Pith. sign in

REVIEW 2 major objections 5 minor 77 references

Quantum geometry of a lattice Chern band can turn FCI and anti-FCI scatterings into a Josephson coupling that drives superconductivity next to a fractional Chern insulator.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

T0 review · grok-4.5

2026-07-14 17:38 UTC pith:BUM363N7

load-bearing objection Clean analytic mechanism: FCI × aFCI OPE generates Josephson coupling and an SC lobe next to FCI, with Tc rising with the geometric ratio r; two-wire anisotropic limit is the main caveat. the 2 major comments →

arxiv 2606.21166 v2 pith:BUM363N7 submitted 2026-06-19 cond-mat.supr-con cond-mat.str-el

Quantum-Geometry-Induced Superconductivity near a Fractional Chern Insulator

Haoyu Hu , Lei Chen This is my paper
classification cond-mat.supr-con cond-mat.str-el PACS 74.20.Mn73.43.Nq71.10.Pm
keywords fractional Chern insulatorquantum geometryanti-FCIcoupled-wire constructionbosonizationsuperconductivitycharge-density waverenormalization group
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.

Partially filled Chern bands host fractional Chern insulators, charge-density waves, and superconductivity as nearby phases, yet how they compete and why superconductivity appears in lattice systems (but not in the continuum Landau-level limit) has been unclear. This paper gives an analytic answer in the anisotropic limit by treating the band as coupled wires and bosonizing. The lattice geometry supplies a second scattering channel, the anti-FCI process, whose strength tracks the quantum metric. When both FCI and anti-FCI channels are active, their operator product expansion generates an effective Josephson coupling between neighboring wires; the same fusion can produce charge-density-wave locking. Renormalization-group flow then shows a superconducting regime sitting next to the FCI, with the estimated transition temperature rising as the quantum geometry is enhanced. The same mechanism works at filling 2/3 after particle-hole conjugation. The result supplies a microscopic reason why quantum geometry can organize the competition among FCI, superconductivity, and charge order in moiré Chern bands.

Core claim

When both the fractional-Chern-insulator and anti-FCI scattering channels are present in a partially filled Chern band, their operator-product expansion generates an effective Josephson coupling that drives a superconducting instability adjacent to the FCI phase; the same fusion can stabilize charge-density-wave order, and the superconducting temperature scale is enhanced by the quantum geometry that strengthens the anti-FCI channel.

What carries the argument

Operator product expansion of the FCI and anti-FCI vertex operators: fusing e^{i\Theta_FCI} with e^{\pm i\Theta_aFCI} produces the superconducting (Josephson) and charge-density-wave vertices whose relevance is then tracked by a two-wire perturbative renormalization-group flow.

Load-bearing premise

The strongly anisotropic two-wire limit is assumed to be enough to capture the bulk competition among FCI, anti-FCI, superconductivity and charge-density-wave channels.

What would settle it

In a microscopic lattice Chern-band model, raise the ratio r = t'_x / t_x that controls quantum geometry while holding filling at 1/3; if the superconducting temperature scale does not rise and no superconducting regime appears next to the FCI in the phase diagram, the claimed mechanism fails.

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

If this is right

  • Superconductivity is expected next to an FCI whenever the lattice Chern band carries a sizable quantum metric that activates the anti-FCI channel.
  • Increasing the quantum geometry (larger r) should raise the estimated superconducting transition temperature.
  • The same FCI–anti-FCI fusion produces charge-density-wave order for other ranges of Luttinger parameters.
  • At filling 2/3 the analogous fusion yields next-nearest-neighbor Josephson coupling and a related superconducting instability.

Where Pith is reading between the lines

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

  • Materials platforms that already show both FCI and superconductivity (twisted MoTe2, rhombohedral graphene multilayers) may be testing this quantum-geometry-driven Josephson mechanism.
  • If the anti-FCI channel can be tuned independently of the FCI gap, one could map a continuous FCI-to-SC transition controlled by a single geometric parameter.
  • The same OPE logic may generate pair-density-wave or other intertwined orders once longer-range wire couplings are restored.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 5 minor

Summary. The manuscript presents an analytic coupled-wire/bosonization study of a partially filled C=1 Chern band at ν=1/3 (and briefly ν=2/3). In the anisotropic limit, quantum geometry is tuned by r=t'_x/t_x, which generates an aFCI scattering channel alongside the usual FCI channel. Operator product expansion of the FCI and aFCI vertices produces Josephson (SC) and CDW couplings; one-loop RG in a two-wire truncation then yields a phase diagram with SC and CDW regimes adjacent to FCI, and an estimated T_SC that rises with r. The appendices supply the microscopic generation of g_FCI and g_aFCI from U and inter-wire hoppings, the bosonization dictionary, and the full RG equations.

