Pith. sign in

REVIEW 1 major objections 5 minor 101 references

Composite-Boson Ansatz for Fractional Quantum Hall Manifolds of Lattice Bosons at Generic Fillings $\nu<1/2$

T0 review · 1 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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

Pith's one-line read Attaching two vortices to each boson yields a trial wave-function basis that reproduces the low-energy quasihole manifolds of the Hofstadter–Bose–Hubbard model at every filling below one half.

desk verdict Careful and mostly right, but the CB identification with the finite-U HBH manifold is only tested inside lowest-band projection; deserves review with a request for a direct full-Hamiltonian check. read the letter →

arxiv 2608.10747 v1 pith:NWJFUBRG submitted 2026-08-11 cond-mat.quant-gas cond-mat.str-el

Umut Can Turhan , A. Levent Subaşi , Rifat Onur Umucal{i}lar This is my paper · ORCID
classification cond-mat.quant-gascond-mat.str-el MSC 81V7082B20 PACS 73.43.-f73.43.Cd67.85.-d
keywords fractionalquantumHalleffectcompositebosonsHofstadter-Bose-HubbardmodelLaughlinquasiholestorusthetafunctionsmany-bodyChernnumberbraidingphaselattice
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

This paper tries to show that the low-energy manifolds of the Hofstadter–Bose–Hubbard model at all fillings $\nu<1/2$ can be described by a simple composite-boson wave-function basis. The idea is to attach two vortices to every boson, leaving $N_d=N_\phi-2N$ effective orbitals, and to occupy those reduced-flux orbitals in every possible way, with magnetic translations trimming the naive occupation count to the known manifold dimension. The paper verifies that this basis spans the exact projected low-energy space with subspace fidelities above $0.9996$, matches its energy spectrum, and reproduces the many-body Chern number, fractional density depletion, and an Aharonov–Bohm-free braiding phase that identify these manifolds as lattice fractional quantum Hall states. If the ansatz is right, it converts a set of exclusion-rule counting formulas into explicit wave functions, making the quasihole structure of lattice fractional Hall systems directly manipulable.

What carries the argument

The load-bearing object is the reduced-flux composite-boson ansatz: a many-body trial state built as a symmetrized product of $N$ single-particle torus $\theta$-function orbitals at reduced flux $N_d=N_\phi-2N$, multiplied by the square of the odd Jacobi $\theta$ function $\vartheta_{1/2,1/2}((z_i-z_j)/L_1|\tau)$ and by a center-of-mass $\theta$ factor $F_{\mathrm{CM}}(Z)=\vartheta_{a,0}(2Z/L_1|2\tau)$. The counting mechanism is the organization of occupation patterns of $N$ bosons in $N_d$ orbitals into cyclic orbits under a common shift of all orbital labels; each orbit of size $s$ contributes $s/n$ internal directions with $n=N_d/g$ and $g=\gcd(N,N_d)$, and the $q_{\mathrm{COM}}=N_\phi/g$ center-of-mass translations multiply the count, yielding $D=\binom{N_d+N-1}{N}N_\phi/N_d$. Overlap-matrix ranks verify this prediction orbit by orbit, so the machinery converts bosonic occupations of reduced-flux orbitals into a basis for the physical low-energy manifold.

What would settle it

A concrete check: pick a filling not listed, for example $(N,N_\phi)=(4,11)$ or $(3,11)$, compute the exact lowest-band-projected low-energy spectrum and the overlap matrix of the composite-boson trial family, and see whether the numerical rank equals $\binom{N_d+N-1}{N}N_\phi/N_d$ and whether the worst principal angle with the exact manifold is as small as the $>0.9996$ fidelities reported here; any case where the rank is larger or the fidelity falls sharply would falsify the generic-filling claim.

