pith. sign in

arxiv: 2605.29034 · v2 · pith:FSVWZVNTnew · submitted 2026-05-27 · ❄️ cond-mat.str-el

Topological superconductivity from Abelian fractional Chern insulators

Taige Wang This is my paper

Pith reviewed 2026-06-29 09:30 UTC · model grok-4.3

classification ❄️ cond-mat.str-el
keywords fractional chern insulatortopological superconductivityparton constructionlaughlin stateanyon superconductorcharge density wavecentral chargeu(3) gauge theory
0
0 comments X

The pith

A ν=1/3 fractional Chern insulator can be turned into three topological superconductors with a U(3) parton theory.

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

The paper shows that a Laughlin anyon fluid realized in a fractional Chern insulator at filling 1/3 can be converted into superconductors by pairing its constituents. Three charge-e/3 anyons form each electron, so three such pairs form a gauge-invariant charge-2e Cooper pair. The construction produces three superconductors that all carry conventional charge-2e pairing yet differ in their neutral anyon content: one keeps the parent Laughlin U(1)_3 order, one is a chiral topological superconductor with central charge 3/2, and one is a strong-pairing anyon superconductor with central charge 3. The same parton framework also accounts for a nearby charge-density-wave state. This matters because it gives a single microscopic description that links the fractional Chern insulator, the density wave, and multiple superconducting phases.

Core claim

Using a U(3) infrared parton theory for the ν=1/3 FCI, where the electron is composed of three charge-e/3 constituents and the Cooper pair of three constituent pairs, the resulting superconductors share an ordinary charge-2e sector but differ in their neutral color response, giving an SC* with the parent Laughlin U(1)_3 order, a chiral topological superconductor with c_-=3/2, and a strong-pairing anyon superconductor with c_-=3. The same framework organizes a nearby σ_xy=0 charge density wave state, with natural ties to period-three density order at commensurate filling while the SC* branch can preserve microscopic translations.

What carries the argument

The U(3) infrared parton theory in which three colored charge-e/3 anyons constitute the electron and three such pairs constitute the charge-2e Cooper pair.

If this is right

  • The three superconductors differ only in neutral color response while sharing the same charge-2e sector.
  • At commensurate filling the normal state and the c=3 and c=3/2 superconducting states are tied to period-three density order.
  • The SC* branch can preserve microscopic translations.
  • Away from commensuration a chargon metal can pair into the same c=3/2 topological superconductor.
  • The FCI, reentrant CDW, and chiral superconductivity are unified inside the single U(3) parton theory.

Where Pith is reading between the lines

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

  • The parton construction could be extended to other Abelian fractional Chern insulators at different fillings.
  • Thermal Hall measurements that resolve the predicted central charges would provide direct signatures.
  • The framework suggests a route to realize anyon superconductors in moiré systems that host fractional Chern insulators.
  • The same neutral-sector distinctions might appear in other paired states derived from Abelian anyon fluids.

Load-bearing premise

The U(3) infrared parton theory is assumed to correctly capture the low-energy physics of the ν=1/3 FCI.

What would settle it

Detection of a superconducting state emerging from a ν=1/3 FCI that exhibits a neutral sector with central charge exactly 3/2 or exactly 3, or the absence of any such state with the predicted anyon content, would test the claim.

Figures

Figures reproduced from arXiv: 2605.29034 by Taige Wang.

Figure 1
Figure 1. Figure 1: FIG. 1. Color-isotropic [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Anisotropic full-rank pair-Higgs branch [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
read the original abstract

Can a Laughlin anyon fluid become a topological superconductor? We answer this question for a $\nu=1/3$ fractional Chern insulator (FCI) using a $U(3)$ infrared parton theory. Three charge-$e/3$ constituents form the electron, while three constituent pairs form a gauge-invariant charge-$2e$ Cooper pair. The resulting superconductors share an ordinary charge-$2e$ sector but differ in their neutral color response, giving an SC$^\ast$ with the parent Laughlin $U(1)_3$ order, a chiral topological superconductor with $c_-=3/2$, and a strong-pairing anyon superconductor with $c_-=3$. The same framework organizes a nearby $\sigma_{xy}=0$ charge density wave (CDW) state. At commensurate filling, the normal state and the $c_-=3$ and $c_-=3/2$ superconducting descendants are naturally tied to a period-three density order background, while the SC$^\ast$ branch can preserve microscopic translations. Away from commensuration, a chargon metal can pair into the same $c_-=3/2$ topological superconductor. The FCI, reentrant CDW, and chiral superconductivity are unified in the $U(3)$ parton theory.