Significance. If the controlled anisotropic construction is accepted, the work supplies a concrete microscopic mechanism—FCI × aFCI fusion into a Josephson vertex—linking lattice quantum geometry to superconductivity near an FCI, a connection suggested by recent moiré experiments and numerics but previously lacking an analytic derivation. Strengths include: (i) explicit perturbative generation of g_aFCI ∝ r from the microscopic Hamiltonian (App. V); (ii) standard but carefully written OPE and one-loop RG (Eqs. 7–9, 11; App. VI) in which SC/CDW are generated dynamically rather than inserted by hand; (iii) a falsifiable trend that T_SC increases with the geometric parameter r (Fig. 4b). The result is a useful organizing principle for the FCI–SC–CDW interplay in Chern bands away from the LLL limit.

major comments (2)
  1. The phase diagram and T_SC estimate (Fig. 4; “Perturbative RG”; App. VI) are obtained in a two-wire truncation with open boundaries. While this is standard and sufficient to show that the FCI×aFCI product generates a relevant Josephson vertex, the manuscript should state more carefully what is thereby established for a bulk multi-wire array: nearest-neighbor Josephson locking can produce 2D phase coherence, but the two-wire gap and critical exponents need not coincide with those of the infinite array. A short paragraph quantifying this limitation (or citing multi-wire ladder results) would make the central claim’s scope precise without changing the derivation.
  2. Eq. (2) and Fig. 2 show that Q_xx diverges logarithmically as r o1, where the indirect gap closes. The T_SC enhancement with r (Fig. 4b) is therefore meaningful only inside the window where the anisotropic C=1 band remains gapped and the wire construction is controlled (ty=M≫|tx|>|t'_x|). The text should mark the range of r used in Fig. 4b and note that the geometric boost saturates or the description fails before the gap-closing point; otherwise the claim that quantum geometry generically enhances T_SC can be over-read.
minor comments (5)
  1. Abstract and Introduction: “anti-FCI (aFCI)” is introduced as a named phase; a one-sentence clarification that aFCI is the lattice-specific scattering channel of Ref. [31] (not necessarily a stable bulk phase in the present RG) would avoid confusion.
  2. Eq. (5) and App. V: the proportionality g_aFCI∝g_FCI r is clear, but the common prefactor involving Q_yy(k_F)/E_H^{2} is left schematic; a brief estimate of its magnitude relative to the bandwidth would help readers judge whether the bare y_FCI=0.1 used in the RG is realistic.
  3. Fig. 4(a): the axes (K_e,0, K_o,0) and the color/label scheme for FCI/SC/CDW/gapless regions should be stated explicitly in the caption; currently one must infer them from the main text.
  4. Filling ν=2/3 (final section): the next-nearest-neighbor character of the induced Josephson coupling is interesting but undeveloped; either a short RG remark or a clear statement that a full analysis is left for future work would improve balance.
  5. Typos/notation: “moiré” accents are inconsistent in the abstract vs. body; “nu=2/3” in the App. VII heading should be “ν=2/3”; Klein-factor products in Eqs. (S100)–(S108) would benefit from a uniform ordering convention.

Circularity Check

0 steps flagged

No significant circularity: SC/CDW vertices and RG product terms are generated dynamically from microscopically derived FCI/aFCI couplings; nothing is fitted to the target or forced by self-citation.

full rationale

The paper constructs a minimal anisotropic Chern-band model (Eq. 1), projects on-site U plus inter-wire hoppings tx, t'x to obtain gFCI ∝ U^{2} Qyy(kF) tx/EH^{2} and gaFCI ∝ gFCI r (Eq. 5 and App. V), bosonizes the resulting vertices (Eq. 6), and shows by ordinary OPE that their product produces Josephson (SC) and CDW operators (Eqs. 7–9). The second-order RG equations (Eq. 11 / App. VI) then generate ySC and yCDW from the product yFCI yaFCI even when the bare SC/CDW couplings vanish; the phase diagram and TSC(r) follow by integrating those flows. All steps are standard bosonization/RG manipulations inside a controlled two-wire anisotropic limit that is stated explicitly. No parameter is fitted to SC or CDW data, no uniqueness theorem is imported from the authors’ prior work, and the sole external reference for the aFCI channel ([31]) is independent. The derivation is therefore self-contained and non-circular.

Axiom & Free-Parameter Ledger

3 free parameters · 4 axioms · 1 invented entities

The central claim rests on the validity of the anisotropic coupled-wire limit, the microscopic generation of aFCI by t′x, the two-wire truncation of the RG, and the standard bosonization/OPE machinery. Free parameters are the bare Luttinger parameters and the ratio of bare couplings chosen for the phase-diagram scan; no experimental fitting is performed. aFCI is not invented here but is elevated to a dynamical player.

