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The paper proposes that doping SU(3)-symmetric spin liquids yields charge-6e superconductors, in which the condensate is a pair of charge-3e fermionic trions enforced by the Z3 center of SU(3).

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2026-08-01 01:08 UTC pith:FQYM4F23

load-bearing objection A serious field-theoretic construction of charge-6e superconductors from doped SU(3) spin liquids; the chiral routes hold up, but the time-reversal-symmetric branch rests on an undemonstrated Z3 QSL parent. the 3 major comments →

arxiv 2607.25909 v1 pith:FQYM4F23 submitted 2026-07-28 cond-mat.str-el cond-mat.mes-hallcond-mat.supr-con

Charge-6e superconductivity from doping SU(3) spin liquids

Yan-Qi Wang , Boran Zhou , Hui Yang , Zhi-Qiang Gao This is my paper
classification cond-mat.str-el cond-mat.mes-hallcond-mat.supr-con
keywords charge-6e superconductivitySU(3) spin liquidsZ3 quantum spin liquidchiral spin liquidsparton constructioncenter enforcementnon-Abelian topological orderflux quantization
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.

The paper proposes a route to superconductivity in which the fundamental condensed object carries charge 6e rather than 2e. The engine is the Z3 center of SU(3) symmetry: three electrons can form a charge-3e trion, but that trion is a fermion and cannot condense, so the minimal symmetry-preserving boson is a pair of trions with charge 6e. Using parton constructions, the authors show that doping three classes of SU(3) spin liquids—a Z3 quantum spin liquid and the chiral spin liquids SU(3)_1, SU(6)_1, and SU(3)_2—realizes this scenario, yielding both time-reversal-symmetric and chiral charge-6e superconductors, the last with non-Abelian h/(6e) vortices. If correct, the phases have distinctive experimental signatures: flux quantization in units of h/(6e) and Josephson oscillations at frequency 6eV/h, with no charge-2e or charge-4e order.

Core claim

The central claim is that the Z3 center of SU(3) enforces charge-6e superconductivity in doped SU(3)-symmetric spin liquids. In a bilayer triangular-lattice Hubbard model with SU(3) spin symmetry, doping a Z3 quantum spin liquid—obtained by condensing a holon trimer—produces a trion orthogonal metal whose charge-3e fermionic trions pair into a time-reversal-symmetric charge-6e superconductor. Doping SU(3)_1 and SU(6)_1 chiral spin liquids produces chiral charge-6e superconductors with and without residual Abelian topological order, respectively; doping the non-Abelian SU(3)_2 chiral spin liquid gives a chiral charge-6e superconductor intertwined with SO(3)_-3 topological order and non-Abelia

What carries the argument

The load-bearing mechanism is center enforcement by the Z3 center of SU(3). Because the center is Z3, an SU(3)-singlet bound state of three electrons (a trion) carries charge 3e and is fermionic; any lower-charge condensate would break SU(3), which is forbidden at finite temperature by the Mermin–Wagner theorem. The paper implements this in two complementary parton constructions: holon fields for layer-symmetric states, where a condensed holon trimer Higgs a U(1) gauge field down to Z3 and pairing of trions yields charge 6e; and bosonic spinon plus fermionic d partons for spin-symmetric chiral states, where the bosons at effective filling -1/2 or -1 form bosonic Laughlin or integer quantum H

Load-bearing premise

The load-bearing premise is that a Z3 quantum spin liquid—formed by condensation of a holon trimer—is actually the ground state of the bilayer SU(3) Hubbard model at intermediate doping, which the paper invokes through ring-exchange energetics but does not demonstrate; if that parent state does not exist, the time-reversal-symmetric charge-6e route loses its starting point.

What would settle it

A numerical tensor-network simulation of the bilayer SU(3) Hubbard model at intermediate doping that finds no holon-trimer condensate, or that finds dominant charge-2e pairing or a Fermi liquid upon doping, would falsify the central proposal. Experimentally, observing h/(2e) flux quantization or a 2eV/h Josephson frequency in a candidate material, rather than h/(6e) and 6eV/h, would rule out charge-6e superconductivity.

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

If this is right

  • Flux quantization in a charge-6e superconductor occurs in units of h/(6e), not h/(2e); a Little–Parks experiment on a ring would show period h/(6e).
  • Josephson tunneling between two such superconductors would oscillate at frequency 6eV/h, tripling the conventional 2eV/h AC Josephson frequency.
  • The time-reversal-symmetric route from the Z3 quantum spin liquid yields a charge-6e superconductor with no chiral edge structure, distinguishable from a conventional superconductor by the absence of charge-2e and charge-4e long-range order.
  • The chiral routes predict quantized thermal Hall response and edge modes whose structure depends on the parent chiral spin liquid: residual Abelian order for SU(3)_1 doping, no intrinsic topological order for SU(6)_1 doping, and non-Abelian SO(3)_-3 order with non-Abelian vortex fusion for SU(3)_2 doping.
  • Because the Z3 center mechanism requires only SU(3) rather than SU(4), charge-6e superconductivity could arise in systems with smaller symmetry than the previously studied route to charge-4e superconductivity.

