Pith. sign in

REVIEW 3 major objections 4 minor 62 references

Accelerated quantum Monte Carlo simulations of the attractive Hubbard model on the kagome lattice

T0 review · 3 major / 4 minor · reviewed 2026-08-05 · deepseek-v4-flash

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

Pith's one-line read A fast-Fourier-transform acceleration doubles the reach of determinant quantum Monte Carlo on kagome lattices, revealing a superfluid quantum critical point at |U_c|=4.79(1) with exponents ν=0.88(3), ζ=−0.68(2), η=0.55(7), and showing the p

desk verdict A clean algorithmic advance for DQMC on composite lattices, with physics results that need a convergence check before the exponents are quoted. read the letter →

arxiv 2608.03894 v1 pith:SQ4MSTC5 submitted 2026-08-04 cond-mat.str-el cond-mat.quant-gas

Jie Zhang , Xiang Li , Yu Wang This is my paper · ORCID
classification cond-mat.str-elcond-mat.quant-gas
keywords kagomelatticeattractiveHubbardmodeldeterminantquantumMonteCarloFFTaccelerationsuperfluidcriticalitycharge-density-waveorderfinite-sizescalingDiracfilling
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 claims that a fast-Fourier-transform (FFT) acceleration of propagator multiplications, combined with the delay-update algorithm, makes determinant quantum Monte Carlo (DQMC) simulations of the kagome-lattice Hubbard model feasible on lattices twice as large as previously accessible (up to L=24). On these sizes, the attractive Hubbard model at Dirac filling ρ=2/3 exhibits a zero-temperature superfluid quantum critical point at |U_c|=4.79(1) with critical exponents ν=0.88(3), ζ=−0.68(2), and η=0.55(7), extracted from correlation-ratio crossings and data collapses. The paper also reports that the previously proposed triangle-rule charge-density-wave order at Γ extrapolates to zero in the thermodynamic limit and is likely a finite-size effect, and that the √3×√3 CDW at the K point also vanishes. If correct, the results resolve a longstanding ambiguity about competing orders in the attractive kagome Hubbard model and demonstrate a general acceleration strategy for DQMC on composite lattices.

What carries the argument

The FFT block-diagonalization of the kinetic matrix on composite lattices: for a lattice with n sublattices and L^d unit cells, the spinless kinetic matrix K is conjugated into a block-diagonal matrix Λ^{n×n} with L^d independent n×n blocks (Eq. 4). Propagator multiplication then becomes forward FFT on each sublattice component, block-diagonal multiply, and inverse FFT, lowering complexity from O(N^3) to O(N^2 log N) for the matrix-matrix product. The delay-update algorithm accelerates the auxiliary-field update part, and their combination shifts the practical bottleneck so that the total cost in the experimentally relevant size range (12≤L≤27) scales as N^2.49.

What would settle it

Measure the pairing correlation ratio R_Pair at β=40, 60, and 80 for the smallest (L=12) and largest (L=24) systems at |U| around 4.8; if the crossing point |U_c| shifts by more than the quoted statistical error, the β=30 convergence assumption is violated. Alternatively, compute D(Γ) at |U|=8 for L=18 and L=24 with β=60; if D(Γ) stops decreasing with L, the triangle-rule CDW is not a pure finite-size effect.

Watch

Extended reading notes

Core claim

The central discovery is that the kinetic-energy propagator e^{-Δτ K} on a composite lattice can be applied in O(N log N) per column by Fourier-transforming each sublattice component separately, multiplying by the block-diagonal momentum-space matrix Λ^{n×n}, and transforming back. Combined with delay updates for the auxiliary-field sweeps, this reduces the practical cost of projector DQMC and enables ground-state simulations of the kagome lattice with L=24. Using these larger sizes, the authors find a clean crossing of the pairing correlation ratio R_Pair at |U_c|=4.79(1) with critical exponents ν=0.88(3), ζ=−0.68(2), η=0.55(7). They further extrapolate the CDW order parameters D(Γ) and D(K

Load-bearing premise

The simulations assume that a projection length of β=30 is converged to the ground state for all system sizes and couplings, and the Trotter time step is never reported; if finite-β or finite-Δτ effects differ across sizes, the critical-point and CDW extrapolations could be biased.

