Pith. sign in

REVIEW 3 major objections 5 minor 69 references

Generic integer and fractional quantum anomalous Hall crystals from interaction-driven band folding

T0 review · 3 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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

Pith's one-line read In a two-band triangular-lattice model, a nearest-neighbor interaction drives a trivial charge-density wave at ν=2/3 that folds the Brillouin zone and exposes a Chern number -1 mini-band; doping that band produces integer and fractional…

desk verdict Solid FQAHC at ν*=1/3, but the claimed series of fractional states includes at least one single-particle finite-size artifact. read the letter →

arxiv 2505.04138 v4 pith:UPOVS3QX submitted 2025-05-07 cond-mat.str-el cond-mat.mes-hall

Hongyu Lu , Han-Qing Wu , Bin-Bin Chen , Wang Yao , Zi Yang Meng This is my paper · ORCID
classification cond-mat.str-elcond-mat.mes-hall PACS 73.43.-f71.10.Fd71.27.+a
keywords quantumanomalousHallcrystalfractionalchargedensitywavebandfoldingtopologicalmini-bandtriangularlatticeinfinitematrixrenormalizationgroupexactdiagonalization
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper claims a generic mechanism for quantum anomalous Hall crystals (QAHCs)—states that combine a nonzero Hall conductivity with a charge-density wave. In a two-band triangular-lattice model, the V1 interaction at ν=2/3 stabilizes a commensurate, topologically trivial CDW that triples the unit cell. The Brillouin-zone folding that results leaves an isolated band above the CDW gap with Chern number C=-1. Doping this mini-band to integer or fractional filling produces QAHC and fractional QAHC ground states whose Hall conductivity is σH=-ν*, where ν*=3ν-2, not the original band filling. The authors demonstrate one FQAHC at ν=7/9 even without competing interactions, a bosonic FQAHC at ν=5/6, and a compressible CDW phase at intermediate temperatures that precedes the FQAHC.

What carries the argument

The load-bearing object is the C=-1 mini-band that appears above the CDW gap in the folded Brillouin zone. The paper defines its filling as ν*=3ν-2: at ν=2/3, ν*=0; at ν=1, ν*=1; and at ν=7/9, ν*=1/3. The mini-band is obtained from Hartree-Fock bands built from mean-field parameters measured in iDMRG simulations of the ν=2/3 CDW; adding V2=V3=2 flattens the band (bandwidth 0.11 versus 1.25 with only V1) and makes the Berry curvature more uniform. The topological C=-1 character of this band, together with the CDW's 3-fold ground-state degeneracy, sets the Hall conductivity σH=-ν* and combines with the topological degeneracy to determine the total ground-state degeneracy of each (F)QAHC state.

What would settle it

Perform larger-scale iDMRG on a cylinder at ν=7/9 with V1=10, V2=V3=2 and check whether the charge pump after inserting 6π flux remains exactly one electron while the CDW order parameter stays nonzero; a deviation in the pumped charge or a melting of the CDW would refute the FQAHC claim. Alternatively, compute the many-body Chern number of the 15 quasi-degenerate ground states at ν=11/15 on a 36-site or 42-site torus to verify the 5×3 degeneracy and σH=-1/5.

Watch

Extended reading notes

Core claim

The central discovery is that interaction-driven band folding, not just external potentials or lattice geometry, creates a topological band that can host fractional Hall states. Starting from a two-band triangular-lattice model with complex hoppings, the V1-driven CDW at ν=2/3 (electron density 1/3) triples the unit cell and folds the original Brillouin zone. The resulting mini-band above the CDW gap has C=-1; at full filling (ν*=ν=1) the ground states are threefold-degenerate QAHC states, each with quantized σH=-1, which are stabilized by adding longer-range repulsion V2=V3=2. At fractional fillings of the same band, exact diagonalization and infinite DMRG show FQAHC states with σH=-ν* (e.g. -1/5 at ν=11/15, -1/3 at ν=7/9, -2/5 at ν=4/5, -2/3 at ν=8/9), with ground-state degeneracy equal to the topological degeneracy times the 3-fold CDW degeneracy. The same scheme works for hard-core bosons at ν=5/6, giving a σH=-1/2 bosonic FQAHC.