Watch

Extended reading notes

Core claim

The central claim is that every low-energy manifold of the Hofstadter–Bose–Hubbard model at fillings $\nu<1/2$ has an explicit microscopic wave-function description in terms of composite bosons. Starting from $N_\phi$ flux quanta and $N$ bosons, attaching two vortices per boson leaves $N_d=N_\phi-2N$ reduced-flux orbitals; the trial states are symmetrized products of $\theta$-function orbitals at this reduced flux, multiplied by the squared odd Jacobi $\theta$ function $\vartheta^2_{1/2,1/2}((z_i-z_j)/L_1|\tau)$ and a center-of-mass $\theta$ factor, with the whole reference state complex-conjugated to match the physical Hofstadter chirality. The naive occupation count $\binom{N_d+N-1}{N}$ is overcomplete, but organizing occupation patterns into cyclic orbits under a common shift of orbital labels and including the $q_{\mathrm{COM}}=N_\phi/\gcd(N,N_d)$ center-of-mass sectors predicts the physical rank $D=\binom{N_d+N-1}{N}N_\phi/N_d$. Numerical overlap matrices confirm this rank orbit by orbit for the systems $(N,N_\phi)=(2,8),(2,10),(3,9),(3,10)$, subspace fidelities with the exact projected low-energy manifold exceed $0.9996$, and variational Monte Carlo finds the predicted ranks $D=11$ and $49$ for $(5,11)$ and $(6,14)$. The same manifolds carry Chern number $C_{\mathrm{MB}}/D=\nu$, pinned added-flux excitations deplete the density by $\nu_{\mathrm{eff}}$, and an Aharonov–Bohm-free braid gives phase $2\nu_{\mathrm{eff}}$.

Load-bearing premise

The argument assumes that projecting the Hofstadter–Bose–Hubbard model onto its lowest band at $U/t=2$, or replacing it by the hard-core real-space limit in the Monte Carlo comparison, leaves the low-energy manifold essentially intact, so that the continuum theta-function trial states evaluated on lattice sites still span the physically relevant space.

Editorial extensions

If this is right

  • The known quasihole counting formula for the sub-half-filled Hofstadter–Bose–Hubbard model is realized by explicit wave functions: each low-energy manifold is spanned by the composite-boson trial family, not just counted by it.
  • For the systems checked, the composite-boson span reproduces the lowest-band-projected spectrum and the exact low-energy subspace, so the ansatz can serve as a variational tool for lattice fractional quantum Hall states.
  • The unpinned manifolds carry total many-body Chern number $C_{\mathrm{MB}}=\binom{N_d+N-1}{N_d}$, so the averaged Chern number per state is exactly the filling $\nu=N/N_\phi$, confirming their topological character.
  • A localized added-flux excitation inside these manifolds depletes the density by $Q=\nu_{\mathrm{eff}}$ and acquires a braiding phase $2\nu_{\mathrm{eff}}$ in an Aharonov–Bohm-free loop, meaning the quasihole response tracks the effective filling.
  • Variational Monte Carlo evaluation without explicit lowest-band projection finds the predicted ranks and narrow manifolds for $(N,N_\phi)=(5,11)$ and $(6,14)$, indicating the ansatz extends beyond the smallest systems.

Reading between the lines

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

  • Inference: if the reduced-flux occupation organization is the correct internal label for these manifolds, the same occupations should predict finer structure, such as the decomposition of the entanglement spectrum or per-sector Berry curvature, rather than only the total dimension; this is testable in the reported systems.
  • Inference: because the braiding phase and depletion track $\nu_{\mathrm{eff}}$ rather than a fixed denominator, the finite-size manifolds behave as Laughlin-quasihole multiplets at an effective filling; explicit adiabatic continuity between these lattice states and the continuum quasihole sector would be a natural check not performed in the paper.
  • Inference: the same cyclic-orbit counting should apply to fermionic Hofstadter models with an odd number of attached vortices, where a composite-fermion analogue would predict the rank of Jain-type quasihole manifolds; the paper treats only two-vortex bosons.
  • Inference: pinning and braiding protocols could be designed using the composite-boson occupations as a guide, for example choosing pin configurations that isolate a single reduced-flux occupation sector, which may simplify future experimental or numerical probes of these lattice states.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

1 major / 5 minor

Summary. The paper proposes a composite-boson (CB) trial wavefunction ansatz for the low-energy manifolds of the Hofstadter–Bose–Hubbard (HBH) model on a torus at fillings ν < 1/2. The ansatz attaches two vortices to each boson, reducing the flux to N_d = N_φ − 2N, and constructs trial states from occupation patterns of N bosons in N_d reduced-flux orbitals, multiplied by a squared Jastrow factor and a center-of-mass theta function. The authors show that organizing the occupation patterns into cyclic orbits and including the center-of-mass multiplicity reproduces the known degeneracy formula D = binom(N_d+N−1,N) N_φ/N_d. They validate the ansatz via orbit-resolved numerical ranks, subspace fidelities > 0.9996 for the lowest-band-projected Hamiltonian, spectral comparisons, many-body Chern numbers satisfying C_MB/D = ν, fractional density depletion Q = ν_eff, and an Aharonov–Bohm-free braiding phase consistent with 2ν_eff. VMC extends the rank and energetic analysis to larger systems in the hard-core limit.