free parameters (3)
  • r = t′x / tx
    Geometric control parameter that sets the relative strength gaFCI / gFCI and is scanned to show Tc enhancement; chosen by hand in the model.
  • bare yFCI(l=0) = 0.1, yaFCI(l=0) = 0.05
    Initial dimensionless couplings used to generate the RG phase diagram (Fig. 4); values chosen for illustration, not fitted to data.
  • Ke,0 , Ko,0
    Initial Luttinger parameters that determine scaling dimensions and which instability wins; scanned over a range of repulsive values.
axioms (4)
  • domain assumption Strongly anisotropic limit ty = M ≫ |tx| > |t′x| allows a controlled coupled-wire description of the Chern band.
    Stated in “Model and quantum geometry” and used throughout; without it the continuum Landau-level limit is recovered and aFCI vanishes.
  • ad hoc to paper Two-wire truncation of the multi-wire array is sufficient to capture the competition among FCI, aFCI, SC and CDW.
    Explicitly adopted in “Perturbative RG” and Appendix VI for analytic tractability; multi-wire effects are not controlled.
  • standard math Standard Abelian bosonization and one-loop operator-product expansions remain valid for the partially filled Chern band.
    Used to obtain Eqs. (6)–(9) and the RG flow (11); textbook technique for 1D systems.
  • domain assumption On-site inter-orbital repulsion U plus inter-wire hoppings generate the FCI and aFCI vertices at third order in perturbation theory.
    Derived in Appendix V; assumes high-energy electrons can be integrated out at energy EH.
invented entities (1)
  • anti-FCI (aFCI) scattering channel as a dynamical competitor that fuses with FCI into Josephson coupling no independent evidence
    purpose: Provides the second interaction vertex whose product with FCI generates the SC and CDW instabilities under RG.
    The channel itself was identified in Ref. [31]; the present work elevates it to the generator of SC via OPE and shows its geometric enhancement. Independent evidence is the microscopic derivation from t′x, but no external experimental confirmation of the fusion mechanism is given.

pith-pipeline@v1.1.0-grok45 · 36476 in / 3088 out tokens · 30157 ms · 2026-07-14T17:38:17.379551+00:00 · methodology

0 comments
read the original abstract

Recent moir\'e experiments and numerical studies of interacting Chern bands have revealed fractional Chern insulators, charge-density-wave order, and superconductivity as proximate correlation-driven phases in topological systems. How these phases compete or intertwine, and how quantum geometry shapes their interplay, remain open questions. Here we present an analytic study of competing correlation-driven phases in a partially filled Chern band using a coupled-wire construction and bosonization. The key ingredient is the coexistence of interaction channels that favor, respectively, a fractional Chern insulator (FCI) and a closely related anti-FCI (aFCI) state. The aFCI channel is specific to lattice Chern bands and is enhanced by the quantum geometry of the underlying band structure. We show that when both FCI and aFCI scattering channels are present, their interplay generates an effective coupling that drives a superconducting instability near the FCI phase. The same mechanism can also favor a charge-density-wave phase, depending on microscopic parameters. Using a perturbative renormalization-group analysis, we obtain the phase diagram and identify a superconducting regime adjacent to the FCI phase. We further estimate the superconducting transition temperature and show that it is enhanced by quantum geometry. Our results establish quantum geometry as an organizing principle for the interplay among FCI, aFCI, and superconducting correlations.

Figures

Figures reproduced from arXiv: 2606.21166 by Haoyu Hu, Lei Chen.

Figure 1
Figure 1. Figure 1: FIG. 1. (a) Two nearest-neighbor inter-wire hopping processes with [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. (a) Scattering processes associated with FCI, aFCI, and [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. RG phase diagram and estimated SC transition temperature. [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗

Review history (2 revisions) →

discussion (0)

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

Reference graph

Works this paper leans on

77 extracted references · 3 linked inside Pith

  1. [1]

    Fractional chern insula- tor,

    N. Regnault and B. Andrei Bernevig, “Fractional chern insula- tor,” Phys. Rev. X1, 021014 (2011)

    2011

  2. [2]

    Frac- tional quantum hall effect in the absence of landau levels,

    D. N. Sheng, Zheng-Cheng Gu, Kai Sun, and L. Sheng, “Frac- tional quantum hall effect in the absence of landau levels,” Na- ture Communications2, 389 (2011)

    2011

  3. [3]

    Fractional quantum hall physics in topological flat bands,

    Siddharth A. Parameswaran, Rahul Roy, and Shivaji L. Sondhi, “Fractional quantum hall physics in topological flat bands,” Comptes Rendus Physique14, 816–839 (2013), topological in- sulators / Isolants topologiques

    2013

  4. [4]

    High- temperature fractional quantum hall states,

    Evelyn Tang, Jia-Wei Mei, and Xiao-Gang Wen, “High- temperature fractional quantum hall states,” Phys. Rev. Lett. 106, 236802 (2011)

    2011

  5. [5]

    Nearly flatbands with nontrivial topology,

    Kai Sun, Zhengcheng Gu, Hosho Katsura, and S. Das Sarma, “Nearly flatbands with nontrivial topology,” Phys. Rev. Lett. 106, 236803 (2011)

    2011

  6. [6]

    Fractional quantum hall states at zero magnetic field,

    Titus Neupert, Luiz Santos, Claudio Chamon, and Christopher Mudry, “Fractional quantum hall states at zero magnetic field,” Phys. Rev. Lett.106, 236804 (2011)

    2011

  7. [7]

    Band geometry of fractional topological insula- tors,

    Rahul Roy, “Band geometry of fractional topological insula- tors,” Phys. Rev. B90, 165139 (2014)

    2014

  8. [8]

    Frac- tional chern insulators and theW ∞ algebra,

    S. A. Parameswaran, R. Roy, and S. L. Sondhi, “Frac- tional chern insulators and theW ∞ algebra,” Phys. Rev. B85, 241308(R) (2012)

    2012

  9. [9]