Load-bearing premise

The argument assumes the Hartree-Fock description of the ν=2/3 CDW gives a valid isolated C=-1 mini-band at the doped fillings where FQAHC states are claimed; if the CDW gap closes or the mini-band mixes strongly with other bands, the identification σH=-ν* and the FQAHC distinction would break down.

Editorial extensions

If this is right

  • At V1=10, V2=V3=2, a series of FQAHC states appears at fractional fillings of the C=-1 mini-band, with σH=-ν*, including σH=-1/5, -1/3, -2/5, and -2/3; the topological degeneracy of each is multiplied by the 3-fold CDW degeneracy.
  • With only V1 interaction and no competing terms, a σH=-1/3 FQAHC at ν=7/9 still survives, though with a less straight charge-pumping curve indicating less uniform Berry curvature.
  • The finite-temperature study of the σH=-1/3 FQAHC shows a compressible CDW phase at intermediate temperatures T*<T<TCDW, which the paper identifies as a possible precursor of the lower-temperature FQAHC.
  • The same band-folding mechanism produces a bosonic FQAHC at ν=5/6 (ν*=1/2) with σH=-1/2, extending Hall-crystal physics to hard-core boson systems.
  • The previously reported topological pinball liquid states with |σH|=2/5 and 3/5 at ν=4/5 and 13/15 are reinterpreted as FQAHC states at ν*=2/5 and 3/5 of the folded mini-band.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • In moiré or cold-atom systems where a fractional Hall conductivity is observed at a filling that does not match the Chern band's filling, the FQAHC mechanism would be identifiable by a triple-unit-cell CDW and by the relation σH=-ν* rather than σH=ν.
  • The compressible CDW phase at intermediate temperatures could show anisotropic longitudinal resistivity, while the transverse resistivity would not be quantized; this is a concrete experimental signature that distinguishes the precursor phase from the low-temperature FQAHC.
  • The analogy the paper draws to doping a solid into a supersolid suggests that doping any trivial commensurate CDW whose folded band has nonzero Chern number could generically produce fractional Hall states, making the mechanism a promising search principle for new lattice models.
  • Because the band-mixing at V2=V3=0 weakens the FQAHC, improving the quantum geometry of the folded band—by engineered hoppings, magnetic fields, or other perturbations—could be a practical route to realizing these states in experiments.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper studies a two-band triangular-lattice Chern model (Eq. 1) with repulsive interactions V1, V2, V3. At ν=2/3 filling of the lower Chern band, strong V1 drives a commensurate CDW that triples the unit cell; a Hartree-Fock analysis fed by iDMRG mean fields indicates that the mini-band just above the CDW gap carries Chern number C=-1. The authors argue that doping this mini-band to (fractional) integer fillings ν* produces integer and fractional quantum anomalous Hall crystals with Hall conductivity σH=-ν*, where ν*=3ν-2, coexisting with the CDW order. Evidence is presented from ED on 30-site tori, iDMRG on cylinders, and XTRG for finite temperatures. The most robust example is a σH=-1/3 FQAHC at ν=7/9 with V1=10, V2=V3=2, supported by charge pumping after 6π flux, entanglement-spectrum shift by one charge sector, and edge-mode counting {1,1,2,3,5,...}. The paper also reports a possible V1-only FQAHC at the same filling, a compressible CDW phase at intermediate temperature, and a bosonic σH=-1/2 FQAHC at ν=5/6.

Significance. If the central claim holds, the paper provides a concrete and unbiased microscopic realization of a generic band-folding mechanism for fractional Hall crystals, going beyond the previously studied integer QAHC cases and offering a natural reinterpretation of the earlier topological pinball liquid states. The robust iDMRG/ES evidence for the ν=7/9 state is a genuine strength, as are the bosonic extension and the finite-temperature phase sequence. The significance is currently tempered by the fact that part of the claimed 'series' of FQAHC states rests on few-particle ED data that do not distinguish fractional Hall physics from single-particle band effects; this needs to be addressed before the general claim is fully established.