Significance. If correct, the paper provides a rare microscopic wavefunction description for a broad family of lattice FQH manifolds, going beyond counting rules. The orbit-resolved rank tests are a strong, parameter-free check: each cyclic orbit's rank matches the prediction, including shortened orbits, and the predicted center-of-mass sectors are separately verified. The subspace fidelities above 0.9996 and consistent Chern numbers across many system sizes are convincing within the projected model. The use of VMC to test the same ansatz without explicit lowest-band projection in the hard-core limit is a valuable robustness check, and the disorder/asymmetric-pinning tests for the braiding phase are thoughtful. The main limitation is that the direct validation is performed within the lowest-band projection at U/t = 2, while the abstract and conclusion refer to the full HBH model without qualification.

major comments (1)
  1. [Numerical Results; Supplemental Material Secs. 5.1 and 5.2] The central claim that the CB span describes the low-energy manifolds of the HBH model at U/t = 2 is not directly tested against the full Hamiltonian. The exact spectra and the subspace fidelities in Table IV are computed for the lowest-band-projected Hamiltonian only, and the only unprojected calculation (Supplemental Sec. 5.2) uses the hard-core limit U → ∞, which changes both the Hilbert space and the interaction. Consequently, the agreement reported in Fig. 1 and in Table IV could in principle be an artifact of the lowest-band projection rather than evidence that the CB states describe the genuine low-energy manifold of Eq. (2). Because the abstract and conclusion state the framework applies to the HBH model without projecting, this gap is load-bearing. I request that, for at least the smaller systems (for example (N,N_φ) = (2,8) or (3,9)), the authors compute the overlap or subspace fidelity between the CB span and the low-energy manifold of the full HBH Hamiltonian at U/t = 2, or otherwise demonstrate quantitatively that the lowest-band projection is accurate at this interaction strength.
minor comments (5)
  1. [Fig. 1] The ratio δ/Δ is used in the caption and text but is not defined in the caption; please define it there or refer to its definition in the main text.
  2. [Main text, above Eq. (5)] The sentence "The number of nonzero eigenvalues of the trial-state overlap matrix confirms this physical rank" does not state the numerical threshold for "nonzero"; the threshold appears later in Supplemental Sec. 2.4 (singular-value ratio > 10^-10) and should be mentioned here.
  3. [Supplemental Material Sec. 2.2] The cyclic-orbit reduction is motivated by a continuous-translation identity that shifts all orbital labels by one, but the numerical trial states are generated by discrete one-site magnetic translations; the paper should state more explicitly that the orbit structure is a predicted reduction that is verified numerically rather than a mathematical proof for the discrete case.
  4. [Main text, 'Fractional density depletion' paragraph] The phrase "N_loc added flux quanta are localized by repulsive potentials" is potentially confusing; N_loc is implemented by changing L2 (and thus the total flux) and pinning the resulting quasiholes, so it would be clearer to say "localized flux insertions" instead of "added flux quanta" to avoid implying extra particles.
  5. [Supplemental Material Table II] The column headers n_O and s_O are not defined in the table caption; they are defined in the surrounding text, but a brief definition in the caption would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the CB trial-space rank, spectra, Chern numbers, depletion, and braiding are independently computed quantities that the paper verifies against, rather than defines by, the known degeneracy formula.

full rationale

The paper's degeneracy formula is admittedly reorganized from earlier work, but the central rank claim is not equivalent to that formula by construction. The ansatz wavefunctions are explicitly defined from reduced-flux theta orbitals, a squared Jastrow factor, and a center-of-mass factor; the naive number of raw states is C_B q_COM, which is larger than D. The reduction to D rests on linear dependencies among the trial states, which are tested directly through orbit-resolved singular-value analysis (SM Table II) rather than imposed by hand. The spectra are obtained by variational diagonalization of the lowest-band-projected Hamiltonian within the CB span and compared with exact projected spectra; the reported subspace fidelities are computed from principal angles, not fitted. The many-body Chern numbers, fractional depletion, and braiding phases are calculated from exact diagonalization or VMC without using the CB rank as an input, and the agreement with C_MB/D = nu and theta_B = 2 nu_eff is a numerical result rather than a definitional identity. The self-citation to the authors' earlier HBH analysis supplies the expected counting and diagnostics, but the present overlap ranks and Chern-number calculations independently reproduce them, so the citation is not load-bearing. Possible failure of the lowest-band projection or of the hard-core VMC equivalence would be a correctness risk, not a circularity, and the paper itself labels the VMC comparison as a robustness check across related Hamiltonians.