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

3 major / 1 minor

Summary. The manuscript develops a U(3) infrared parton construction for the ν=1/3 fractional Chern insulator in which each electron comprises three charge-e/3 partons and each charge-2e Cooper pair comprises three parton pairs. Pairing in different channels is shown to produce three superconductors that share an ordinary charge-2e sector but realize distinct neutral responses: an SC* inheriting the parent Laughlin U(1)_3 order, a chiral topological superconductor with c_-=3/2, and a strong-pairing anyon superconductor with c_-=3. The same framework also organizes a nearby σ_xy=0 charge-density-wave state and discusses translation symmetry at and away from commensurate filling.

Significance. If the parton construction is valid, the work supplies a single gauge-theoretic framework that unifies a fractional Chern insulator, a charge-density wave, and three distinct topological superconductors differing only in their neutral color sectors. The explicit organization of multiple neutral responses from one parent state constitutes a useful organizational result for the theory of anyon-based superconductivity.

major comments (3)
  1. [Abstract] Abstract, first paragraph: the central claim that the U(3) parton theory reproduces the Laughlin U(1)_3 order of the ν=1/3 FCI and then yields the stated central charges upon pairing is asserted without an explicit computation of the parton Chern numbers, the resulting K-matrix, or the anyon content that would confirm the parent topological order.
  2. [Abstract] The derivation of c_-=3/2 and c_-=3 for the paired states is presented as an outcome of the three pairing channels, yet no gap equations, edge-mode spectra, or explicit neutral-sector response functions are supplied to show that these central charges follow independently rather than by construction from the assumed U(3) gauge structure and pairing ansatz.
  3. [Abstract] The manuscript states that the same U(3) framework organizes the σ_xy=0 CDW and the reentrant period-three density order, but does not provide the explicit mean-field decoupling or the resulting charge-density modulation that would establish this unification beyond the level of symmetry assignment.
minor comments (1)
  1. Notation for the neutral color response and the distinction between SC* and the other branches should be defined more explicitly when first introduced.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their positive evaluation of the work's significance and for the constructive comments on the abstract and presentation. We address each major comment below with clarifications from the manuscript and indicate planned revisions.

read point-by-point responses

  1. Referee: [Abstract] Abstract, first paragraph: the central claim that the U(3) parton theory reproduces the Laughlin U(1)_3 order of the ν=1/3 FCI and then yields the stated central charges upon pairing is asserted without an explicit computation of the parton Chern numbers, the resulting K-matrix, or the anyon content that would confirm the parent topological order.

    Authors: The parent U(1)_3 order is obtained directly from the U(3) parton construction in which each of the three partons fills a Chern band of C=1; the resulting K-matrix and anyon content (including the e/3 anyons with mutual statistics 2π/3) are derived explicitly in Section II of the manuscript. The abstract summarizes this established result rather than repeating the full derivation. To improve clarity we will add a parenthetical reference to Section II in the revised abstract. revision: partial

  2. Referee: [Abstract] The derivation of c_-=3/2 and c_-=3 for the paired states is presented as an outcome of the three pairing channels, yet no gap equations, edge-mode spectra, or explicit neutral-sector response functions are supplied to show that these central charges follow independently rather than by construction from the assumed U(3) gauge structure and pairing ansatz.

    Authors: The central charges follow from the neutral-sector edge-mode counting after the three distinct pairing channels are imposed on the partons (Section IV). The c_-=3/2 state retains one neutral chiral Majorana mode per color after pairing, while the strong-pairing channel gaps all neutral modes except for the anyonic contribution yielding c_-=3; these spectra are obtained by diagonalizing the paired parton Hamiltonian and are independent of the specific gap magnitude. Gap equations are not solved because the construction is mean-field by design. We will add a short sentence in the abstract noting that the central charges are obtained from the neutral edge spectrum. revision: partial

  3. Referee: [Abstract] The manuscript states that the same U(3) framework organizes the σ_xy=0 CDW and the reentrant period-three density order, but does not provide the explicit mean-field decoupling or the resulting charge-density modulation that would establish this unification beyond the level of symmetry assignment.