Where Pith is reading between the lines

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

  • The same center-enforcement logic applied to SU(N) with odd N would predict minimal symmetry-preserving bosons of charge 2Ne (a pair of N-electron fermionic singlets), making charge-6e a special case of a larger odd-N family.
  • The chiral routes are less dependent on the contested Z3 quantum spin liquid parent: if the bilayer model fails to stabilize the Z3 QSL, doping the chiral spin liquid states could still yield charge-6e superconductivity, so the overall programme has independent legs.
  • Level-rank duality suggests exact SU(3) symmetry may not be required in a real material: any topological order admitting a U(1)_-3 or U(2)_-3,-6 description could serve as a parent, so charge-6e superconductivity might be searched for in kagome metals or moiré systems with threefold structure.
  • The intermediate trion orthogonal metal is a concrete finite-doping prediction: angle-resolved photoemission or quantum-oscillation probes could look for a Fermi surface with one-third of the expected volume, carried by charge-3e quasiparticles.

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

3 major / 5 minor

Summary. The paper proposes that doping SU(3)-symmetric spin liquids can produce superconductivity with charge 6e. The authors introduce a bilayer triangular-lattice SU(3) Hubbard model and use two parton constructions. In the layer-symmetric route, a holon-trimer condensate is assumed to produce a Z3 quantum spin liquid; pairing the resulting charge-3e fermionic trions gives a time-reversal-symmetric charge-6e superconductor. In the spin-symmetric routes, doping SU(3)_1, SU(6)_1, and SU(3)_2 chiral spin liquids is claimed to give chiral charge-6e superconductors with residual Abelian order, no intrinsic topological order, and SO(3)_{-3} non-Abelian order, respectively. The paper also lists several descendant phases and proposes experimental signatures such as h/(6e) flux quantization and 6eV/h Josephson oscillations. The Chern–Simons/K-matrix manipulations are standard and internally coherent, but the parent states and the required bosonic fractional quantum Hall states of the doped partons are largely assumed rather than derived.

Significance. If the parent-state assumptions are granted, the paper provides a coherent and fairly comprehensive classification of charge-6e superconducting phases from SU(3) spin liquids, extending the earlier charge-4e SU(4) program. It explicitly gives topological responses, K-matrices, chiral central charges, and anyon content for each route, and it identifies concrete experimental signatures. The center-enforcement idea — that the Z3 center of SU(3) forces the minimal symmetry-preserving charged boson to be 6e — is appealing and is made concrete. However, the central predictive claim is conditional on several unproven inputs: the existence of the Z3 QSL parent at intermediate doping, and the formation of bosonic Laughlin/IQH/U(2) states by the z partons upon doping. No energetics are given to show that charge-6e SC is selected over symmetry-breaking charge-2e SC or other competing orders. The strength is the internal consistency of the formal derivations; the weakness is the gap between the proposed phases and the microscopic model.