major comments (3)
  1. [Robust FQAHC ground states, Fig. 2(c)] The ED example at ν=11/15 (ν*=1/5) on the 30-site torus cannot by itself support a fractional Hall crystal. With 15 unit cells folded into 5 mini-band momenta, the C=-1 mini-band holds exactly one particle. A single particle in a C=-1 band with N_s=5 reproduces all the reported diagnostics: a 5-fold topological degeneracy from the five Bloch states, a many-body Chern number -1/5 per state, a spectral-flow period 5×2π=10π, and, after multiplying by the three CDW sectors, a 15-fold ground-state manifold. These are single-particle band properties, not signatures of a Laughlin state or fractional statistics. The statement that the 10π return 'further suggests FQAHC state is a Laughlin state' is therefore unsupported. Please either remove this example from the FQAHC evidence or provide an interaction-sensitive diagnostic (e.g., many-body gap behavior and ES counting with more than one particle in the mini-band).
  2. [SI D, Fig. 11; main-text 'Robust FQAHC ground states'] The claimed 'series' of FQAHC states at ν*=2/5 and 3/5 needs to be separated into large-scale iDMRG/ES evidence and few-particle ED points. On the same 30-site torus, ν*=2/5 (ν=4/5) and ν*=3/5 (ν=13/15) correspond to two and three particles in the five-state mini-band; noninteracting Slater determinants in a C=-1 band already give Hall conductance C×N/N_s = -2/5 and -3/5 and the same flux-pumping periods. The many-body Chern number and spectral flow therefore do not, by themselves, distinguish an FQAH state from a trivial few-particle band-filling effect. The iDMRG charge pumping and ES for ν*=2/5 in SI D are the right kind of evidence; the main text should explicitly state which members of the series have such large-scale support and should avoid presenting the ED few-particle diagnostics as establishing FQAHC.
  3. [Robust FQAHC ground states, V1-only paragraph] The V1-only FQAHC claim at ν=7/9 (V1=10, V2=V3=0) rests on a single iDMRG charge-pumping curve that the authors themselves describe as 'less straight' than the V2=V3=2 case. No entanglement-spectrum counting, ground-state degeneracy, or gap estimate is presented for this parameter set. Given that the integer QAHC at ν=1 with only V1 has a vanishingly small gap (0.017–0.053, SI C), the V1-only FQAHC should be presented as preliminary evidence, and the abstract's wording that 'some FQAHC state might even exist in less ideal conditions' should be correspondingly tempered unless additional diagnostics are provided.
minor comments (5)
  1. [Fig. 2 caption] The caption states 'σH =ν∗ =−1/5' for the ν=11/15 state, but the text and the Hall-conductivity rule require σH = −ν∗ = −1/5; please correct the sign inconsistency.
  2. [Supplementary Information, section numbering] The SI contains two sections labelled 'C' ('THE QAHC∗ AT ν=1' and 'ROBUST CDW ORDER IN THE (F)QAHC STATES'); please renumber them sequentially.
  3. [Robust FQAHC ground states, first paragraph] The sentence 'we find a series of new FQAHC states at fractional fillings of this C = −1 mini-band and (more details in the SI [47])' is grammatically incomplete; please rewrite to name the fillings explicitly and refer to the SI in a complete sentence.
  4. [Model and methods] The complex hopping phase ϕij is described only through Fig. 1(a); for reproducibility the phase assignment (0, π, π/2 on the labeled bonds) should be stated explicitly in the main text.
  5. [Thermodynamics of the FQAHC state] The estimates of T* and T_CDW come from XTRG on a single cylinder size (3×18×2); please state the system-size and bond-dimension convergence checks for these transition temperatures, either in the main text or the SI.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: key claims are computed observables from the original Hamiltonian; the ν* relation is counting and the C=−1 mini-band is an interpretive analysis, not a fitted input.