    Geometric sta- bility of topological lattice phases,

    T. S. Jackson, Gunnar M ¨oller, and Rahul Roy, “Geometric sta- bility of topological lattice phases,” Nature Communications6, 8629 (2015)

    2015

  10. [10]

    Position-momentum duality and fractional quantum hall effect in chern insulators,

    Martin Claassen, Ching Hua Lee, Ronny Thomale, Xiao-Liang Qi, and Thomas P. Devereaux, “Position-momentum duality and fractional quantum hall effect in chern insulators,” Phys. Rev. Lett.114, 236802 (2015)

    2015

  11. [11]

    Exact landau level description of geometry and in- teraction in a flatband,

    Jie Wang, Jennifer Cano, Andrew J. Millis, Zhao Liu, and Bo Yang, “Exact landau level description of geometry and in- teraction in a flatband,” Phys. Rev. Lett.127, 246403 (2021)

    2021

  12. [12]

    Signatures of frac- tional quantum anomalous hall states in twisted mote2,

    Jiaqi Cai, Eric Anderson, Chong Wang, Xiaowei Zhang, Xi- aoyu Liu, William Holtzmann, Yinong Zhang, Fengren Fan, Takashi Taniguchi, Kenji Watanabe, Ying Ran, Ting Cao, Liang Fu, Di Xiao, Wang Yao, and Xiaodong Xu, “Signatures of frac- tional quantum anomalous hall states in twisted mote2,” Nature 622, 63–68 (2023)

    2023

  13. [13]

    Thermodynamic evi- dence of fractional chern insulator in moir´e mote2,

    Yihang Zeng, Zhengchao Xia, Kaifei Kang, Jiacheng Zhu, Patrick Kn ¨uppel, Chirag Vaswani, Kenji Watanabe, Takashi Taniguchi, Kin Fai Mak, and Jie Shan, “Thermodynamic evi- dence of fractional chern insulator in moir´e mote2,” Nature622, 69–73 (2023)

    2023

  14. [14]

    Observation of frac- tionally quantized anomalous hall effect,

    Heonjoon Park, Jiaqi Cai, Eric Anderson, Yinong Zhang, Jiayi Zhu, Xiaoyu Liu, Chong Wang, William Holtzmann, Chaowei Hu, Zhaoyu Liu, Takashi Taniguchi, Kenji Watanabe, Jiun-Haw Chu, Ting Cao, Liang Fu, Wang Yao, Cui-Zu Chang, David Cobden, Di Xiao, and Xiaodong Xu, “Observation of frac- tionally quantized anomalous hall effect,” Nature622, 74–79 (2023)

    2023

  15. [15]

    Observation of integer and fractional quantum anomalous hall effects in twisted bilayer mote2,

    Fan Xu, Zheng Sun, Tongtong Jia, Chang Liu, Cheng Xu, Chushan Li, Yu Gu, Kenji Watanabe, Takashi Taniguchi, Bing- bing Tong, Jinfeng Jia, Zhiwen Shi, Shengwei Jiang, Yang Zhang, Xiaoxue Liu, and Tingxin Li, “Observation of integer and fractional quantum anomalous hall effects in twisted bilayer mote2,” Phys. Rev. X13, 031037 (2023)

    2023

  16. [16]

    Local probe of bulk and edge states in a fractional chern insulator,

    Zhurun Ji, Heonjoon Park, Mark E. Barber, Chaowei Hu, Kenji Watanabe, Takashi Taniguchi, Jiun-Haw Chu, Xiaodong Xu, and Zhi-Xun Shen, “Local probe of bulk and edge states in a fractional chern insulator,” Nature635, 578–583 (2024)

    2024

  17. [17]

    Direct magnetic imaging of fractional chern insulators in twisted mote2,

    Evgeny Redekop, Canxun Zhang, Heonjoon Park, Jiaqi Cai, Eric Anderson, Owen Sheekey, Trevor Arp, Grigory Babikyan, Samuel Salters, Kenji Watanabe, Takashi Taniguchi, Martin E. Huber, Xiaodong Xu, and Andrea F. Young, “Direct magnetic imaging of fractional chern insulators in twisted mote2,” Nature 635, 584–589 (2024)

    2024

  18. [18]

    Ferromagnetism and topology of the higher flat band in a fractional chern insulator,

    Heonjoon Park, Jiaqi Cai, Eric Anderson, Xiao-Wei Zhang, Xiaoyu Liu, William Holtzmann, Weijie Li, Chong Wang, Chaowei Hu, Yuzhou Zhao, Takashi Taniguchi, Kenji Watan- abe, Jihui Yang, David Cobden, Jiun-haw Chu, Nicolas Reg- nault, B. Andrei Bernevig, Liang Fu, Ting Cao, Di Xiao, and Xiaodong Xu, “Ferromagnetism and topology of the higher flat band in a ...