major comments (3)
  1. [Sec. III C and Eq. (10)] The time-reversal-symmetric charge-6e route is built on an assumed parent. The text states that holon-trimer condensation 'can in principle be favored at intermediate doping' by an SU(3) ring exchange, but no calculation supports this. Appendix B's x=1 parton mean-field (Fig. 1) finds a trimer VBS and CSLs, not a Z3 QSL, and no finite-doping effective model is derived. In Eq. (10) the 6e order parameter is an input: it is defined as three f-parton Cooper pairs times the square of the holon trimer. The paper should either derive the Z3 QSL (or give a concrete microscopic regime) or explicitly state that the Z3 branch is a conditional construction.
  2. [Sec. V and Secs. IV A–IV C] The doped phases are not shown to be energetically selected. At T=0 the SU(3) symmetry can be spontaneously broken, and the abstract itself lists SU(3)-breaking charge-2e superconductors as possible phases; the paper never computes why the charge-6e channel should win. Likewise, in Sec. IV A the 'more interesting possibility' that z forms a bosonic Laughlin state at ν=-1/2 is adopted because it is 'natural'; z condensation and pair condensation are listed as alternatives with no energy comparison. The authors concede in Sec. V that 'it would be valuable to establish the energetic stability' of the proposed phases. Without at least a qualitative energetic argument, the title's 'from doping SU(3) spin liquids' overstates the result.
  3. [Sec. IV A–C, Eqs. (18), (21), (28)] The chiral charge-6e SCs depend on assumed bosonic topological states of the doped z partons: a bosonic Laughlin state at ν=-1/2, a bosonic IQH state at total filling -2, and a U(2)_{-1,2} state. No microscopic mechanism or parameter regime is provided to favor these states over the condensed or pair-condensed alternatives also discussed. Since these inputs are the decisive steps that produce charge-6e order, the chiral results are formal classifications rather than predictions for Eq. (1). This should be stated clearly, or the missing step should be filled.
minor comments (5)
  1. [Eq. (9)] After the GL(2,Z) redefinition, the text says 'integrating out α' yields the response. The derivation actually requires integrating out a (or setting α=0 after the constraint dα=0) to obtain the (6/2π) α̃ dA term. The final result is correct, but the wording is confusing.
  2. [Eq. (3)] The term 'holon trimer' is misleading because h_σ creates two-electron states; the charge assignments of h and f after Eq. (3) should be stated explicitly so that the charge-3e trion operator is unambiguous.
  3. [Sec. IV A, after Eq. (19)] The chiral central charge is first given as -6 for the K-matrix sector and then as -3 for the total state. The distinction between the topological sector and the total central charge including the gravitational Chern–Simons term should be spelled out.
  4. [References] Reference [100] is listed as 'To appear' with no arXiv number. If a preprint exists, it should be cited properly; otherwise the note should be phrased as a personal communication.
  5. [General notation] Terms such as 'spin C connection' and 'spin-layer symmetric' are used without definition. A brief explanation in Sec. II or a footnote would improve readability.

Circularity Check

0 steps flagged

No significant circularity; central derivations are conditional topological calculations, while the main weakness is an unproven parent-state assumption, not a circular input.

full rationale

The paper does not fit a parameter and rename it a prediction, nor does it define the target result into its inputs. In the Z3-route, the charge-6e order parameter (Eq. 10) is written after the topological response derivation (Eqs. 8–9): the charge-6e result follows from a GL(2,Z) transformation of CS levels 2 (f-pairing) and 3 (holon-trimer condensate), so the 6e is a consequence of the assumed parent and pairing sectors, not a hidden fit. Similarly, the chiral routes derive 6e from Streda-filling arguments and explicit CS Lagrangians (Eqs. 14, 18–19, 21–22, 27–29); the bosonic Laughlin/IQH states are stated ansätze, not quantities fitted to the 6e outcome. The main vulnerability is not circularity but the unsupported existence of the parent Z3 QSL: Sec. III C says the trimer condensation “can in principle be favored at intermediate doping 0<x<1 by an SU(3) ring exchange” [52], and Sec. V concedes “It would be valuable to establish the energetic stability of the proposed charge-6e superconducting phases in microscopic models.” These are open assumptions; they weaken the claim’s support but do not make the derivation equal to its inputs. Self-citations to [13] (SU(N) center enforcement) and [11] (non-Abelian vortex properties) are present, but the center-enforcement argument is restated in the introduction using SU(3) representation theory, so the self-citation is not load-bearing. Under the required standard, no circular step can be exhibited from the paper’s own equations.

Axiom & Free-Parameter Ledger

0 free parameters · 8 axioms · 0 invented entities

No new fundamental particles, forces, or symmetries are proposed; the trions, holon trimers, and non-Abelian vortices are emergent composites/defects from the parton constructions. The paper's main postulates are the existence and form of parent topological states, captured in the axioms above.

axioms (8)
  • standard math SU(3) center Z3 enforces that the minimal SU(3)-singlet local boson is a pair of trions (charge 6e).
    Group theory of SU(3) representations; used throughout to motivate charge-6e SC.
  • domain assumption Parton decompositions c = h† f (Eq. 3) and c = z† d (Eq. 11) faithfully capture the low-energy Hilbert space with dynamical U(1) / U(2) gauge redundancy.
    Standard parton construction; validity depends on the Mott regime u>>t,J⊥.
  • ad hoc to paper A holon-trimer condensation produces an SU(3)-symmetric Z3 QSL at intermediate doping.
    Asserted in Sec. III C; no microscopic derivation or numerical evidence.
  • domain assumption The J-K3 model at x=1 realizes SU(3)_1 and SU(6)_1 CSLs for K3/J≈0.42–5.96.
    Supported only by variational parton mean-field (Appendix B), not exact methods.
  • domain assumption Upon doping, d fermions stay in their C=1 or C=2 Chern bands and the Streda formula relates flux and density; z bosons sit at effective filling ν=-1/2 or -1.
    Linear response assumption; underpins the doped-state constructions in Sec. IV.
  • ad hoc to paper The z bosons form a bosonic Laughlin state at ν=-1/2 (SU(3)_1 case) or a bosonic IQH state at total filling -2 (SU(6)_1 case).
    Chosen as 'natural' simplest states; not derived from interactions.
  • standard math Level-rank duality SU(3)_2 ⊠ sVec ≅ U(2)_-3,-6 + 12Ω_g (Refs. [84,85]).
    Known duality used to construct the non-Abelian doping route.
  • ad hoc to paper For the non-Abelian case, z_{l,s} forms a U(2)_-1,2 bosonic CS state.
    Described as 'the simplest choice'; consistent with symmetry but not derived.