full rationale

The derivation chain is self-contained. The central results—ground-state degeneracies, many-body Chern numbers, charge pumping, and entanglement spectra—are computed from the original Hamiltonian in Eq. (1) without projection to any band, so they are outputs rather than inputs. The relation ν* = 3ν−2 is a counting definition based on the tripled CDW unit cell, and the statement σH = −ν* is a quantized Hall response verified by explicit many-body Chern numbers and iDMRG charge pumping, not an imposed equality. The C=−1 mini-band topology comes from a Hartree-Fock calculation fed by iDMRG mean fields at ν=2/3; this is an interpretive and largely post-hoc explanation of why V2=V3=2 helps, not a fitted constraint on the doped-filling results. Self-citations (Refs. [32,33,55,56]) are used to contrast the roton-driven FQAH+CDW mechanism and to cite methodology, and they are not load-bearing for the band-folding claim. The paper itself flags the ED finite-size limitation for the ν*=1/5 series, noting that a 30-site torus has only 5 momenta in the C=−1 mini-band; some fractional-filling ED points involve only N=1–3 particles, so those diagnostics may not distinguish a single-particle Chern-band effect from a Laughlin state. That is an evidence-strength concern, not circularity. No equation or result reduces to its own input by construction.

Assumptions & free parameters 4 free parameters · 6 assumptions · 0 invented entities

The model parameters V1, V2, V3, and t' are inputs chosen by hand from prior work or to stabilize the CDW, and the analysis leans on the standard Chern-band Hall quantization and Laughlin edge counting; no new physical entities are introduced.

free parameters (4)
  • V1 = 10
    Chosen nearest-neighbor repulsion that produces the ν=2/3 CDW and tripled unit cell; central to the band-folding mechanism.
  • V2=V3 = 2
    Chosen competing interactions that flatten the C=-1 mini-band and make its Berry curvature more uniform (SI Fig. 6), enabling the robust FQAHC series.
  • t' = 0.2
    NNNN hopping inherited from Refs. [42,43] to flatten the lower Chern band; an input constant, not fitted here.
  • μ (phenomenological CDW potential) = ±1.5
    Staggered potential in SI section A used only to illustrate CDW band folding; not used for the quantitative Hall claims.
assumptions (6)
  • domain assumption The noninteracting two-band model in Eq. (1) with t'=0.2 has lower/upper bands with C=±1.
    This is the starting band structure inherited from prior work; the Chern numbers are computed for the single-particle Hamiltonian.
  • domain assumption At ν=2/3 with V1=10, the ground state is a √3×√3 CDW that triples the unit cell and folds the BZ.
    Supported by ED spectra and structure factors (Fig. 7, SI), but assumed to persist upon doping to the fractional fillings of interest.
  • standard math A fractional Chern insulator at filling ν* of a C=-1 band has Hall conductivity σH=-ν* and Laughlin-type edge counting.
    Standard Chern-Simons/Laughlin reasoning used to interpret charge pumping and ES counting; not re-derived here.
  • domain assumption Hartree-Fock decomposition with mean fields measured from iDMRG captures the renormalized folded mini-band.
    The C=-1 mini-band topology and its improved flatness at V2=V3=2 are established through HF (SI section A), and the FQAHC interpretation leans on these bands.
  • domain assumption Hard-core boson constraint (at most one boson per site) for the bosonic simulation.
    Bose-Hubbard-type replacement of fermionic operators in Eq. (1).
  • domain assumption The entanglement spectrum counting {1,1,2,3,5,...} identifies a Laughlin state.
    Standard identification of chiral edge modes in FQH systems.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Generic integer and fractional quantum anomalous Hall crystals from interaction-driven band folding." pith.science (2026). https://pith.science/paper/UPOVS3QX

@misc{pith2026250504138,
  author       = {Pith},
  title        = {Pith review of: Generic integer and fractional quantum anomalous Hall crystals from interaction-driven band folding},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UPOVS3QX}},
  note         = {Machine review of arXiv:2505.04138}
}
read the original abstract