    Authors: The σ_xy=0 CDW arises by condensing the U(3) parton density operators at wave-vector 2π/3, which is worked out via mean-field decoupling in Section V; the resulting charge modulation is period-three and carries zero Hall conductivity by construction. The reentrant order at commensurate filling is likewise obtained from the same parton bilinears. We agree that a concise description of the decoupling and the explicit density profile would strengthen the abstract and will add one sentence summarizing the mean-field result. revision: yes

Circularity Check

0 steps flagged

No circularity detected in provided text

full rationale

The paper introduces a U(3) parton construction as the framework for analyzing the FCI and its superconducting descendants. The abstract describes the setup and states the resulting states, but supplies no equations, no fitted parameters renamed as predictions, and no self-citations that bear the central load. The topological orders are outputs of the chosen ansatz, which is presented transparently as the method rather than derived from an independent first-principles calculation that secretly reduces to the input. No load-bearing step reduces by construction to its own definition or to a prior self-citation. The derivation is therefore self-contained within the stated theoretical model.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the validity of the U(3) parton description for the Laughlin FCI and the assumption that parton pairing produces the listed topological orders; no independent evidence or machine-checked derivation is supplied in the abstract.

axioms (2)
  • domain assumption U(3) infrared parton theory accurately describes the low-energy physics of the ν=1/3 FCI
    The entire unification is built on this effective theory (abstract).
  • domain assumption Three charge-e/3 constituents form the electron and three constituent pairs form a gauge-invariant charge-2e Cooper pair
    Stated explicitly as the starting point of the construction.

pith-pipeline@v0.9.1-grok · 5760 in / 1755 out tokens · 37562 ms · 2026-06-29T09:30:20.108214+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Superconductivity and non-Fermi liquid metals in a charge-1/3 anyon fluid

    cond-mat.str-el 2026-06 unverdicted novelty 5.0

    Doping a fractional Chern insulator yields an anyon fluid that can form an SC* superconductor with residual Z2 order or a non-Fermi liquid Z3 orthogonal metal.

Reference graph

Works this paper leans on

60 extracted references · 28 canonical work pages · cited by 1 Pith paper · 15 internal anchors

  1. [1]

    J. M. Leinaas and J. Myrheim, Il Nuovo Cimento B37, 1 (1977)

    1977

  2. [2]

    Wilczek, Physical Review Letters49, 957 (1982)

    F. Wilczek, Physical Review Letters49, 957 (1982)

    1982

  3. [3]

    Arovas, J

    D. Arovas, J. R. Schrieffer, and F. Wilczek, Physical Review Letters53, 722 (1984)

    1984

  4. [4]

    R. B. Laughlin, Physical Review Letters60, 2677 (1988)

    1988

  5. [5]

    R. B. Laughlin, Science242, 525 (1988)

    1988

  6. [6]

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

    1989

  7. [7]

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

    1989

  8. [8]

    Lee and M

    D.-H. Lee and M. P. A. Fisher, Physical Review Letters 63, 903 (1989)

    1989

  9. [9]

    B. I. Halperin, Helvetica Physica Acta65, 215 (1992)

    1992

  10. [10]

    Tang, J.-W

    E. Tang, J.-W. Mei, and X.-G. Wen, Physical Review Letters106, 236802 (2011)

    2011

  11. [11]

    K. Sun, Z. Gu, H. Katsura, and S. Das Sarma, Physical Review Letters106, 236803 (2011)

    2011

  12. [12]

    Neupert, L

    T. Neupert, L. Santos, C. Chamon, and C. Mudry, Physi- cal Review Letters106, 236804 (2011)

    2011

  13. [13]

    D. N. Sheng, Z.-C. Gu, K. Sun, and L. Sheng, Nature Communications2, 389 (2011)

    2011

  14. [14]

    Regnault and B

    N. Regnault and B. A. Bernevig, Physical Review X1, 021014 (2011)

    2011

  15. [15]

    J. Cai, E. Anderson, C. Wang, X. Zhang, X. Liu, W. Holtz- mann, Y. Zhang, F. Fan, T. Taniguchi, K. Watanabe, Y. Ran, T. Cao, L. Fu, D. Xiao, W. Yao, and X. Xu, Nature622, 63 (2023)

    2023

  16. [16]

    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 X 13, 031037 (2023)

    2023

  17. [17]

    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)

    2023

  18. [18]

    Z. Lu, T. Han, Y. Yao, A. P. Reddy, J. Yang, J. Seo, K. Watanabe, T. Taniguchi, L. Fu, and L. Ju, Nature 626, 759 (2024), arXiv:2309.17436 [cond-mat.mes-hall]

    work page arXiv 2024

  19. [19]

    Waters, A

    D. Waters, A. Okounkova, R. Su, B. Zhou, J. Yao, K. Watanabe, T. Taniguchi, X. Xu, Y.-H. Zhang, J. Folk, and M. Yankowitz, Physical Review X15, 011045 (2025), arXiv:2408.10133 [cond-mat.mes-hall]

    work page arXiv 2025

  20. [20]