Editorial extensions

If this is right

  • DQMC on kagome lattices can now reach L=24, doubling the previous limit and enabling reliable finite-size scaling of correlation ratios at the Dirac filling.
  • The attractive Hubbard model at ρ=2/3 has a superfluid quantum critical point at |U_c|=4.79(1) with exponents ν=0.88(3), ζ=−0.68(2), η=0.55(7), giving a benchmark for future studies.
  • The triangle-rule CDW order proposed in earlier DQMC work does not survive in the thermodynamic limit; the √3×√3 CDW seen in the Holstein model is also absent in the attractive Hubbard model.
  • The FFT block-diagonalization applies to other composite lattices (triangular, honeycomb) and to models beyond the attractive Hubbard model, such as SU(2N) and three-component Hubbard models.

Reading between the lines

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

  • The same FFT acceleration should carry over to finite-temperature DQMC and to repulsive Hubbard models, since it relies only on the kinetic propagator structure; testing this would broaden the method's reach beyond the ground-state attractive case.
  • If the reported exponents are confirmed, the transition at the Dirac filling may belong to a known universality class; comparing ν≈0.88 and η≈0.55 with analytic or large-scale results for chiral XY or Gross-Neveu-Yukawa fixed points would be a natural next step.
  • The vanishing of D(K) in the attractive Hubbard model while a √3×√3 CDW appears in the Holstein model suggests that the phonon-mediated interaction in the Holstein model, rather than the pure Hubbard attraction, is what stabilizes that order; checking this by adding phonon degrees of freedom to the Hubbard model could isolate the mechanism.
  • The N^2.49 scaling is a crossover artifact: for lattices beyond L≈27 the delay update dominates and the cost returns to O(N^3), so the method's advantage is strongest in the experimentally relevant size regime.
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

3 major / 4 minor

Summary. The manuscript presents an FFT-based acceleration scheme for propagator multiplications in projector determinant quantum Monte Carlo (DQMC) on composite lattices, combining it with delay updates. The method is applied to the attractive Hubbard model on the kagome lattice at Dirac filling. The authors report a superfluid transition at |U_c|=4.79(1) with critical exponents ν=0.88(3), ζ=-0.68(2), η=0.55(7) from finite-size scaling on lattices up to L=24, and conclude that the previously proposed triangle-rule CDW order does not survive in the thermodynamic limit. They also characterize an effective computational scaling of N^2.49 for experimentally relevant system sizes.

Significance. The algorithmic claim is well motivated and, if correct, represents a practical advance: the block-diagonalization derivation in Eqs. (2)-(6) is clean and the reduction of propagator multiplication from O(N^3) to O(N^2 log N) is a genuine mathematical identity. Combining this with delay updates is a timely contribution. The physics claims—reliable critical exponents at the Dirac filling and the absence of triangle-rule CDW—are significant for the kagome Hubbard model, but the quantitative conclusions rest on a single, unverifiable projection length (β=30) and an unreported Trotter step. Without convergence evidence, the reported exponents and phase-diagram conclusions remain provisional. No code or data are released, which limits reproducibility but is not a flaw in the derivation itself.

