result (the distinct c_-=-1/2 edge or hidden weak-pairing superconductivity); the zero-field mapping is an input choice whose consequences are then analyzed, the gauge-fluctuation glue follows from the anyon statistics in the mapped theory rather than tautologically assuming the outcome, and the contrast to Laughlin's integer-c_- picture is presented as an independent distinction arising from the weak-pairing phase. No self-citations are load-bearing for the uniqueness or ansatz, no parameters are fitted and relabeled as predictions, and the overall argument remains self-contained against external benchmarks such as symmetry considerations and known CF pairing instabilities. This is the expected honest non-finding for a theoretical construction paper.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The construction rests on standard anyon statistics and the lattice-induced pocket degeneracy without introducing new free parameters or entities from the abstract description.

axioms (2)
  • domain assumption Anyons with charge e/3 obey exchange statistics θ=-π/3
    Invoked as the starting point for the gas of anyons in the doped FCI setup.
  • domain assumption Projective lattice translations enforce three degenerate anyon pockets at filling 1/3 or 2/3
    Used to enable the flux-attachment construction with vanishing average statistical flux.

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Works this paper leans on

50 extracted references · 50 canonical work pages · 5 internal anchors

  1. [1]

    Wilczek, Phys

    F. Wilczek, Phys. Rev. Lett.49, 957 (1982)

    work page 1982

  2. [2]

    R. B. Laughlin, Phys. Rev. Lett.60, 2677 (1988)

    work page 1988

  3. [3]

    A. L. Fetter, C. B. Hanna, and R. B. Laughlin, Phys. Rev. B39, 9679(R) (1989)

    work page 1989

  4. [4]

    Y .-H. Chen, F. Wilczek, E. Witten, and B. I. Halperin, Interna- tional Journal of Modern Physics B3, 1001 (1989)

    work page 1989

  5. [5]

    B. I. Halperin, Phys. Rev. Lett.52, 1583 (1984)

    work page 1984

  6. [6]

    Arovas, J

    D. Arovas, J. R. Schrieffer, and F. Wilczek, Phys. Rev. Lett.53, 722 (1984)

    work page 1984

  7. [7]

    Z. D. Shi and T. Senthil, Phys. Rev. X15, 031069 (2025)

    work page 2025

  8. [8]

    Divic, V

    S. Divic, V . Cr ´epel, T. Soejima, X.-Y . Song, A. J. Mil- lis, M. P. Zaletel, and A. Vishwanath, Proceedings of the National Academy of Sciences122, e2426680122 (2025), https://www.pnas.org/doi/pdf/10.1073/pnas.2426680122

    work page doi:10.1073/pnas.2426680122 2025

  9. [9]

    Z. Han, T. Wang, Z. Dong, M. P. Zaletel, and A. Vishwanath, arXiv preprint arXiv:2508.14894 (2025)

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  10. [10]

    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, Nature622, 63 (2023)

    work page 2023

  11. [11]

    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, Nature622, 74 (2023)

    work page 2023

  12. [12]

    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, Physical Review X13, 031037 (2023)

    work page 2023

  13. [13]

    W. Li, C. Wang Beach, C. Hu, T. Taniguchi, K. Watanabe, J.-H. Chu, A. Imamo˘glu, T. Cao, D. Xiao, and X. Xu, Nature651, 48 (2026)

    work page 2026

  14. [14]

    The conventional fractional quantum Hall states in Landau lev- els also support anyons, but due to the strong magnetic field, the anyons are immobile and lead to Hall plateaus

    work page

  15. [15]

    F. Xu, Z. Sun, J. Li, C. Zheng, C. Xu, J. Gao, T. Jia, K. Watanabe, T. Taniguchi, B. Tong,et al., arXiv preprint arXiv:2504.06972 (2025)

    work page internal anchor Pith review arXiv 2025

  16. [16]

    P. A. Nosov, Z. Han, and E. Khalaf, Phys. Rev. Lett.136, 106501 (2026)

    work page 2026

  17. [17]

    Z. D. Shi and T. Senthil, Proceedings of the Na- tional Academy of Sciences122, e2520608122 (2025), https://www.pnas.org/doi/pdf/10.1073/pnas.2520608122

    work page doi:10.1073/pnas.2520608122 2025

  18. [18]

    Pichler, C

    F. Pichler, C. Kuhlenkamp, M. Knap, and A. Vishwanath, New- ton2(2026)

    work page 2026

  19. [19]

    Gattu and J

    M. Gattu and J. K. Jain, Phys. Rev. Lett.135, 236601 (2025)

    work page 2025

  20. [20]

    Q. Xu, G. Ji, Y . Wang, H. Q. Trung, and B. Yang, Phys. Rev. B 112, 235112 (2025)

    work page 2025

  21. [21]

    Q. Li, P. A. Nosov, T. Wang, and E. Khalaf, arXiv preprint arXiv:2603.24701 (2026)

    work page arXiv 2026

  22. [22]

    Anyon molecules in fractional quantum Hall states

    T. Wang and M. P. Zaletel, arXiv preprint arXiv:2604.09798 (2026)

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  23. [23]

    Bardeen, L

    J. Bardeen, L. N. Cooper, and J. R. Schrieffer, Phys. Rev.108, 1175 (1957)

    work page 1957

  24. [24]