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

No new physical entities are introduced; the composite boson is a bookkeeping of flux attachment. The free parameters listed are numerical protocol choices, not fitted constants. The central claim rests on standard theta-function mathematics, the validity of the lowest-band projection, and the quasihole-localization assumption for the braiding protocol.

free parameters (4)
  • Pinning strength for charge diagnostic = V/t = 1
    Used for all panels of the density depletion calculation; no V-sweep is reported, so the plateau Q=nu_eff could depend on this choice.
  • Pinning strengths for braiding = V1 = V2 = 0.8t (clean protocol)
    The braiding phase is extracted at one absolute strength; asymmetric-pinning and disorder robustness are tested only in the smaller N=2 system.
  • Chern number twist mesh = N_theta = 21
    The many-body Chern number is computed on a 21x21 mesh; no convergence study versus mesh size is shown.
  • VMC sampling schedule = 150 bins, 1000-1500 configurations per bin, Gram cutoff 1e-8
    Numerical controls for the Monte Carlo rank and energy estimates; the paper reports jackknife errors but does not show sensitivity to these hyperparameters.
assumptions (4)
  • standard math Theta-function quasiperiodicity and the analyticity-induced N_phi-dimensional single-particle space on the torus
    Used to build the reduced-flux orbitals and center-of-mass factor (Supplemental Material Sec. 1).
  • domain assumption The lowest Hofstadter band is the relevant Hilbert space; projection onto it captures the low-energy manifold at U/t=2
    Used in the exact projected ED and Fig. 1 comparisons; no direct check of band mixing is provided.
  • domain assumption Hard-core limit U -> infinity is representative of the low-energy manifold at moderate U
    Used in VMC for (5,11) and (6,14); the comparison to projected U/t=2 is explicitly called a robustness check, not a variational error estimate.
  • domain assumption Moving pinning potentials create well-localized quasiholes whose Aharonov-Bohm phases cancel in the commutator loop
    Underlies the braiding phase interpretation; tested indirectly via disorder and asymmetric pinning in a smaller system.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Composite-Boson Ansatz for Fractional Quantum Hall Manifolds of Lattice Bosons at Generic Fillings $\nu<1/2$." pith.science (2026). https://pith.science/paper/NWJFUBRG

@misc{pith2026260810747,
  author       = {Pith},
  title        = {Pith review of: Composite-Boson Ansatz for Fractional Quantum Hall Manifolds of Lattice Bosons at Generic Fillings $\nu<1/2$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NWJFUBRG}},
  note         = {Machine review of arXiv:2608.10747}
}
abstract

Hofstadter systems provide a lattice route to fractional quantum Hall physics, but their low-energy manifolds often lack simple wave-function descriptions. Here we present a composite-boson ansatz for a broad family of finite-size lattice-split Laughlin-quasihole manifolds in the low-flux Hofstadter--Bose--Hubbard model on a torus at generic fillings \(\nu<1/2\). Attaching two vortices to each boson yields reduced-flux orbitals from which we construct a many-body trial basis. Its translation-resolved rank reproduces the expected low-energy-manifold dimension and provides a composite-boson interpretation of the established quasihole counting previously inferred from generalized exclusion rules and thin-torus arguments. For smaller systems, diagonalization within the lowest-band-projected ansatz span closely reproduces the exact projected spectrum, while variational Monte Carlo extends the rank and energetic analysis to larger systems for which explicit subspace construction becomes costly. To probe the fractional quantum Hall character of these manifolds, we calculate their many-body Chern-numbers and show that localized added-flux excitations exhibit quasihole-like behavior, with fractional density depletion and an Aharonov--Bohm-free braiding phase that both track the effective filling. Together, these results establish a microscopic composite-boson framework for organizing a broad family of sub-half-filled fractional quantum Hall manifolds of lattice bosons.

Figures

Figures reproduced from arXiv: 2608.10747 by the authors.