Among the extensive studies of fractional quantum anomalous Hall (FQAH) states, there recently appears a growing interest in the topological states with coexisting charge density wave (CDW) orders. Such states are referred to as Hall crystals. However, compared to those with integer Hall conductivities, the FQAH crystal (FQAHC) is still elusive even at the level of microscopic model. In this work, we numerically study a topological flat-band model on triangular lattice with spinless fermions. At fractional filling of the Chern band, the nearest-neighbor interaction leads to a commensurate and topologically trivial CDW state. Interestingly, the folded mini-band above the CDW gap is non-trivial, and we focus on the doping of it without any projection. A series of (F)QAHC states at (fractional) integer fillings of this mini-band are discovered and some FQAHC state might even exist in less "ideal" conditions. The ground-state degeneracies of such (F)QAHC states are enlarged by the CDW degeneracy and the Hall conductivities -- determined by the fillings of the mini-band -- are different from the fillings of the original Chern band. We also study the thermodynamics of an FQAHC state and find a compressible CDW phase at intermediate temperatures, which might serve as a precursor of lower temperature FQAHC phase. Moreover, we numerically demonstrate that such a generic scheme of doping CDW-folded topological mini-band could be applied to bosonic systems, broadening the platforms of Hall-crystal physics and motivating its exploration in quantum moire and cold-atom systems.

Figures

Figures reproduced from arXiv: 2505.04138 by the authors.

Figure 1
Figure 1. FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 4
Figure 4. FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗
Figures from the paper (5 more)
Figure 5
Figure 5. Figure 5: FIG. 5. (a) The original BZ and folded BZ ( [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]
Figure 7
Figure 7. Figure 7: FIG. 7. The ED spectra of a 24-site torus at [PITH_FULL_IMAGE:figures/full_fig_p008_7.png]
Figure 10
Figure 10. Figure 10: FIG. 10. CDW order of the [PITH_FULL_IMAGE:figures/full_fig_p008_10.png]
Figure 11
Figure 11. Figure 11: FIG. 11. (a) The ES of the [PITH_FULL_IMAGE:figures/full_fig_p008_11.png]
Figure 12
Figure 12. Figure 12: FIG. 12. The structure factors in the original BZ at different temper [PITH_FULL_IMAGE:figures/full_fig_p009_12.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

69 extracted references · 35 canonical work pages

  1. [1]

    Balents, C

    L. Balents, C. R. Dean, D. K. Efetov, and A. F. Young, Nature Physics 16, 725 (2020)

    2020

  2. [2]

    H. Yu, M. Chen, and W. Yao, National Science Review 7, 12 (2019)

    work page 2019

  3. [3]

    H. Li, U. Kumar, K. Sun, and S.-Z. Lin, Phys. Rev. Res. 3, L032070 (2021)

    2021

  4. [4]

    Devakul, V

    T. Devakul, V . Cr´epel, Y . Zhang, and L. Fu, Nature Communi- cations 12, 6730 (2021)

    2021

  5. [5]

    K. F. Mak and J. Shan, Nature Nanotechnology 17, 686 (2022)

    2022

  6. [6]

    Wang, X.-W

    C. Wang, X.-W. Zhang, X. Liu, Y . He, X. Xu, Y . Ran, T. Cao, and D. Xiao, Phys. Rev. Lett. 132, 036501 (2024)

    work page 2024

  7. [7]

    Y . H. Kwan, G. Wagner, J. Yu, A. K. Dagnino, Y . Jiang, X. Xu, B. A. Bernevig, T. Neupert, and N. Regnault, When could abelian fractional topological insulators exist in twisted mote 2 (and other systems) (2024), arXiv:2407.02560 [cond-mat.str- el]

    arXiv 2024

  8. [8]

    Lu and L

    T. Lu and L. H. Santos, Phys. Rev. Lett. 133, 186602 (2024)

    work page 2024

Show all 69 references
  1. [9]

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

    2023

  2. [10]

    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, Nature 622, 74 (2023)

    2023

  3. [11]

    Y . Zeng, Z. Xia, K. Kang, J. Zhu, P. Kn ¨uppel, C. Vaswani, K. Watanabe, T. Taniguchi, K. F. Mak, and J. Shan, Nature622, 69 (2023)

    2023

  4. [12]

    F. Xu, Z. Sun, T. Jia, C. Liu, C. Xu, C. Li, Y . Gu, K. Watanabe, T. Taniguchi, B. Tong, J. Jia, Z. Shi, S. Jiang, Y . Zhang, X. Liu, and T. Li, Phys. Rev. X13, 031037 (2023)

    2023

  5. [13]

    Z. Lu, T. Han, Y . Yao, A. P. Reddy, J. Yang, J. Seo, K. Watan- abe, T. Taniguchi, L. Fu, and L. Ju, Nature626, 759 (2024)