    D. M. Eagles, Phys. Rev.186, 456 (1969)

    work page 1969

  25. [25]

    Leggett, Modern Trends in the Theory of Condensed Matter, Proc

    A. Leggett, Modern Trends in the Theory of Condensed Matter, Proc. XVI Karpacz Winter School of Theoretical Physics, 1980 (1980)

    work page 1980

  26. [26]

    Read and D

    N. Read and D. Green, Phys. Rev. B61, 10267 (2000)

    work page 2000

  27. [27]

    New directions in the pursuit of Majorana fermions in solid state systems

    J. Alicea, Reports on Progress in Physics75, 076501 (2012), arXiv:1202.1293 [cond-mat.supr-con]

    work page internal anchor Pith review Pith/arXiv arXiv 2012

  28. [28]

    Guerci, A

    D. Guerci, A. Abouelkomsan, and L. Fu, arXiv preprint arXiv:2602.15106 (2026)

    work page arXiv 2026

  29. [29]

    Wang and M

    T. Wang and M. P. Zaletel, arXiv preprint arXiv:2507.07921 (2025)

    work page arXiv 2025

  30. [30]

    Cheng, M

    M. Cheng, M. Zaletel, M. Barkeshli, A. Vishwanath, and P. Bonderson, Physical Review X6, 041068 (2016)

    work page 2016

  31. [31]

    S. C. Zhang, T. H. Hansson, and S. Kivelson, Phys. Rev. Lett. 62, 980 (1989)

    work page 1989

  32. [32]

    S. C. Zhang, International Journal of Modern Physics B6, 25 (1992)

    work page 1992

  33. [33]

    B. I. Halperin, P. A. Lee, and N. Read, Phys. Rev. B47, 7312 (1993)

    work page 1993

  34. [34]

    Lopez and E

    A. Lopez and E. Fradkin, Phys. Rev. B44, 5246 (1991)

    work page 1991

  35. [35]

    N. E. Bonesteel, Phys. Rev. Lett.82, 984 (1999)

    work page 1999

  36. [36]

    M. A. Metlitski, D. F. Mross, S. Sachdev, and T. Senthil, Phys. 7 Rev. B91, 115111 (2015)

    work page 2015

  37. [37]

    R. B. Laughlin, Phys. Rev. Lett.50, 1395 (1983)

    work page 1983

  38. [38]

    M. E. Peskin, Annals of Physics113, 122 (1978)

    work page 1978

  39. [39]

    Dasgupta and B

    C. Dasgupta and B. I. Halperin, Phys. Rev. Lett.47, 1556 (1981)

    work page 1981

  40. [40]

    Wen, Advances in Physics44, 405 (1995)

    X.-G. Wen, Advances in Physics44, 405 (1995)

    work page 1995

  41. [41]

    C. L. Kane and M. P. A. Fisher, Phys. Rev. B55, 15832 (1997)

    work page 1997

  42. [42]

    Kitaev, Annals of Physics321, 2 (2006), january Special Issue

    A. Kitaev, Annals of Physics321, 2 (2006), january Special Issue

    work page 2006

  43. [43]

    Lotri ˇc and S

    T. Lotri ˇc and S. H. Simon, Phases of itinerant anyons in laugh- lin’s quantum hall states on a lattice (2026), arXiv:2603.22389 [cond-mat.str-el]

    work page arXiv 2026

  44. [44]

    gravitational Chern-Simons term

    The2CS g term is referred to as a “gravitational Chern-Simons term” but it simply capturesc − = 1

    work page

  45. [45]

    all chargeeexcitations that are not anyons, must be fermions [50]

    More precisely, these are spin c connections, whose flux quan- tization condition is fixed by the spin-charge relation, i.e. all chargeeexcitations that are not anyons, must be fermions [50]

    work page

  46. [46]

    Other proposals that do capture the correctc − were found to break translation symmetry [43]

    Some works have invoked more elaborate SU(2) parton con- structions to accommodate half integer chiral central chargec − [? ?], but the simplest such construction still fails to correctly capture the precise valuec − =−1/2for 2/3 filling seen in numerics. Other proposals that do capture the correctc − were found to break translation symmetry [43]

    work page

  47. [47]

    Lu and A

    Y .-M. Lu and A. Vishwanath, Phys. Rev. B86, 125119 (2012)

    work page 2012

  48. [48]

    Senthil and M

    T. Senthil and M. Levin, Phys. Rev. Lett.110, 046801 (2013)

    work page 2013

  49. [49]

    Song, Y .-H

    X.-Y . Song, Y .-H. Zhang, and T. Senthil, Physical Review B 109, 10.1103/physrevb.109.085143 (2024)

    work page doi:10.1103/physrevb.109.085143 2024

  50. [50]

    Gapped Boundary Phases of Topological Insulators via Weak Coupling

    N. Seiberg and E. Witten, Gapped boundary phases of topo- logical insulators via weak coupling (2016), arXiv:1602.04251 [cond-mat.str-el]. END MA TTER Strong-pairing phase vs Laughlin’s theory– In this let- ter, we mainly focus on1/3anyon gas. We have adopted the picture that in the CF strong-pairing phase, two1/3anyons bind into an anyon molecule withθ= ...

    work page internal anchor Pith review Pith/arXiv arXiv 2016