Figure 1
Figure 1. compares the exact and CB-trial-space spectra of the lowest-band-projected Hamiltonian. The energy￾subgroup multiplicities agree in every panel, partly re￾flecting the magnetic-translation symmetry shared by the two calculations, and several energies coincide in (a) and (b); only in (c) is the ordering of the two lowest subgroups reversed. Because δ/∆ ≤ 3.0 × 10−3 in all panels, with δ the manifold bandwidth and ∆ t… view at source ↗
Figure 2
Figure 2. FIG. 2. Accumulated density depletion versus [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. AB-free quasihole braid at [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (1 more)
Figure 1
Figure 1. Figure 1: FIG. 1. Robustness diagnostics for the Aharonov–Bohm-free cross-shaped braid of [PITH_FULL_IMAGE:figures/full_fig_p023_1.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

101 extracted references · 58 canonical work pages

  1. [1]

    D. C. Tsui, H. L. Stormer, and A. C. Gossard, Phys. Rev. Lett.48, 1559 (1982)

    1982

  2. [2]

    Yoshioka,The Quantum Hall Effect, Springer Series in Solid-State Sciences, Vol

    D. Yoshioka,The Quantum Hall Effect, Springer Series in Solid-State Sciences, Vol. 133 (Springer, Berlin, Hei- delberg, 2002)

    2002

  3. [3]

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

    1983

  4. [4]

    de Picciotto, M

    R. de Picciotto, M. Reznikov, M. Heiblum, V. Umansky, G. Bunin, and D. Mahalu, Nature389, 162 (1997)

    1997

  5. [5]

    Saminadayar, D

    L. Saminadayar, D. C. Glattli, Y. Jin, and B. Etienne, Phys. Rev. Lett.79, 2526 (1997)

    1997

  6. [6]

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

    1977

  7. [8]

    Stern, Annals of Physics323, 204 (2008)

    A. Stern, Annals of Physics323, 204 (2008)

    2008

  8. [9]

    Nayak, S

    C. Nayak, S. H. Simon, A. Stern, M. Freedman, and S. Das Sarma, Rev. Mod. Phys.80, 1083 (2008)

    2008

Show all 101 references
  1. [10]

    Bartolomei, M

    H. Bartolomei, M. Kumar, R. Bisognin, A. Marguerite, J.-M. Berroir, E. Bocquillon, B. Pla¸ cais, A. Cavanna, Q. Dong, U. Gennser, Y. Jin, and G. F` eve, Science368, 173 (2020)

    2020

  2. [11]

    Nakamura, S

    J. Nakamura, S. Liang, G. C. Gardner, and M. J. Manfra, Nat. Phys.16, 931 (2020)

    2020

  3. [12]

    J. K. Jain,Composite Fermions(Cambridge University Press, Cambridge, 2007)

    2007

  4. [13]

    T. H. Hansson, M. Hermanns, S. H. Simon, and S. F. Viefers, Rev. Mod. Phys.89, 025005 (2017)

    2017

  5. [14]

    Kapit and E

    E. Kapit and E. Mueller, Phys. Rev. Lett.105, 215303 (2010)

    2010

  6. [15]

    Glasser, J

    I. Glasser, J. I. Cirac, G. Sierra, and A. E. B. Nielsen, Phys. Rev. B94, 245104 (2016)

    2016

  7. [16]

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

    2005

  8. [17]

    Hafezi, A

    M. Hafezi, A. S. Sørensen, E. Demler, and M. D. Lukin, Phys. Rev. A76, 023613 (2007)

    2007

  9. [18]

    M¨ oller and N

    G. M¨ oller and N. R. Cooper, Phys. Rev. Lett.103, 105303 (2009)

    2009

  10. [19]

    M¨ oller and N

    G. M¨ oller and N. R. Cooper, Phys. Rev. Lett.115, 126401 (2015)

    2015

  11. [20]

    Regnault and B

    N. Regnault and B. A. Bernevig, Phys. Rev. X1, 021014 (2011)

    2011

  12. [21]

    Z. Liu, E. J. Bergholtz, H. Fan, and A. M. L¨ auchli, Phys. Rev. Lett.109, 186805 (2012)

    2012

  13. [22]

    B. A. Bernevig and N. Regnault, Phys. Rev. B85, 075128 (2012)

    2012

  14. [23]

    T. Liu, C. Repellin, B. A. Bernevig, and N. Regnault, Phys. Rev. B87, 205136 (2013)

    2013

  15. [24]

    Sterdyniak, C

    A. Sterdyniak, C. Repellin, B. A. Bernevig, and N. Reg- nault, Phys. Rev. B87, 205137 (2013)

    2013

  16. [25]

    McGreevy, B

    J. McGreevy, B. Swingle, and K.-A. Tran, Phys. Rev. B 85, 125105 (2012)

    2012

  17. [26]

    S. A. Parameswaran, R. Roy, and S. L. Sondhi, Comptes Rendus Physique14, 816 (2013)

    2013

  18. [27]