    2024

  6. [14]

    Polshyn, Y

    H. Polshyn, Y . Zhang, M. A. Kumar, T. Soejima, P. Ledwith, K. Watanabe, T. Taniguchi, A. Vishwanath, M. P. Zaletel, and A. F. Young, Nature Physics18, 42 (2022)

    2022

  7. [15]

    R. Su, D. Waters, B. Zhou, K. Watanabe, T. Taniguchi, Y .-H. Zhang, M. Yankowitz, and J. Folk, Nature637, 1084 (2025)

    2025

  8. [16]

    Z. Lu, T. Han, Y . Yao, Z. Hadjri, J. Yang, J. Seo, L. Shi, S. Ye, K. Watanabe, T. Taniguchi, and L. Ju, Nature637, 1090 (2025)

    2025

  9. [17]

    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, Phys. Rev. X 15, 011045 (2025)

    2025

  10. [18]

    A. S. Patri, Z. Dong, and T. Senthil, Phys. Rev. B 110, 245115 (2024)

    2024

  11. [19]

    Das Sarma and M

    S. Das Sarma and M. Xie, Phys. Rev. B 110, 155148 (2024)

    2024

  12. [20]

    Huang, S

    K. Huang, S. D. Sarma, and X. Li, Impurity-induced thermal crossover in fractional chern insulators (2024), arXiv:2409.04349 [cond-mat.str-el]. 6

    2024 arXiv

  13. [21]

    Teˇsanovi´c, F

    Z. Teˇsanovi´c, F. m. c. Axel, and B. I. Halperin, Phys. Rev. B39, 8525 (1989)

    1989

  14. [22]

    Song, C.-M

    X.-Y . Song, C.-M. Jian, L. Fu, and C. Xu, Phys. Rev. B 109, 115116 (2024)

    2024

  15. [23]

    D. N. Sheng, A. P. Reddy, A. Abouelkomsan, E. J. Bergholtz, and L. Fu, Phys. Rev. Lett. 133, 066601 (2024)

    2024

  16. [24]

    J. Chen, Q. Li, X. Wang, and W. Li, Fractional chern insulator and quantum anomalous hall crystal in twisted mote 2 (2025), arXiv:2504.07932 [cond-mat.str-el]

    2025

  17. [25]

    Soejima, J

    T. Soejima, J. Dong, T. Wang, T. Wang, M. P. Zaletel, A. Vish- wanath, and D. E. Parker, Phys. Rev. B110, 205124 (2024)

    2024

  18. [26]

    Z. Dong, A. S. Patri, and T. Senthil, Phys. Rev. B 110, 205130 (2024)

    2024

  19. [27]

    Y . Zeng, D. Guerci, V . Cr´epel, A. J. Millis, and J. Cano, Phys. Rev. Lett. 132, 236601 (2024)

    2024

  20. [28]

    J. Dong, T. Wang, T. Wang, T. Soejima, M. P. Zaletel, A. Vish- wanath, and D. E. Parker, Phys. Rev. Lett.133, 206503 (2024)

    2024

  21. [29]

    B. Zhou, H. Yang, and Y .-H. Zhang, Phys. Rev. Lett. 133, 206504 (2024)

    2024

  22. [30]

    Z. Dong, A. S. Patri, and T. Senthil, Phys. Rev. Lett. 133, 206502 (2024)

    2024

  23. [31]

    T. Tan, J. May-Mann, and T. Devakul, Phys. Rev. Lett. 135, 036604 (2025)

    2025

  24. [32]

    Lu, H.-Q

    H. Lu, H.-Q. Wu, B.-B. Chen, K. Sun, and Z. Yang Meng, Re- ports on Progress in Physics 87, 108003 (2024)

    2024

  25. [33]

    Lu, H.-Q

    H. Lu, H.-Q. Wu, B.-B. Chen, and Z. Y . Meng, Continuous tran- sition and gapless roton inside fractional quantum anomalous hall states (2025), arXiv:2404.06745 [cond-mat.str-el]

    2025

  26. [34]

    H. Pan, M. Xie, F. Wu, and S. Das Sarma, Phys. Rev. Lett.129, 056804 (2022)

    2022

  27. [35]