    F. Xu, Z. Sun, J. Li, C. Zheng, C. Xu, J. Gao, T. Jia, Y. Su, K. Watanabe, T. Taniguchi, B. Tong, L. Lu, J. Jia, Z. Shi, S. Jiang, J. Lin, Y. Zhang, Y. Zhang, S. Lei, X. Liu, and T. Li, Signatures of Unconventional Superconductivity near Reentrant and Fractional Quantum Anomalous Hall Insulators (2025), arXiv:2504.06972 [cond-mat.mes-hall]. 6

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  21. [21]

    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. Watanabe, T. Taniguchi, P. Xiong, D. M. Zumb¨ uhl, L. Fu, and L. Ju, Nature643, 654 (2025), arXiv:2408.15233 [cond-mat.mes- hall]

    work page arXiv 2025

  22. [22]

    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, Nature 639, 342 (2025), arXiv:2408.12584 [cond-mat.mes-hall]

    work page arXiv 2025

  23. [23]

    Wang and M

    T. Wang and M. P. Zaletel, Chiral Superconductivity near a Fractional Chern Insulator (2025), arXiv:2507.07921 [cond-mat.str-el]

    work page arXiv 2025

  24. [24]

    Guerci, A

    D. Guerci, A. Abouelkomsan, and L. Fu, Physical Re- view Letters135, 186601 (2025), arXiv:2506.10938 [cond- mat.str-el]

    work page arXiv 2025

  25. [25]

    Guerci, A

    D. Guerci, A. Abouelkomsan, and L. Fu, Topological Su- perconductivity with Emergent Vortex Lattice in Twisted Semiconductors (2026), arXiv:2602.15106 [cond-mat.supr- con]

    work page arXiv 2026

  26. [26]

    Z. D. Shi and T. Senthil, Physical Review X15, 031069 (2025), arXiv:2409.20567 [cond-mat.str-el]

    work page arXiv 2025

  27. [27]

    P. A. Nosov, Z. Han, and E. Khalaf, Physical Review Let- ters136, 106501 (2026), arXiv:2506.02108 [cond-mat.str- el]

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  28. [28]

    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 Sciences of the United States of America122, e2426680122 (2025), arXiv:2410.18175 [cond-mat.str-el]

    work page arXiv 2025

  29. [29]

    Pichler, C

    F. Pichler, C. Kuhlenkamp, M. Knap, and A. Vishwanath, Newton2, 100340 (2026), arXiv:2506.08000 [cond-mat.str- el]

    work page arXiv 2026

  30. [30]

    Topological Charge-2ne Superconductors

    Z.-Q. Gao, Y.-Q. Wang, H. Yang, and C. Wu, Topological Charge-2ne Superconductors (2025), arXiv:2512.21325 [cond-mat.str-el]

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  31. [31]

    Gao, Y.-Q

    Z.-Q. Gao, Y.-Q. Wang, Y.-H. Zhang, and H. Yang, Pri- mary Charge-4e Superconductivity from Doping a Fea- tureless Mott Insulator (2026), arXiv:2602.03925 [cond- mat.str-el]

    work page arXiv 2026

  32. [32]

    Ma and Y.-C

    R. Ma and Y.-C. He, Physical Review Research2, 033348 (2020)

    2020

  33. [33]

    P. W. Anderson, Science235, 1196 (1987)

    1987

  34. [34]

    Baskaran, Z

    G. Baskaran, Z. Zou, and P. W. Anderson, Solid State Communications63, 973 (1987)

    1987

  35. [35]

    Kotliar and J

    G. Kotliar and J. Liu, Physical Review B38, 5142 (1988)

    1988

  36. [36]

    Z_2 Gauge Theory of Electron Fractionalization in Strongly Correlated Systems

    T. Senthil and M. P. A. Fisher, Physical Review B62, 7850 (2000), arXiv:cond-mat/9910224

    work page internal anchor Pith review Pith/arXiv arXiv 2000

  37. [37]

    P. A. Lee, N. Nagaosa, and X.-G. Wen, Reviews of Modern Physics78, 17 (2006), arXiv:cond-mat/0410445

    work page internal anchor Pith review Pith/arXiv arXiv 2006

  38. [38]

    Quantum Orders and Symmetric Spin Liquids

    X.-G. Wen, Physical Review B65, 165113 (2002), arXiv:cond-mat/0107071

    work page internal anchor Pith review Pith/arXiv arXiv 2002

  39. [39]

    Zhang, Physical Review B113, 155110 (2026), arXiv:2506.00110 [cond-mat.str-el]

    Y.-H. Zhang, Physical Review B113, 155110 (2026), arXiv:2506.00110 [cond-mat.str-el]

    work page arXiv 2026

  40. [40]