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read the original abstract

We propose doping $SU(3)$-symmetric spin liquids as a route toward charge-$6e$ superconductivity. This generalizes the idea of constructing charge-$4e$ superconductivity from doped $SU(4)$-symmetric phases. As a concrete platform, we study a bilayer triangular-lattice Hubbard model with $SU(3)$ spin symmetry and interlayer antiferromagnetic exchange. Using complementary parton constructions, we analyze doped $\mathbb{Z}_3$ quantum spin liquid and $SU(3)$-related chiral spin liquids. Doping a $\mathbb{Z}_3$ quantum spin liquid can produce an orthogonal metal with a gauge invariant fermi surface of charge-$3e$ fermionic trions. Pairing these trions gives a time-reversal-symmetric charge-$6e$ superconductor. Doping Abelian $SU(3)_1$ and $SU(6)_1$ chiral spin liquids yields chiral charge-$6e$ superconductors with and without residual Abelian topological order, respectively. Doping a non-Abelian $SU(3)_2$ chiral spin liquid leads to a non-Abelian chiral charge-$6e$ superconductor intertwined with $SO(3)_{-3}$ topological order and supporting non-Abelian $h/(6e)$ superconducting vortices. We also identify several other phases, including $\mathbb{Z}_3$ orthogonal metal, quantum anomalous Hall (crystal) phases enriched by $\mathbb{Z}_3$ or $\mathbb{Z}_2$ topological order, $SU(3)$-breaking charge-$2e$ superconductors, composite fermi liquid coupled to non-Abelian gauge field, and descendant chiral spin liquids. Our results identify doped $SU(3)$ spin liquids as a natural setting where symmetry, fractionalization, and topology cooperate to produce charge-$6e$ superconductivity.

Figures

Figures reproduced from arXiv: 2607.25909 by Boran Zhou, Hui Yang, Yan-Qi Wang, Zhi-Qiang Gao.

Figure 1
Figure 1. Figure 1: FIG. 1. Mean-field phase diagram of Eq. ( [PITH_FULL_IMAGE:figures/full_fig_p011_1.png] view at source ↗

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Works this paper leans on

100 extracted references · 30 linked inside Pith

  1. [1]

    Babaev and B

    E. Babaev and B. Svistunov, Hydrodynamics of bor- romean counterfluids, Phys. Rev. Lett.133, 026001 (2024)

    2024

  2. [2]

    Zeng, L.-H

    M. Zeng, L.-H. Hu, H.-Y. Hu, Y.-Z. You, and C. Wu, High-order time-reversal symmetry breaking normal state, Science China Physics, Mechanics & Astronomy 67, 237411 (2024)

    2024

  3. [3]

    G. E. Volovik, Fermionic quartet and vestigial gravity, JETP Letters119, 330 (2024)

    2024

  4. [4]

    M. O. Soldini, M. H. Fischer, and T. Neupert, Charge- 4esuperconductivity in a hubbard model, Phys. Rev. B 109, 214509 (2024)

    2024

  5. [5]

    Dai and Q.-F

    Y.-X. Dai and Q.-F. Sun, Andreev reflection in normal metal/charge-4esuperconductor junctions, Phys. Rev. B109, 144504 (2024)

    2024

  6. [6]

    Y.-M. Wu, C. Murthy, and S. A. Kivelson, Possible slid- ing regimes in twisted bilayer wte 2, Phys. Rev. Lett. 133, 246501 (2024)

    2024

  7. [7]

    Samoilenka and E

    A. Samoilenka and E. Babaev, Microscopic theory of electron quadrupling condensates, Phys. Rev. Res.8, 013139 (2026)

    2026

  8. [8]

    Zou, Z.-Q

    X. Zou, Z.-Q. Wan, and H. Yao, Emergence of charge- 4esuperconductivity from 2d nematic superconductors (2025), arXiv:2510.26720 [cond-mat.supr-con]

    arXiv 2025

  9. [9]

    Z. D. Shi, Z. Han, S. Raghu, and A. Vishwanath, Charge-4esuperconductor with parafermionic vortices: A path to universal topological quantum computation (2026), arXiv:2602.06963 [cond-mat.str-el]

    arXiv 2026

  10. [10]