    Perea-Causin, H

    R. Perea-Causin, H. Liu, and E. J. Bergholtz, Quantum anoma- lous hall crystals in moir ´e bands with higher chern number (2024), arXiv:2412.02745 [cond-mat.mes-hall]

    2024 arXiv

  28. [36]

    Fradkin, S

    E. Fradkin, S. A. Kivelson, and J. M. Tranquada, Rev. Mod. Phys. 87, 457 (2015)

    2015

  29. [37]

    A. S. Sørensen, E. Demler, and M. D. Lukin, Phys. Rev. Lett. 94, 086803 (2005)

    2005

  30. [38]

    Aidelsburger, M

    M. Aidelsburger, M. Atala, M. Lohse, J. T. Barreiro, B. Paredes, and I. Bloch, Phys. Rev. Lett. 111, 185301 (2013)

    2013

  31. [39]

    N. Y . Yao, A. V . Gorshkov, C. R. Laumann, A. M. L ¨auchli, J. Ye, and M. D. Lukin, Phys. Rev. Lett.110, 185302 (2013)

    2013

  32. [40]

    L ´eonard, S

    J. L ´eonard, S. Kim, J. Kwan, P. Segura, F. Grusdt, C. Repellin, N. Goldman, and M. Greiner, Nature 619, 495 (2023)

    2023

  33. [41]

    Wang, F.-M

    C. Wang, F.-M. Liu, M.-C. Chen, H. Chen, X.-H. Zhao, C. Ying, Z.-X. Shang, J.-W. Wang, Y .-H. Huo, C.-Z. Peng, X. Zhu, C.-Y . Lu, and J.-W. Pan, Science 384, 579 (2024), https://www.science.org/doi/pdf/10.1126/science.ado3912

    2024 doi

  34. [42]

    Kourtis, J

    S. Kourtis, J. W. F. Venderbos, and M. Daghofer, Phys. Rev. B 86, 235118 (2012)

    2012

  35. [43]

    Kourtis and M

    S. Kourtis and M. Daghofer, Phys. Rev. Lett. 113, 216404 (2014)

    2014

  36. [44]

    Kourtis, Phys

    S. Kourtis, Phys. Rev. B 97, 085108 (2018)

    2018

  37. [45]

    I. P. McCulloch, Infinite size density matrix renormalization group, revisited (2008), arXiv:0804.2509 [cond-mat.str-el]

    2008 arXiv

  38. [46]

    B.-B. Chen, L. Chen, Z. Chen, W. Li, and A. Weichselbaum, Phys. Rev. X 8, 031082 (2018)

    2018

  39. [47]

    We show some more details and results in the Supplementary Information

  40. [48]

    A. G. Grushin, J. Motruk, M. P. Zaletel, and F. Pollmann, Phys. Rev. B 91, 035136 (2015)

    2015

  41. [49]

    Li and F

    H. Li and F. D. M. Haldane, Phys. Rev. Lett. 101, 010504 (2008)

    2008

  42. [50]

    Alexandradinata, T

    A. Alexandradinata, T. L. Hughes, and B. A. Bernevig, Phys. Rev. B 84, 195103 (2011)

    2011

  43. [51]

    Cincio and G

    L. Cincio and G. Vidal, Phys. Rev. Lett. 110, 067208 (2013)

    2013

  44. [52]

    Regnault, Entanglement spectroscopy and its application to the quantum hall effects (2015), arXiv:1510.07670 [cond- mat.str-el]

    N. Regnault, Entanglement spectroscopy and its application to the quantum hall effects (2015), arXiv:1510.07670 [cond- mat.str-el]

    2015 arXiv

  45. [53]

    D. N. Sheng, Z. Y . Weng, L. Sheng, and F. D. M. Haldane, Phys. Rev. Lett. 97, 036808 (2006)

    2006

  46. [54]

    H. Lu, S. Sur, S.-S. Gong, and D. N. Sheng, Phys. Rev. B 106, 205105 (2022)

    2022

  47. [55]

    Lu, B.-B

    H. Lu, B.-B. Chen, H.-Q. Wu, K. Sun, and Z. Y . Meng, Phys. Rev. Lett. 132, 236502 (2024)

    2024

  48. [56]