major comments (3)
  1. [Sec. IV, Figs. 3-5] All data used for the finite-size scaling are taken at projection length β=30, with no β-dependence shown for any observable. The FSS forms in Eqs. (9)-(10) assume ground-state scaling, but finite-β contamination is generally size-dependent: for L=24, a Dirac-like gap ΔE ~ v_F/L (with v_F ~ t) gives ΔE β ~ 1.2, so the largest lattice can have substantially larger excited-state contributions than L=12. Such size-dependent projection error would bias |U_c| and the exponents. A β-convergence study for representative L (e.g., L=12, 18, 24) and observables (R_Pair, P, D(Γ), D(K)) is required before the quoted error bars can be considered reliable.
  2. [Eq. (5) and all simulations] The Trotter time step Δτ is never specified. Since Eq. (5) defines the discrete-time propagator and all reported results depend on this parameter, the manuscript should state the value used and demonstrate convergence (e.g., by comparing Δτ and Δτ/2 for a representative system). Without this, the possibility of a Trotter-discretization bias in the extracted critical parameters cannot be assessed.
  3. [Sec. IV, Fig. 4 and Sec. V, Fig. 6] The χ2 minimization procedure is not fully documented. The manuscript reports the minimum of χ2 but does not give the reduced χ2 value, the number of data points, the fit ranges, or the sensitivity to the polynomial order. A fourth-order polynomial with five parameters can overfit, and the claim of a 'very high' collapse quality should be quantified. This is particularly important because the central conclusion of 'reliable' exponents depends on the FSS fits, not just on the algorithmic speedup.
minor comments (4)
  1. [Throughout] The text contains several typographical errors (e.g., 'attract ive' in the title, 'partical', 'hebavior', 'deffinition', 'acclerated'). A careful proofreading pass is needed.
  2. [Fig. 2 caption] The range 12 ≤ L ≤ 27 is mentioned, but the actual values of L used in the runtime analysis are not listed. Please specify the data points and the number of cores used for each point.
  3. [Eq. (7)] The definition of δk is vague ('the smallest available neighboring wave vector'). Please specify it explicitly, e.g., in terms of the reciprocal lattice vectors, to reproduce the correlation ratio.
  4. [Uncited material] No information is given about the boundary conditions (presumably periodic), the number of independent samples used for error bars, or the autocorrelation times. These are standard details that should be included for reproducibility.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the superfluid exponents and CDW extrapolations are outputs of the DQMC simulations; the FFT acceleration is a mathematical identity, and the self-citations are methodological, not load-bearing.

full rationale

The paper's central physics claims are derived from fresh simulation data, not from its inputs. The FFT acceleration is justified by the block-diagonalization of the kinetic matrix under the discrete Fourier transform (Eqs. 2-6); this is a mathematical identity and does not presuppose the target phase diagram. The superfluid critical point and exponents (|U_c|=4.79(1), ν=0.88(3), ζ=-0.68(2), η=0.55(7)) are extracted from measured correlation ratios and order parameters via χ² minimization and standard finite-size scaling forms (Eqs. 7-10, Fig. 4). No parameter is fitted to a subset of data and then renamed a prediction: the scaling variables are the independent variables, and the critical parameters are the fitted outputs. Likewise, the CDW conclusion follows from direct finite-size extrapolations of D(Γ) and D(K) (Fig. 6), not from an assumed order. The paper does cite the same group's prior work (e.g., Ref. [40] for the symmetric pairing structure factor, and Refs. [50-56] for possible future applications), but these citations are methodological and not load-bearing; the scaling ansatz is also referenced to independent external works [36,41,42]. The unstated Trotter step and the fixed projection length β=30 are convergence/numerical-error concerns, not circularity. No equation or fitted quantity reduces by construction to another fitted quantity or to a self-citation. Therefore the circularity score is 0.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

No new particles, mediators, forces, or dimensions are postulated. The free parameters are numerical convergence settings rather than hidden fitted constants. The critical parameters U_c, ν, ζ, and η are reported outputs of the fits, not inputs to a derivation.

free parameters (2)
  • Projection length β = 30
    Fixed for all ground-state simulations; no convergence sweep reported, so finite-β bias is uncontrolled.
  • Trotter time step Δτ
    Required in Eq. (5) but never specified in the text; controls Trotter error and is essential for reproduction.
assumptions (5)
  • standard math The kinetic matrix K is translationally invariant and exactly block-diagonalized by the discrete Fourier transform over L^d unit cells times identity on sublattices.
    Eqs. (2)-(4); standard Fourier diagonalization; relies on periodic boundary conditions.
  • domain assumption The PQMC trial wave function constructed from the lowest N_occ non-interacting eigenstates has finite overlap with the true attractive-Hubbard ground state for the fillings studied.
    Sec. II, 'the filling number is controlled by the trial wave function'; standard for DQMC but not separately justified here.
  • ad hoc to paper A projection length β=30 yields converged ground-state properties for all L and U.
    Sec. IV and V; stated without a convergence check.
  • domain assumption The finite-size scaling ansatz P=L^{ζ/ν} P̃(δ|U|L^{1/ν}) and RPair=R̃(δ|U|L^{1/ν}) with dynamical exponent z=1 applies at the Dirac filling.
    Eqs. (9)-(10); z=1 is taken from Refs [43-46].
  • domain assumption The CDW field D(Q) defined in Eq. (11) with complex sublattice phases at Γ detects triangle-rule order without cancellation under Monte Carlo averaging.
    Sec. V; the non-cancellation statement is asserted, not proven.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Accelerated quantum Monte Carlo simulations of the attractive Hubbard model on the kagome lattice." pith.science (2026). https://pith.science/paper/SQ4MSTC5