    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]

    Pith/arXiv arXiv 2025

  11. [11]

    Gao, Y.-Q

    Z.-Q. Gao, Y.-Q. Wang, H. Yang, and C. Wu, Topological charge-2ne superconductors (2026), arXiv:2512.21325 [cond-mat.str-el]

    Pith/arXiv arXiv 2026

  12. [12]

    Zhang, Y.-H

    L. Zhang, Y.-H. Zhang, and X.-Y. Song, Charge-4e anyon superconductor from doping SU(4) 1 chiral spin liquid (2025), arXiv:2508.12370 [cond-mat.str-el]

    Pith/arXiv arXiv 2025

  13. [13]

    Gao, Y.-Q

    Z.-Q. Gao, Y.-Q. Wang, Y.-H. Zhang, and H. Yang, Primary charge-4e superconductivity from doping a fea- tureless mott insulator (2026), arXiv:2602.03925 [cond- mat.str-el]

    arXiv 2026

  14. [14]

    S.-H. Shi, Z. Wu, J. Hu, and Z.-X. Li, High- temperature charge-4e superconductivity in su(4) inter- acting fermions (2026), arXiv:2604.15056 [cond-mat.str- el]

    Pith/arXiv arXiv 2026

  15. [15]

    Z.-Q. Wan, H. Jiang, X. Zou, S. Zhang, and S.-K. Jian, Quantum charge-4e superconductivity and deconfined pseudocriticality in the attractive su(4) hubbard model (2026), arXiv:2604.14289 [cond-mat.str-el]

    Pith/arXiv arXiv 2026

  16. [16]

    C.-T. Gao, C. Lu, Y.-B. Liu, Z. Pan, and F. Yang, Charge-4e/6e superconductivity and chiral metal from 3d chiral superconductor (2026), arXiv:2604.12328 [cond-mat.supr-con]

    Pith/arXiv arXiv 2026

  17. [17]

    E. Berg, E. Fradkin, and S. A. Kivelson, Charge-4e su- perconductivity from pair-density-wave order in certain high-temperature superconductors, Nature Physics5, 830 (2009)

    2009

  18. [18]

    Radzihovsky and A

    L. Radzihovsky and A. Vishwanath, Quantum liquid crystals in an imbalanced fermi gas: Fluctuations and fractional vortices in larkin-ovchinnikov states, Phys. Rev. Lett.103, 010404 (2009)

    2009

  19. [19]

    Radzihovsky, Fluctuations and phase transitions in larkin-ovchinnikov liquid-crystal states of a population- imbalanced resonant fermi gas, Phys

    L. Radzihovsky, Fluctuations and phase transitions in larkin-ovchinnikov liquid-crystal states of a population- imbalanced resonant fermi gas, Phys. Rev. A84, 023611 (2011)

    2011

  20. [20]

    Dou¸ cot and J

    B. Dou¸ cot and J. Vidal, Pairing of cooper pairs in a fully frustrated josephson-junction chain, Phys. Rev. Lett. 88, 227005 (2002)

    2002

  21. [21]

    Babaev, A

    E. Babaev, A. Sudbø, and N. W. Ashcroft, A supercon- ductor to superfluid phase transition in liquid metallic hydrogen, Nature431, 666–668 (2004)

    2004

  22. [22]

    Radzihovsky, J

    L. Radzihovsky, J. Park, and P. B. Weichman, Su- perfluid transitions in bosonic atom-molecule mixtures near a feshbach resonance, Phys. Rev. Lett.92, 160402 (2004)

    2004

  23. [23]

    Radzihovsky, P

    L. Radzihovsky, P. B. Weichman, and J. I. Park, Su- perfluidity and phase transitions in a resonant bose gas, Annals of Physics323, 2376–2451 (2008)

    2008

  24. [24]

    E. V. Herland, E. Babaev, and A. Sudbø, Phase transi- tions in a three dimensionalu(1)×u(1) lattice london superconductor: Metallic superfluid and charge-4esu- perconducting states, Phys. Rev. B82, 134511 (2010)

    2010

  25. [25]

    Zhou and Z

    S. Zhou and Z. Wang, Chern fermi pocket, topologi- cal pair density wave, and charge-4e and charge-6e su- perconductivity in kagom´ e superconductors, Nat. Com- mun.13, 7288 (2022)

    2022

  26. [26]

    S.-K. Jian, Y. Huang, and H. Yao, Charge-4esuper- conductivity from nematic superconductors in two and three dimensions, Phys. Rev. Lett.127, 227001 (2021)

    2021

  27. [27]