    2025

  19. [19]

    Fractional quantum anomalous hall effect in multilayer graphene,

    Zhengguang Lu, Tonghang Han, Yuxuan Yao, Aidan P. Reddy, Jixiang Yang, Junseok Seo, Kenji Watanabe, Takashi Taniguchi, Liang Fu, and Long Ju, “Fractional quantum anomalous hall effect in multilayer graphene,” Nature626, 759–764 (2024)

    2024

  20. [20]

    Evidence of the fractional quantum spin hall effect in moir´e mote2,

    Kaifei Kang, Bowen Shen, Yichen Qiu, Yihang Zeng, Zhengchao Xia, Kenji Watanabe, Takashi Taniguchi, Jie Shan, and Kin Fai Mak, “Evidence of the fractional quantum spin hall effect in moir´e mote2,” Nature628, 522–526 (2024)

    2024

  21. [21]

    Hidden states and dynamics of fractional fillings in twisted mote2 bilayers,

    Yiping Wang, Jeongheon Choe, Eric Anderson, Weijie Li, Julian Ingham, Eric A. Arsenault, Yiliu Li, Xiaodong Hu, Takashi Taniguchi, Kenji Watanabe, Xavier Roy, Dmitri Basov, Di Xiao, Raquel Queiroz, James C. Hone, Xiaodong Xu, and X.-Y . Zhu, “Hidden states and dynamics of fractional fillings in twisted mote2 bilayers,” Nature641, 1149–1155 (2025)

    2025

  22. [22]

    Chern insulators at integer and fractional filling in moir ´e pentalayer graphene,

    Dacen Waters, Anna Okounkova, Ruiheng Su, Boran Zhou, Jiang Yao, Kenji Watanabe, Takashi Taniguchi, Xiaodong Xu, Ya-Hui Zhang, Joshua Folk, and Matthew Yankowitz, “Chern insulators at integer and fractional filling in moir ´e pentalayer graphene,” Phys. Rev. X15, 011045 (2025)

    2025

  23. [23]

    Signatures of chi- ral superconductivity in rhombohedral graphene,

    Tonghang Han, Zhengguang Lu, Zach Hadjri, Lihan Shi, Zhenghan Wu, Wei Xu, Yuxuan Yao, Armel A. Cotten, Omid 6 Sharifi Sedeh, Henok Weldeyesus, Jixiang Yang, Junseok Seo, Shenyong Ye, Muyang Zhou, Haoyang Liu, Gang Shi, Zhenqi Hua, Kenji Watanabe, Takashi Taniguchi, Peng Xiong, Do- minik M. Zumb¨uhl, Liang Fu, and Long Ju, “Signatures of chi- ral supercond...

    2025

  24. [24]

    Signatures of unconventional superconductivity near reentrant and fractional quantum anomalous hall insulators,

    Fan Xu, Zheng Sun, Jiayi Li, Ce Zheng, Cheng Xu, Jingjing Gao, Tongtong Jia, Yanfei Su, Kenji Watanabe, Takashi Taniguchi, Bingbing Tong, Li Lu, Jinfeng Jia, Zhiwen Shi, Shengwei Jiang, Junhao Lin, Yuanbo Zhang, Yang Zhang, Shiming Lei, Xiaoxue Liu, and Tingxin Li, “Signatures of unconventional superconductivity near reentrant and fractional quantum anoma...

    Pith/arXiv arXiv 2026

  25. [25]

    Extended quan- tum anomalous hall states in graphene/hbn moir´e superlattices,

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

    2025

  26. [26]

    Twist-angle evo- lution from valley-polarized fractional topological phases to valley-degenerate superconductivity in twisted bilayer mote2,

    Zheng Sun, Fan Xu, Jiayi Li, Yifan Jiang, Jingjing Gao, Cheng Xu, Tongtong Jia, Kehao Cheng, Jinyang Zhang, Wanghao Tian, Kenji Watanabe, Takashi Taniguchi, Jinfeng Jia, Shengwei Jiang, Yang Zhang, Yuanbo Zhang, Shim- ing Lei, Xiaoxue Liu, and Tingxin Li, “Twist-angle evo- lution from valley-polarized fractional topological phases to valley-degenerate sup...

    arXiv 2026

  27. [27]

    Displacement field- controlled fractional chern insulators and charge density waves in a graphene/hbn moir´e superlattice,

    Samuel H. Aronson, Tonghang Han, Zhengguang Lu, Yuxuan Yao, Jackson P. Butler, Kenji Watanabe, Takashi Taniguchi, Long Ju, and Raymond C. Ashoori, “Displacement field- controlled fractional chern insulators and charge density waves in a graphene/hbn moir´e superlattice,” Phys. Rev. X15, 031026 (2025)

    2025

  28. [28]

    Fractional quantization in insulators from hall to chern,

    B. A. Bernevig, L. Fu, L. Ju, A. H. MacDonald, K. F. Mak, and J. Shan, “Fractional quantization in insulators from hall to chern,” Nature Physics21, 1702–1713 (2025)

    2025

  29. [29]

    Anomalous hall metal and fractional chern insulator in twisted transition metal dichalco- genides,

    Valentin Cr ´epel and Liang Fu, “Anomalous hall metal and fractional chern insulator in twisted transition metal dichalco- genides,” Phys. Rev. B107, L201109 (2023)

    2023

  30. [30]