    Projective Construction of Non-Abelian Quantum Hall Liquids

    X.-G. Wen, Physical Review B60, 8827 (1999), arXiv:cond-mat/9811111

    work page internal anchor Pith review Pith/arXiv arXiv 1999

  41. [41]

    In the quotient group of Eq

    In a lifted SU (3) ×U (1) notation, a charge- −3 Higgs might appear to leave a separate Z3 gauge group. In the quotient group of Eq. (3), that subgroup is identified with the SU (3) center. Determinant locking therefore leaves the local neutral algebra su(3) while preserving the correct U(3) flux lattice and line selection rule

  42. [42]

    Level/rank Duality and Chern-Simons-Matter Theories

    P.-S. Hsin and N. Seiberg, Journal of High Energy Physics 2016, 095 (2016), arXiv:1607.07457 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  43. [43]

    Commensurability, excitation gap and topology in quantum many-particle systems on a periodic lattice

    M. Oshikawa, Physical Review Letters84, 1535 (2000), arXiv:cond-mat/9911137

    work page internal anchor Pith review Pith/arXiv arXiv 2000

  44. [44]

    M. B. Hastings, Physical Review B69, 104431 (2004), arXiv:cond-mat/0305505

    work page internal anchor Pith review Pith/arXiv arXiv 2004

  45. [45]

    Filling constraints for spin-orbit coupled insulators in symmorphic and non-symmorphic crystals

    H. Watanabe, H. C. Po, A. Vishwanath, and M. P. Zale- tel, Proceedings of the National Academy of Sciences of the United States of America112, 14551 (2015), arXiv:1505.04193

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  46. [46]

    Paired states of fermions in two dimensions with breaking of parity and time-reversal symmetries, and the fractional quantum Hall effect

    N. Read and D. Green, Physical Review B61, 10267 (2000), arXiv:cond-mat/9906453

    work page internal anchor Pith review Pith/arXiv arXiv 2000

  47. [47]

    Witten, Communications in Mathematical Physics121, 351 (1989)

    E. Witten, Communications in Mathematical Physics121, 351 (1989)

    1989

  48. [48]

    Kitaev, Annals of Physics321, 2 (2006)

    A. Kitaev, Annals of Physics321, 2 (2006)

    2006

  49. [49]

    F. A. Bais, P. Bouwknegt, M. Surridge, and K. Schoutens, Nuclear Physics B304, 371 (1988)

    1988

  50. [50]

    Lacki and P

    J. Lacki and P. Zaugg, Physics Letters B228, 223 (1989)

    1989

  51. [51]

    Chern-Simons-matter dualities with $SO$ and $USp$ gauge groups

    O. Aharony, F. Benini, P.-S. Hsin, and N. Seiberg, Journal of High Energy Physics2017, 072 (2017), arXiv:1611.07874 [cond-mat.str-el]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  52. [52]

    Z. D. Shi and T. Senthil, Non-Abelian Topological Su- perconductivity from Melting Abelian Fractional Chern Insulators (2025), arXiv:2512.17996 [cond-mat.str-el]

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  53. [53]

    Z.-D. Fan, A. Vishwanath, and Z. Wang, Hidden Weak- Pairing Superconductivity of Non-Interacting Anyons Obeying 1 3 Statistics (2026), arXiv:2605.19036 [cond- mat.str-el]

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  54. [54]

    W. Ji, R. A. Lanzetta, Z. Zhou, and C. Wang, Self- Dual Higgs Transitions: Toric Code and Beyond (2026), arXiv:2601.20945 [cond-mat.str-el]

    work page arXiv 2026

  55. [55]

    Li and F

    H. Li and F. D. M. Haldane, Physical Review Letters 101, 010504 (2008)

    2008

  56. [56]

    M. P. Zaletel, R. S. K. Mong, and F. Pollmann, Physical Review Letters110, 236801 (2013)

    2013

  57. [57]

    Q. Niu, D. J. Thouless, and Y.-S. Wu, Physical Review B31, 3372 (1985)

    1985

  58. [58]

    D. J. Van Harlingen, Reviews of Modern Physics67, 515 (1995)

    1995

  59. [59]

    C. C. Tsuei and J. R. Kirtley, Reviews of Modern Physics 72, 969 (2000)

    2000

  60. [60]

    Banerjee, M

    M. Banerjee, M. Heiblum, V. Umansky, D. E. Feldman, Y. Oreg, and A. Stern, Nature559, 205 (2018). End Matter Magnetic translations and PSG—Here we justify the translation convention used in the main text. If the U(3) partons are taken as microscopic Hofstadter degrees of freedom, the magnetic-translation algebra must be kept explicitly. After determinant ...

    2018