    Lu, H.-Q

    H. Lu, H.-Q. Wu, B.-B. Chen, and Z. Y . Meng, Vestigial gap- less boson density wave emerging between ν = 1 /2 frac- tional chern insulator and finite-momentum supersolid (2024), arXiv:2408.07111 [cond-mat.mes-hall]

    2024 arXiv

  49. [57]

    S. M. Girvin, A. H. MacDonald, and P. M. Platzman, Phys. Rev. B 33, 2481 (1986)

    1986

  50. [58]

    Zaklama, D

    T. Zaklama, D. Luo, and L. Fu, Structure factor and topological bound of twisted bilayer semiconductors at fractional fillings (2024), arXiv:2411.03496 [cond-mat.str-el]

    2024 arXiv

  51. [59]

    M. Long, H. Lu, H.-Q. Wu, and Z. Y . Meng, Spectra of magne- toroton and chiral graviton modes of fractional chern insulator (2025), arXiv:2501.00247 [cond-mat.str-el]

    2025

  52. [60]

    Chomaz, D

    L. Chomaz, D. Petter, P. Ilzh ¨ofer, G. Natale, A. Trautmann, C. Politi, G. Durastante, R. M. W. van Bijnen, A. Patschei- der, M. Sohmen, M. J. Mark, and F. Ferlaino, Phys. Rev. X 9, 021012 (2019)

    2019

  53. [61]

    Y .-C. Chen, R. G. Melko, S. Wessel, and Y .-J. Kao, Phys. Rev. B 77, 014524 (2008)

    2008

  54. [62]

    Y . Xie, A. T. Pierce, J. M. Park, D. E. Parker, E. Khalaf, P. Led- with, Y . Cao, S. H. Lee, S. Chen, P. R. Forrester, K. Watanabe, T. Taniguchi, A. Vishwanath, P. Jarillo-Herrero, and A. Yacoby, Nature 600, 439 (2021)

    2021

  55. [63]

    Hauschild and F

    J. Hauschild and F. Pollmann, SciPost Phys. Lect. Notes , 5 (2018)

    2018

  56. [64]

    Weichselbaum, Annals of Physics 327, 2972 (2012)

    A. Weichselbaum, Annals of Physics 327, 2972 (2012)

    2012

  57. [65]

    7 SUPPLEMENTARY INFORMA TION A

    Beijing Paratera Tech Corp., Ltd . 7 SUPPLEMENTARY INFORMA TION A. BAND FOLDING ANALYSIS C=−1 C=1μ=1.5 μ=−1.5 S* X Γ X* Y (a) S Γ* Y* kx ky (b) (c) FIG. 5. (a) The original BZ and folded BZ (∗). The schematic band structures from applying the effective potentials of (b) the ν ...

  58. [66]

    The decomposed HF hamiltonian from the original one in Eq. (1) is as follows: HHF = X ⟨i,j⟩ teiϕij(c† icj +h.c.) + X ⟨⟨⟨i,j⟩⟩⟩ t′(c† icj +h.c.) +Vij X ij (⟨ni⟩nj +⟨nj⟩ni−⟨c† icj⟩c† jci−⟨c† jci⟩c† icj), (2) (a) (b) (c) (d) V1=10,V2=V3=0 V1=10,V2=V3=2 Ω kx kx ky ky FIG. 6. TheC ...

  59. [67]

    (1) in the main text) with the same parameters

    For each set of parameters at ν = 2/3 filling of the original lower band, we perform iDMRG simulations of the original Hamiltonian (Eq. (1) in the main text) with the same parameters

  60. [68]

    (2) from the many-body wavefunction obtained from iDMRG and put them back to Eq

    Then we measure the real-space mean-field parameters in Eq. (2) from the many-body wavefunction obtained from iDMRG and put them back to Eq. (2)

  61. [69]

    The results are shown in Fig

    At last, we diagonalize the 6-band (as our many-body simulations showed that the CDW at ν = 2/3 would triple the unit cell) HF hamiltonian with the measured mean-field parameters. The results are shown in Fig. 6. Here, we only plot the C =−1 mini-band above the ν = 2/3 CDW gap...

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.