    Fractional quantum anomalous hall states in twisted bilayermote 2 andwse 2,

    Aidan P. Reddy, Faisal Alsallom, Yang Zhang, Trithep Devakul, and Liang Fu, “Fractional quantum anomalous hall states in twisted bilayermote 2 andwse 2,” Phys. Rev. B108, 085117 (2023)

    2023

  31. [31]

    Quantum geometry and stabiliza- tion of fractional chern insulators far from the ideal limit,

    Gal Shavit and Yuval Oreg, “Quantum geometry and stabiliza- tion of fractional chern insulators far from the ideal limit,” Phys. Rev. Lett.133, 156504 (2024)

    2024

  32. [32]

    Moir´e fractional chern in- sulators. i. first-principles calculations and continuum models of twisted bilayermote 2,

    Yujin Jia, Jiabin Yu, Jiaxuan Liu, Jonah Herzog-Arbeitman, Ziyue Qi, Hanqi Pi, Nicolas Regnault, Hongming Weng, B. An- drei Bernevig, and Quansheng Wu, “Moir´e fractional chern in- sulators. i. first-principles calculations and continuum models of twisted bilayermote 2,” Phys. Rev. B109, 205121 (2024)

    2024

  33. [33]

    Moir ´e fractional chern insulators. ii. first-principles calculations and continuum models of rhom- bohedral graphene superlattices,

    Jonah Herzog-Arbeitman, Yuzhi Wang, Jiaxuan Liu, Pok Man Tam, Ziyue Qi, Yujin Jia, Dmitri K. Efetov, Oskar Vafek, Nico- las Regnault, Hongming Weng, Quansheng Wu, B. Andrei Bernevig, and Jiabin Yu, “Moir ´e fractional chern insulators. ii. first-principles calculations and continuum models of rhom- bohedral graphene superlattices,” Phys. Rev. B109, 205122 (2024)

    2024

  34. [34]

    Kwan, Jiabin Yu, Jonah Herzog-Arbeitman, Dmitri K

    Yves H. Kwan, Jiabin Yu, Jonah Herzog-Arbeitman, Dmitri K. Efetov, Nicolas Regnault, and B. Andrei Bernevig, “Moir ´e fractional chern insulators. iii. hartree-fock phase diagram, magic angle regime for chern insulator states, role of moir´e po- tential, and goldstone gaps in rhombohedral graphene superlat- tices,” Phys. Rev. B112, 075109 (2025)

    2025

  35. [35]

    Moir ´e fractional chern in- sulators. iv. fluctuation-driven collapse in multiband exact di- agonalization calculations on rhombohedral graphene,

    Jiabin Yu, Jonah Herzog-Arbeitman, Yves H. Kwan, Nicolas Regnault, and B. Andrei Bernevig, “Moir ´e fractional chern in- sulators. iv. fluctuation-driven collapse in multiband exact di- agonalization calculations on rhombohedral graphene,” Phys. Rev. B112, 075110 (2025)

    2025

  36. [36]

    Fractional chern insulators versus nonmagnetic states in twisted bilayer mote2,

    Jiabin Yu, Jonah Herzog-Arbeitman, Minxuan Wang, Oskar Vafek, B. Andrei Bernevig, and Nicolas Regnault, “Fractional chern insulators versus nonmagnetic states in twisted bilayer mote2,” Phys. Rev. B109, 045147 (2024)

    2024

  37. [37]

    Frac- tional chern insulator states in multilayer graphene moir ´e su- perlattices,

    Zhongqing Guo, Xin Lu, Bo Xie, and Jianpeng Liu, “Frac- tional chern insulator states in multilayer graphene moir ´e su- perlattices,” Phys. Rev. B110, 075109 (2024)

    2024

  38. [38]

    Theory of quantum anomalous hall phases in pentalayer rhombohe- dral graphene moir ´e structures,

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

    2024

  39. [39]

    Anomalous hall crystals in rhombohedral multilayer graphene. i. interaction-driven chern bands and fractional quantum hall states at zero magnetic field,

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

    2024

  40. [40]

    Anomalous hall crystals in rhombohedral multilayer graphene. ii. general mechanism and a minimal model,

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

    2024

  41. [41]

    Fractional quan- tum anomalous hall effect in rhombohedral multilayer graphene in the moir´eless limit,

    Boran Zhou, Hui Yang, and Ya-Hui Zhang, “Fractional quan- tum anomalous hall effect in rhombohedral multilayer graphene in the moir´eless limit,” Phys. Rev. Lett.133, 206504 (2024)

    2024

  42. [42]

    Self- consistent theory of fractional quantum anomalous hall states in rhombohedral graphene,

    Ke Huang, Xiao Li, Sankar Das Sarma, and Fan Zhang, “Self- consistent theory of fractional quantum anomalous hall states in rhombohedral graphene,” Phys. Rev. B110, 115146 (2024)

    2024

  43. [43]

    Parent berry curvature and the ideal anomalous hall crystal,

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

    2024

  44. [44]

    Quantum anomalous hall crystal at fractional filling of moir ´e superlattices,

    D. N. Sheng, Aidan P. Reddy, Ahmed Abouelkomsan, Emil J. Bergholtz, and Liang Fu, “Quantum anomalous hall crystal at fractional filling of moir ´e superlattices,” Phys. Rev. Lett.133, 066601 (2024)