    Y.-C. He, F. Grusdt, A. Kaufman, M. Greiner, and A. Vishwanath, Phys. Rev. B96, 201103 (2017)

    2017

  19. [28]

    Repellin, J

    C. Repellin, J. L´ eonard, and N. Goldman, Phys. Rev. A 102, 063316 (2020)

    2020

  20. [29]

    L´ eonard, S

    J. L´ eonard, S. Kim, J. Kwan, P. Segura, F. Grusdt, C. Repellin, N. Goldman, and M. Greiner, Nature619, 495 (2023)

    2023

  21. [31]

    K. Kudo, T. Kariyado, and Y. Hatsugai, J. Phys. Soc. Jpn.86, 103701 (2017)

    2017

  22. [33]

    Andrews, M

    B. Andrews, M. Mohan, and T. Neupert, Phys. Rev. B 103, 075132 (2021)

    2021

  23. [34]

    Paredes, P

    B. Paredes, P. Fedichev, J. I. Cirac, and P. Zoller, Phys. Rev. Lett.87, 010402 (2001)

    2001

  24. [35]

    Raˇ ci¯ unas, F

    M. Raˇ ci¯ unas, F. N.¯Unal, E. Anisimovas, and A. Eckardt, Phys. Rev. A98, 063621 (2018)

    2018

  25. [36]

    R. O. Umucalılar, Phys. Rev. A98, 063629 (2018)

    2018

  26. [37]

    R. O. Umucalılar, E. Macaluso, T. Comparin, and I. Carusotto, Phys. Rev. Lett.120, 230403 (2018)

    2018

  27. [38]

    Macaluso, T

    E. Macaluso, T. Comparin, R. O. Umucalılar, M. Ger- ster, S. Montangero, M. Rizzi, and I. Carusotto, Phys. Rev. Research2, 013145 (2020)

    2020

  28. [39]

    Wang, X.-Y

    B. Wang, X.-Y. Dong, and A. Eckardt, SciPost Phys.12, 095 (2022)

    2022

  29. [40]

    Kapit, P

    E. Kapit, P. Ginsparg, and E. Mueller, Phys. Rev. Lett. 108, 066802 (2012)

    2012

  30. [41]

    Z. Liu, R. N. Bhatt, and N. Regnault, Phys. Rev. B91, 045126 (2015)

    2015

  31. [42]

    Jaworowski, N

    B. Jaworowski, N. Regnault, and Z. Liu, Phys. Rev. B 99, 045136 (2019)

    2019

  32. [43]

    F. D. M. Haldane, Phys. Rev. Lett.55, 2095 (1985)

    1985

  33. [45]

    Greiter, V

    M. Greiter, V. Schnells, and R. Thomale, Phys. Rev. B 93, 245156 (2016)

    2016

  34. [46]

    B. A. Bernevig and F. D. M. Haldane, Phys. Rev. Lett. 100, 246802 (2008)

    2008

  35. [47]

    Seidel and K

    A. Seidel and K. Yang, Phys. Rev. B84, 085122 (2011)

    2011

  36. [48]

    S. C. Zhang, T. H. Hansson, and S. Kivelson, Phys. Rev. Lett.62, 82 (1989)

    1989

  37. [49]

    N. K. Wilkin and J. M. F. Gunn, Phys. Rev. Lett.84, 6 (2000)

    2000

  38. [50]

    F. J. Pauw, F. A. Palm, U. Schollw¨ ock, A. Bohrdt, S. Paeckel, and F. Grusdt, Phys. Rev. Research6, 023180 (2024)

    2024

  39. [51]

    E. J. Bergholtz, T. H. Hansson, M. Hermanns, and A. Karlhede, Phys. Rev. Lett.99, 256803 (2007)

    2007

  40. [52]

    J. K. Jain and R. K. Kamilla, Int. J. Mod. Phys. B11, 2621 (1997)

    1997

  41. [53]

    J. K. Jain and R. K. Kamilla, Phys. Rev. B55, R4895 (1997)

    1997

  42. [54]

    Hermanns, Phys

    M. Hermanns, Phys. Rev. B87, 235128 (2013)

    2013

  43. [55]

    Pu, Y.-H

    S. Pu, Y.-H. Wu, and J. K. Jain, Phys. Rev. B96, 195302 (2017)

    2017

  44. [56]

    N. R. Cooper and N. K. Wilkin, Phys. Rev. B60, R16279 (1999)

    1999

  45. [57]

    Regnault and T

    N. Regnault and T. Jolicoeur, Phys. Rev. Lett.91, 030402 (2003)

    2003

  46. [58]