    R. M. Fernandes and L. Fu, Charge-4e superconductiv- ity from multicomponent nematic pairing: Application to twisted bilayer graphene, Physical Review Letters 127, 047001 (2021)

    2021

  28. [28]

    Grinenko, D

    V. Grinenko, D. Weston, F. Caglieris, C. Wuttke, C. Hess, T. Gottschall, I. Maccari, D. Gorbunov, S. Zherlitsyn, J. Wosnitza, A. Rydh, K. Kihou, C.- 9 H. Lee, R. Sarkar, S. Dengre, J. Garaud, A. Char- nukha, R. H¨ uhne, K. Nielsch, B. B¨ uchner, H.-H. Klauss, and E. Babaev, State with spontaneously broken time- reversal symmetry above the superconducting ...

    2021

  29. [29]

    Gao, Y.-T

    Z.-Q. Gao, Y.-T. Huang, and D.-H. Lee, Fractional vortices,𭟋 2 gauge theory, and the confinement- deconfinement transition, Phys. Rev. B106, L121105 (2022)

    2022

  30. [30]

    Fetter, C

    A. Fetter, C. Hanna, and R. Laughlin, Random-phase approximation in the fractional-statistics gas, Physical Review B39, 9679 (1989)

    1989

  31. [31]

    R. B. Laughlin, Superconducting ground state of non- interacting particles obeying fractional statistics, Phys. Rev. Lett.60, 2677 (1988)

    1988

  32. [32]

    Lee and M

    D.-H. Lee and M. P. A. Fisher, Anyon superconductivity and the fractional quantum hall effect, Phys. Rev. Lett. 63, 903 (1989)

    1989

  33. [33]

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

    1989

  34. [34]

    Divic, V

    S. Divic, V. Cr´ epel, T. Soejima, X.-Y. Song, A. J. Millis, M. P. Zaletel, and A. Vishwanath, Anyon su- perconductivity from topological criticality in a hof- stadter–hubbard model, Proceedings of the National Academy of Sciences122, e2426680122 (2025)

    2025

  35. [35]

    Pichler, C

    F. Pichler, C. Kuhlenkamp, M. Knap, and A. Vish- wanath, Microscopic mechanism of anyon superconduc- tivity emerging from fractional chern insulators, Newton 2, 100340 (2025)

    2025

  36. [36]

    Z. D. Shi and T. Senthil, Doping a fractional quan- tum anomalous hall insulator, Phys. Rev. X15, 031069 (2025)

    2025

  37. [37]

    Z. D. Shi, C. Zhang, and T. Senthil, Doping lattice non- abelian quantum hall states (2025), arXiv:2505.02893 [cond-mat.str-el]

    Pith/arXiv arXiv 2025

  38. [38]

    Kuhlenkamp, S

    C. Kuhlenkamp, S. Divic, M. P. Zaletel, T. Soe- jima, and A. Vishwanath, Robust superconductivity upon doping chiral spin liquid and chern insulators in a hubbard-hofstadter model (2025), arXiv:2509.02675 [cond-mat.str-el]

    Pith/arXiv arXiv 2025

  39. [39]

    Wu, Competing orders in one dimensional spin 3/2 fermionic systems, Phys

    C. Wu, Competing orders in one dimensional spin 3/2 fermionic systems, Phys. Rev. Lett.95, 266404 (2005)

    2005

  40. [40]

    Lecheminant, E

    P. Lecheminant, E. Boulat, and P. Azaria, Confine- ment and superfluidity in one-dimensional degenerate fermionic cold atoms, Phys. Rev. Lett.95, 240402 (2005)

    2005

  41. [41]

    Capponi, G

    S. Capponi, G. Roux, P. Azaria, E. Boulat, and P. Le- cheminant, Confinement versus deconfinement of cooper pairs in one-dimensional spin-3/2 fermionic cold atoms, Phys. Rev. B75, 100503 (2007)

    2007

  42. [42]

    Capponi, G

    S. Capponi, G. Roux, P. Lecheminant, P. Azaria, E. Boulat, and S. R. White, Molecular superfluid phase in systems of one-dimensional multicomponent fermionic cold atoms, Phys. Rev. A77, 013624 (2008)

    2008

  43. [43]

    G. Roux, S. Capponi, P. Lecheminant, and P. Azaria, Spin 3/2 fermions with attractive interactions in a one- dimensional optical lattice: phase diagrams, entangle- ment entropy, and the effect of the trap, The European Physical Journal B68, 293–308 (2008)

    2008

  44. [44]

    W.-H. Ko, P. A. Lee, and X.-G. Wen, Doped kagome system as exotic superconductor, Phys. Rev. B79, 214502 (2009)

    2009

  45. [45]