    2024

  45. [45]

    Intertwined fractional quantum anomalous hall states and charge density waves,

    Xue-Yang Song, Chao-Ming Jian, Liang Fu, and Cenke Xu, “Intertwined fractional quantum anomalous hall states and charge density waves,” Phys. Rev. B109, 115116 (2024)

    2024

  46. [46]

    Sublattice structure and topology in spontaneously crystallized electronic states,

    Yongxin Zeng, Daniele Guerci, Valentin Cr´epel, Andrew J. Mil- lis, and Jennifer Cano, “Sublattice structure and topology in spontaneously crystallized electronic states,” Phys. Rev. Lett. 132, 236601 (2024)

    2024

  47. [47]

    ”berry trashcan

    B. Andrei Bernevig and Yves H. Kwan, “”berry trashcan” model of interacting electrons in rhombohedral graphene,” (2025), arXiv:2503.09692 [cond-mat.str-el]

    Pith/arXiv arXiv 2025

  48. [48]

    Multi- band exact diagonalization and an iteration approach to search for fractional chern insulators in rhombohedral multilayer graphene,

    Heqiu Li, B. Andrei Bernevig, and Nicolas Regnault, “Multi- band exact diagonalization and an iteration approach to search for fractional chern insulators in rhombohedral multilayer graphene,” Phys. Rev. B112, 075130 (2025)

    2025

  49. [49]

    Chiral superconductivity from spin polarized chern band in twisted mote2,

    Cheng Xu, Nianlong Zou, Nikolai Peshcherenko, Ammar Jahin, Tingxin Li, Shi-Zeng Lin, and Yang Zhang, “Chiral superconductivity from spin polarized chern band in twisted mote2,” Phys. Rev. Lett.135, 266005 (2025)

    2025

  50. [50]

    Quantum-geometric dipole: A topological boost to flavor ferromagnetism in flat bands,

    Lei Chen, Sayed Ali Akbar Ghorashi, Jennifer Cano, and Valentin Cr ´epel, “Quantum-geometric dipole: A topological boost to flavor ferromagnetism in flat bands,” Phys. Rev. Lett. 136, 186602 (2026)

    2026

  51. [51]

    Ef- fects of berry curvature on ideal fractional chern insulator 7 many-body gaps,

    Jingtian Shi, Jennifer Cano, and Nicol ´as Morales-Dur´an, “Ef- fects of berry curvature on ideal fractional chern insulator 7 many-body gaps,” Phys. Rev. Res.8, L022045 (2026)

    2026

  52. [52]

    Anyon superconductivity from topological crit- icality in a hofstadter–hubbard model,

    Stefan Divic, Valentin Cr ´epel, Tomohiro Soejima, Xue-Yang Song, Andrew J. Millis, Michael P. Zaletel, and Ashvin Vishwanath, “Anyon superconductivity from topological crit- icality in a hofstadter–hubbard model,” Proceedings of the National Academy of Sciences122, e2426680122 (2025), https://www.pnas.org/doi/pdf/10.1073/pnas.2426680122

    doi:10.1073/pnas.2426680122 2025

  53. [53]

    Chiral superconductivity near a fractional chern insulator,

    Taige Wang and Michael P. Zaletel, “Chiral superconductivity near a fractional chern insulator,” (2025), arXiv:2507.07921 [cond-mat.str-el]

    Pith/arXiv arXiv 2025

  54. [54]

    From fractionalization to chiral topological superconductivity in a flat chern band,

    Daniele Guerci, Ahmed Abouelkomsan, and Liang Fu, “From fractionalization to chiral topological superconductivity in a flat chern band,” (2025), arXiv:2506.10938 [cond-mat.supr-con]

    arXiv 2025

  55. [55]

    Topological chiral superconductivity in the triangular-lattice hofstadter-hubbard model,

    Feng Chen, Wen O. Wang, Jia-Xin Zhang, Leon Balents, and D. N. Sheng, “Topological chiral superconductivity in the triangular-lattice hofstadter-hubbard model,” Phys. Rev. Lett. 136, 086503 (2026)

    2026

  56. [56]

    Doping a fractional quan- tum anomalous hall insulator,

    Zhengyan Darius Shi and T. Senthil, “Doping a fractional quan- tum anomalous hall insulator,” Phys. Rev. X15, 031069 (2025)

    2025

  57. [57]

    Microscopic mechanism of anyon super- conductivity emerging from fractional chern insulators,

    Fabian Pichler, Clemens Kuhlenkamp, Michael Knap, and Ashvin Vishwanath, “Microscopic mechanism of anyon super- conductivity emerging from fractional chern insulators,” New- ton2(2026)

    2026

  58. [58]

    Anyon superconductivity and plateau transitions in doped fractional quantum anomalous hall insulators,

    Pavel A. Nosov, Zhaoyu Han, and Eslam Khalaf, “Anyon superconductivity and plateau transitions in doped fractional quantum anomalous hall insulators,” Phys. Rev. Lett.136, 106501 (2026)

    2026

  59. [59]

    Bosonization for begin- ners—refermionization for experts,

    Jan V on Delft and Herbert Schoeller, “Bosonization for begin- ners—refermionization for experts,” Annalen der Physik510, 225–305 (1998)