    Regnault and T

    N. Regnault and T. Jolicoeur, Phys. Rev. B69, 235309 (2004)

    2004

  47. [59]

    Chang, N

    C.-C. Chang, N. Regnault, T. Jolicoeur, and J. K. Jain, Phys. Rev. A72, 013611 (2005)

    2005

  48. [60]

    Regnault, C

    N. Regnault, C. C. Chang, T. Jolicoeur, and J. K. Jain, J. Phys. B: At. Mol. Opt. Phys.39, S89 (2006)

    2006

  49. [61]

    Viefers, T

    S. Viefers, T. H. Hansson, and S. M. Reimann, Physical Review A62, 053604 (2000). 7

    2000

  50. [62]

    Mazaheri, G

    T. Mazaheri, G. Ortiz, Z. Nussinov, and A. Seidel, Phys. Rev. B91, 085115 (2015)

    2015

  51. [63]

    Yu and Z

    G. Yu and Z. Zhu, Composite boson theory of fractional chern insulators (2026), arXiv:2602.14184 [cond-mat.str- el]

    2026

  52. [64]

    Ji and J

    G. Ji and J. Wang, Generalized model fractional quantum hall states on lattices (2026), arXiv:2605.13978 [cond- mat.str-el]

    2026 arXiv

  53. [66]

    K. Iyer, A. Feuerpfeil, V. Cr´ epel, N. Regnault, and C. Mora, Dispersion of Anyon Bloch Bands (2026), arXiv:2604.24859 [cond-mat.str-el]

    2026 arXiv

  54. [69]

    M. L. Meyer, G. J. Sreejith, and S. Viefers, Physical Re- view A89, 043625 (2014)

    2014

  55. [71]

    W. M. C. Foulkes, L. Mitas, R. J. Needs, and G. Ra- jagopal, Rev. Mod. Phys.73, 33 (2001)

    2001

  56. [75]

    A. C. Balram, C. T˝ oke, A. W´ ojs, and J. K. Jain, Phys. Rev. B91, 045109 (2015)

    2015

  57. [77]

    Langbein, Phys

    D. Langbein, Phys. Rev.180, 633 (1969)

    1969

  58. [78]

    D. R. Hofstadter, Phys. Rev. B14, 2239 (1976)

    1976

  59. [79]

    Aidelsburger, M

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

    2013

  60. [80]

    Miyake, G

    H. Miyake, G. A. Siviloglou, C. J. Kennedy, W. C. Bur- ton, and W. Ketterle, Phys. Rev. Lett.111, 185302 (2013)

    2013

  61. [81]

    See Supplemental Material for details of the torus boundary conditions and composite-boson ansatz con- struction, the occupation-orbit reduction and energy- subgroup structure, the projected-basis transformation and variational Monte Carlo evaluation, and the many- body Chern-...

  62. [83]

    M. V. Berry, Proc. R. Soc. Lond. A392, 45 (1984)

    1984

  63. [85]

    X. G. Wen, E. Dagotto, and E. Fradkin, Phys. Rev. B 42, 6110 (1990)

    1990

  64. [86]

    Fukui, Y

    T. Fukui, Y. Hatsugai, and H. Suzuki, J. Phys. Soc. Jpn. 74, 1674 (2005)

    2005

  65. [87]

    Efron,The Jackknife, the Bootstrap and Other Re- sampling Plans, CBMS-NSF Regional Conference Series in Applied Mathematics, Vol

    B. Efron,The Jackknife, the Bootstrap and Other Re- sampling Plans, CBMS-NSF Regional Conference Series in Applied Mathematics, Vol. 38 (Society for Industrial and Applied Mathematics, Philadelphia, 1982). Supplemental Material

    1982

  66. [88]

    MAGNETIC-TRANSLATION CONVENTIONS AND BOUNDARY CONDITIONS OF THE COMPOSITE-BOSON ANSATZ This section distinguishes the magnetic-translation convention of the physical Hofstadter problem from the opposite- chirality reference convention used to construct the torus wave functions...

  67. [89]

    COUNTING SKETCH FOR THE REDUCED ANSATZ DEGENERACY This section gives a simple counting interpretation of the degeneracies observed in the composite-boson ansatz. Here we explain why the ansatz naturally organizes the naive reduced-flux configurations into cyclic orbits, and wh...