    Zhang and D

    Y.-H. Zhang and D. Mao, Spin liquids and pseudogap metals in the su(4) hubbard model in a moir´ e superlat- tice, Phys. Rev. B101, 035122 (2020)

    2020

  46. [46]

    Khalaf, P

    E. Khalaf, P. Ledwith, and A. Vishwanath, Symme- try constraints on superconductivity in twisted bi- layer graphene: Fractional vortices, 4econdensates, or nonunitary pairing, Phys. Rev. B105, 224508 (2022)

    2022

  47. [47]

    C˘ alug˘ aru, Y

    D. C˘ alug˘ aru, Y. Jiang, H. Hu, H. Pi, J. Yu, M. G. Vergniory, J. Shan, C. Felser, L. M. Schoop, D. K. Efe- tov, K. F. Mak, and B. A. Bernevig, Moir´ e materials based on m-point twisting, Nature643, 376–381 (2025)

    2025

  48. [48]

    Vasiliou, D

    K. Vasiliou, D. C˘ alug˘ aru, J. S. Hofmann, and S. A. Parameswaran, Hidden antiferromagnetism, persistent valley fluctuations, andu(6) crossovers in triangular- lattice m-point moir´ e materials via determinantal quantum monte carlo (2026), arXiv:2606.12530 [cond- mat.str-el]

    Pith/arXiv arXiv 2026

  49. [49]

    C˘ alug˘ aru, K

    D. C˘ alug˘ aru, K. Vasiliou, H. Hu, B. A. Bernevig, W. Krauth, and S. A. Parameswaran, Mixed- dimensional quantum monte carlo studies of m-point moir´ e materials (2026), arXiv:2606.12508 [cond-mat.str- el]

    Pith/arXiv arXiv 2026

  50. [50]

    J. Ge, P. Wang, Y. Xing, Q. Yin, A. Wang, J. Shen, H. Lei, Z. Wang, and J. Wang, Charge-4eand charge-6e flux quantization and higher charge superconductivity in kagome superconductor ring devices, Phys. Rev. X 14, 021025 (2024)

    2024

  51. [51]

    Zhang, Holon metal, charge density wave metal, and chiral superconductor from doping a fractional chern insulator and an su(3) 1 chiral spin liquid, Phys

    Y.-H. Zhang, Holon metal, charge density wave metal, and chiral superconductor from doping a fractional chern insulator and an su(3) 1 chiral spin liquid, Phys. Rev. B113, 155110 (2026)

    2026

  52. [52]

    Lai, Possible uniform-flux chiral spin liquid states in the su(3) ring-exchange model on the triangular lat- tice, Phys

    H.-H. Lai, Possible uniform-flux chiral spin liquid states in the su(3) ring-exchange model on the triangular lat- tice, Phys. Rev. B87, 205111 (2013)

    2013

  53. [53]

    H. Oh, H. Yang, and Y.-H. Zhang, High-temperature superconductivity from kinetic energy (2025), arXiv:2411.07292 [cond-mat.str-el]

    Pith/arXiv arXiv 2025

  54. [54]

    H. Yang, H. Oh, and Y.-H. Zhang, Strong pairing and symmetric pseudogap metal in a double kondo lattice model: From a nickelate superconductor to a tetralayer optical lattice, Phys. Rev. B111, L241102 (2025)

    2025

  55. [55]

    H. Oh, B. Zhou, and Y.-H. Zhang, Type-iit−jmodel in charge transfer regime in bilayer la3ni2o7 and trilayer la4ni3o10, Phys. Rev. B111, L020504 (2025)

    2025

  56. [56]

    H. Oh, H. Yang, and Y.-H. Zhang, Doping a spin-one mott insulator: possible application to bilayer nickelate (2025), arXiv:2509.02673 [cond-mat.str-el]

    Pith/arXiv arXiv 2025

  57. [57]

    P. A. Lee, N. Nagaosa, and X.-G. Wen, Doping a mott insulator: Physics of high-temperature superconductiv- ity, Rev. Mod. Phys.78, 17 (2006)

    2006

  58. [58]

    Nandkishore, M

    R. Nandkishore, M. A. Metlitski, and T. Senthil, Or- thogonal metals: The simplest non-fermi liquids, Phys. Rev. B86, 045128 (2012)

    2012

  59. [59]

    M. A. Metlitski and S. Sachdev, Quantum phase tran- sitions of metals in two spatial dimensions. i. ising- nematic order, Phys. Rev. B82, 075127 (2010)

    2010

  60. [60]

    M. A. Metlitski and S. Sachdev, Quantum phase transi- tions of metals in two spatial dimensions. ii. spin density wave order, Phys. Rev. B82, 075128 (2010)

    2010

  61. [61]