    1998

  60. [60]

    Thierry Giamarchi, Quantum physics in one dimension (Clarendon press, 2003)

    2003

  61. [61]

    John Cardy, Scaling and renormalization in statistical physics (Cambridge university press, 1996)

    1996

  62. [62]

    Frac- tional quantum hall effect in an array of quantum wires,

    C. L. Kane, Ranjan Mukhopadhyay, and T. C. Lubensky, “Frac- tional quantum hall effect in an array of quantum wires,” Phys. Rev. Lett.88, 036401 (2002)

    2002

  63. [63]

    Quantum hall hierarchy from coupled wires,

    Yohei Fuji and Akira Furusaki, “Quantum hall hierarchy from coupled wires,” Phys. Rev. B99, 035130 (2019)

    2019

  64. [64]

    From luttinger liquid to non- abelian quantum hall states,

    Jeffrey C. Y . Teo and C. L. Kane, “From luttinger liquid to non- abelian quantum hall states,” Phys. Rev. B89, 085101 (2014)

    2014

  65. [65]

    Non-abelian topological insulators from an array of quantum wires,

    Eran Sagi and Yuval Oreg, “Non-abelian topological insulators from an array of quantum wires,” Phys. Rev. B90, 201102(R) (2014)

    2014

  66. [66]

    Coupled-wire constructions: a luttinger liquid approach to topology,

    Tobias Meng, “Coupled-wire constructions: a luttinger liquid approach to topology,” The European Physical Journal Special Topics229, 527–543 (2020)

    2020

  67. [67]

    Weak-coupling phase diagram of the two-chain hubbard model,

    Leon Balents and Matthew P. A. Fisher, “Weak-coupling phase diagram of the two-chain hubbard model,” Phys. Rev. B53, 12133–12141 (1996)

    1996

  68. [68]

    n- chain hubbard model in weak coupling,

    Hsiu-Hau Lin, Leon Balents, and Matthew P. A. Fisher, “n- chain hubbard model in weak coupling,” Phys. Rev. B56, 6569–6593 (1997)

    1997

  69. [69]

    Antiferro- magnetic spin ladders: Crossover between spin s=1/2 and s=1 chains,

    D. G. Shelton, A. A. Nersesyan, and A. M. Tsvelik, “Antiferro- magnetic spin ladders: Crossover between spin s=1/2 and s=1 chains,” Phys. Rev. B53, 8521–8532 (1996)

    1996

  70. [70]

    Pair-density-wave super- conducting order in two-leg ladders,

    Akbar Jaefari and Eduardo Fradkin, “Pair-density-wave super- conducting order in two-leg ladders,” Phys. Rev. B85, 035104 (2012)

    2012

  71. [71]

    Colloquium: Theory of intertwined orders in high temperature superconductors,

    Eduardo Fradkin, Steven A. Kivelson, and John M. Tranquada, “Colloquium: Theory of intertwined orders in high temperature superconductors,” Rev. Mod. Phys.87, 457–482 (2015)

    2015

  72. [72]

    Riemannian structure on manifolds of quantum states,

    JP Provost and G Vallee, “Riemannian structure on manifolds of quantum states,” Communications in Mathematical Physics 76, 289–301 (1980)

    1980

  73. [73]

    Maximally localized wannier functions: Theory and applications,

    Nicola Marzari, Arash A. Mostofi, Jonathan R. Yates, Ivo Souza, and David Vanderbilt, “Maximally localized wannier functions: Theory and applications,” Rev. Mod. Phys.84, 1419–1475 (2012)

    2012

  74. [74]

    Quantum geometry in quantum materials,

    Jiabin Yu, B. Andrei Bernevig, Raquel Queiroz, Enrico Rossi, P¨aivi T ¨orm¨a, and Bohm-Jung Yang, “Quantum geometry in quantum materials,” npj Quantum Materials10, 101 (2025)

    2025

  75. [75]

    Essay: Where can quantum geometry lead us?

    P ¨aivi T¨orm¨a, “Essay: Where can quantum geometry lead us?” Phys. Rev. Lett.131, 240001 (2023)

    2023

  76. [76]

    left”(α= 2)and “right

    See Supplemental Material for further details on the model, a review of the bosonization procedure, and details of the renor- malization group calculations, which include Refs. [31, 59, 61– 63]. 8 Supplementary Materials Haoyu Hu CONTENTS References 5 I Non-interacting Hamiltonian 9 II Coupled wire construction and bosonization 10 III Sliding Luttinger li...

  77. [77]

    (b) Characteristic energy/temperature scale of the SC phase TSC as a function oft ′ x/tx =y aF CI,0 /yF CI,0

    = 0.1,y aF CI(l= 0) = 0.05, andy SC(l= 0) =y CDW (l= 0) = 0. (b) Characteristic energy/temperature scale of the SC phase TSC as a function oft ′ x/tx =y aF CI,0 /yF CI,0 . We take parameters inside the SC phase withK o(l= 0) = 1.5. Largert ′ x/tx indicates stronger quantum geometry, which in turn produces a more stable SC phase.T 0 denotes the UV energy s...