  68. [90]

    Let θ= (θ x,θy)(76) be the two twist angles inserted through the two non-contractible cycles of the torus

    EVALUATION OF THE MANY-BODY CHERN NUMBER We compute the many-body Chern number of a selected low-energy manifold by using the gauge-invariant lattice formulation of the Berry curvature in the space of boundary twists [4]. Let θ= (θ x,θy)(76) be the two twist angles inserted th...

  69. [91]

    quasihole

    NUMERICAL BERRY PHASE FOR BRAIDING PINNED QUASIHOLES We calculate the Berry phase by adiabatically moving the pinning potentials along a closed path and following the low-energy quasihole manifold. Here and below, “quasihole” denotes the localized flux defect created by the pi...

  70. [92]

    VARIATIONAL EVALUATION OF THE ANSATZ SUBSPACE We evaluate the same analytic composite-boson trial-state family in two complementary ways. For the smaller systems, the trial states are explicitly projected onto the many-body basis of the lowest Hofstadter band, and the lowest-b...

  71. [93]

    F. D. M. Haldane and E. H. Rezayi, Phys. Rev. B31, 2529 (1985)

    1985

  72. [94]

    Greiter, V

    M. Greiter, V. Schnells, and R. Thomale, Phys. Rev. B93, 245156 (2016)

    2016

  73. [95]

    Lotrič and S

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

    2026

  74. [96]

    Fukui, Y

    T. Fukui, Y. Hatsugai, and H. Suzuki, J. Phys. Soc. Jpn.74, 1674 (2005)

    2005

  75. [97]

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

    1985

  76. [98]

    Zeybek and R

    Z. Zeybek and R. O. Umucalılar, Physics Letters A445, 128259 (2022)

    2022

  77. [99]

    M. V. Berry, Proc. R. Soc. Lond. A392, 45 (1984). 22

    1984

  78. [100]

    Wilczek and A

    F. Wilczek and A. Zee, Phys. Rev. Lett.52, 2111 (1984)

    1984

  79. [101]

    Vanderbilt,Berry Phases in Electronic Structure Theory: Electric Polarization, Orbital Magnetization and Topological Insulators(Cambridge University Press, Cambridge, 2018)

    D. Vanderbilt,Berry Phases in Electronic Structure Theory: Electric Polarization, Orbital Magnetization and Topological Insulators(Cambridge University Press, Cambridge, 2018)

    2018

  80. [102]

    Kapit, P

    E. Kapit, P. Ginsparg, and E. Mueller, Phys. Rev. Lett.108, 066802 (2012)

    2012

  81. [103]

    X. G. Wen, E. Dagotto, and E. Fradkin, Phys. Rev. B42, 6110 (1990)

    1990

  82. [104]

    Arovas, J

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

    1984

  83. [105]

    M. L. Meyer, O. Liabøtrø, and S. Viefers, Journal of Physics A: Mathematical and Theoretical49, 395201 (2016)

    2016

  84. [106]

    S. S. Mandal and J. K. Jain, Phys. Rev. B66, 155302 (2002)

    2002

  85. [107]

    M. L. Meyer, G. J. Sreejith, and S. Viefers, Physical Review A89, 043625 (2014)

    2014

  86. [108]

    Metropolis, A

    N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, and E. Teller, J. Chem. Phys.21, 1087 (1953)

    1953

  87. [109]

    W. M. C. Foulkes, L. Mitas, R. J. Needs, and G. Rajagopal, Rev. Mod. Phys.73, 33 (2001)

    2001

  88. [110]

    M. P. Nightingale and V. Melik-Alaverdian, Phys. Rev. Lett.87, 043401 (2001)

    2001

  89. [111]

    C. J. Umrigar and C. Filippi, Phys. Rev. Lett.94, 150201 (2005)

    2005

  90. [112]

    Toulouse and C

    J. Toulouse and C. J. Umrigar, J. Chem. Phys.126, 084102 (2007)

    2007

  91. [113]

    J. Wang, S. D. Geraedts, E. H. Rezayi, and F. D. M. Haldane, Phys. Rev. B99, 125123 (2019)

    2019

  92. [114]

    A. C. Balram, C. Tőke, A. Wójs, and J. K. Jain, Phys. Rev. B91, 045109 (2015)

    2015

  93. [115]

    Efron,The Jackknife, the Bootstrap and Other Resampling Plans, CBMS-NSF Regional Conference Series in Applied Mathematics, Vol

    B. Efron,The Jackknife, the Bootstrap and Other Resampling Plans, CBMS-NSF Regional Conference Series in Applied Mathematics, Vol. 38 (Society for Industrial and Applied Mathematics, Philadelphia, 1982)

    1982

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.