    S. A. Hartnoll, D. M. Hofman, M. A. Metlitski, and S. Sachdev, Quantum critical response at the onset of spin-density-wave order in two-dimensional metals, Phys. Rev. B84, 125115 (2011). 10

    2011

  62. [62]

    A. A. Patel and S. Sachdev, dc resistivity at the onset of spin density wave order in two-dimensional metals, Phys. Rev. B90, 165146 (2014)

    2014

  63. [63]

    Sachdev and D

    S. Sachdev and D. Chowdhury, The novel metallic states of the cuprates: Topological fermi liquids and strange metals, Progress of Theoretical and Experimen- tal Physics2016, 12C102 (2016)

    2016

  64. [64]

    Zhang and S

    Y.-H. Zhang and S. Sachdev, Deconfined criticality and ghost fermi surfaces at the onset of antiferromagnetism in a metal, Phys. Rev. B102, 155124 (2020)

    2020

  65. [65]

    A. A. Patel, H. Guo, I. Esterlis, and S. Sachdev, Uni- versal theory of strange metals from spatially random interactions, Science381, 790 (2023)

    2023

  66. [66]

    H. Xu, X. Li, Z. Zhou, X. Wang, L. Wang, C. Wu, and Y. Wang, Trion states and quantum criticality of at- tractive su(3) dirac fermions, Phys. Rev. Res.5, 023180 (2023)

    2023

  67. [67]

    Stepp, E

    J. Stepp, E. Ibarra-Garc ´ ıa-Padilla, R. T. Scalettar, and K. R. A. Hazzard, Trion formation and ordering in the attractive su(3) fermi-hubbard model (2026), arXiv:2506.12300 [cond-mat.quant-gas]

    Pith/arXiv arXiv 2026

  68. [68]

    Z. D. Shi and T. Senthil, Superconductivity and non- fermi liquid metals in a charge-1/3 anyon fluid (2026), arXiv:2606.20403 [cond-mat.str-el]

    Pith/arXiv arXiv 2026

  69. [69]

    Senthil, Fractionalized metals from doped anyons: Application to tmote2 (2026), arXiv:2607.11484 [cond- mat.str-el]

    T. Senthil, Fractionalized metals from doped anyons: Application to tmote2 (2026), arXiv:2607.11484 [cond- mat.str-el]

    Pith/arXiv arXiv 2026

  70. [70]

    Z. Han, A. Vishwanath, and E. Khalaf, Single- component twistedZ 3 orthogonal metal in ane/3-anyon fluid (2026), arXiv:2607.15274 [cond-mat.str-el]

    Pith/arXiv arXiv 2026

  71. [71]

    Zhang, Color superconductors and holon met- als from doping a fractional chern insulator (2026), arXiv:2607.18238 [cond-mat.str-el]

    Y.-H. Zhang, Color superconductors and holon met- als from doping a fractional chern insulator (2026), arXiv:2607.18238 [cond-mat.str-el]

    Pith/arXiv arXiv 2026

  72. [72]

    Mehta, Y

    U. Mehta, Y. Nakajima, and H. Goldman, Coloring in anyon superconductivity (2026), arXiv:2607.19470 [cond-mat.str-el]

    Pith/arXiv arXiv 2026

  73. [73]

    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 quan- tum anomalous hall insulators (2026), arXiv:2504.06972 [cond-mat.mes-hall]

    Pith/arXiv arXiv 2026

  74. [74]

    Dijkgraaf and E

    R. Dijkgraaf and E. Witten, Topological gauge theories and group cohomology, Communications in Mathemat- ical Physics129, 393 (1990)

    1990

  75. [75]

    C. Boos, C. J. Ganahl, M. Lajk´ o, P. Nataf, A. M. L¨ auchli, K. Penc, K. P. Schmidt, and F. Mila, Time- reversal symmetry breaking abelian chiral spin liquid in mott phases of three-component fermions on the trian- gular lattice, Phys. Rev. Res.2, 023098 (2020)

    2020

  76. [76]

    Timmel and X.-G

    A. Timmel and X.-G. Wen, Non-abelian fibonacci quantum hall states in 4-layer rhombohedral stacked graphene (2023), arXiv:2308.09702 [cond-mat.mes-hall]

    Pith/arXiv arXiv 2023

  77. [77]

    Z.-C. Gu, Z. Wang, and X.-G. Wen, Lattice model for fermionic toric code, Phys. Rev. B90, 085140 (2014)

    2014

  78. [78]

    Here only the UV symmetry layerU(1) z is strictly pre- served

  79. [79]

    The standard form of theSU(6) 1 CSL without doping can be similarly derived by first integrating outain Eq. (13)

  80. [80]

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

    1993

Showing first 80 references.