@misc{pith2026260803894,
  author       = {Pith},
  title        = {Pith review of: Accelerated quantum Monte Carlo simulations of the attractive Hubbard model on the kagome lattice},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SQ4MSTC5}},
  note         = {Machine review of arXiv:2608.03894}
}
abstract

The recent discovery of several families of kagome materials and experimental realization of optical kagome lattices have stimulated growing numerical studies of interaction-driven correlated states on the kagome lattice. Among the available numerical approaches, determinant quantum Monte Carlo (DQMC) is a powerful method for investigating such strongly correlated states. However, the accessible system sizes of existing DQMC simulations remain limited, preventing reliable finite-size scaling analyses. Here we develop a general acceleration scheme based on fast Fourier transform (FFT) for propagator multiplications on composite lattices and combine it with the delay-update algorithm, enabling simulations on system sizes twice as large as those of previous DQMC studies, allowing reliable finite-size scaling analyses of the attractive kagome-lattice Hubbard model. Our large-scale simulations reveal the interaction-driven zero-temperature superfluid quantum criticality at the Dirac filling and provide reliable estimates of the associated critical exponents. Besides, we find no evidence that the previously proposed triangle-rule charge-density-wave order survives in the thermodynamic limit, suggesting that it is likely a finite-size effect. Moreover, for system sizes accessible in current two-dimensional optical lattice experiments, the combined FFT and delay-update scheme exhibits an effective computational cost scaling as $N^{2.49}$, substantially below the $\mathcal{O}(N^3)$ computational cost of conventional DQMC simulations.

Figures

Figures reproduced from arXiv: 2608.03894 by the authors.

Figure 2
Figure 2. FIG. 2. (a) Percentages of the computational time occupied b [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 ↗
Figure 3
Figure 3. FIG. 3. (a) The correlation ratio [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figures from the paper (1 more)
Figure 6
Figure 6. Figure 6: FIG. 6. Finite-size extrapolations of the CDW order paramet [PITH_FULL_IMAGE:figures/full_fig_p005_6.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

62 extracted references · 61 canonical work pages

  1. [18]

    K. Y . Chen, N. N. Wang, Q. W. Yin, Y . H. Gu, K. Jiang, Z. J. Tu, C. S. Gong, Y . Uwatoko, J. P. Sun, H. C. Lei, J. P. Hu, and J.-G. Cheng, Double Superconducting Dome and Triple En- hancement of Tc in the Kagome Superconductor CsV3Sb5 un- 7 der High Pressure, Phys. Rev. Lett. 126, 247001 (2021)

    work page 2021

  2. [1]

    (A1) K is Fourier transformed into the diagonal matrix Λ{1×1} tri : K =(F ⊗ F )[I ⊗ Λs + Λs ⊗ I + Λs ⊗ Λs + h.c.] ( F † ⊗ F †) =(F ⊗ F )Λ{1×1} tri ( F † ⊗ F †)

    triangular lattice The spinless kinetic matrix K on the triangular lattice can be expressed as K = I ⊗ Ks + Ks ⊗ I + Ks ⊗ Ks + h.c. (A1) K is Fourier transformed into the diagonal matrix Λ{1×1} tri : K =(F ⊗ F )[I ⊗ Λs + Λs ⊗ I + Λs ⊗ Λs + h.c.] ( F † ⊗ F †) =(F ⊗ F )Λ{1×1} tri ( F † ⊗ F †) . (A2)

    work page

  3. [2]

    honeycomb lattice The spinless kinetic matrix K on the honeycomb lattice can be expressed as K = (I ⊗ I + I ⊗ Ks + Ks ⊗ I) ⊗ δ + h.c., (A3) where the 2-dimensional matrices δ encodes the intra-cell hop- ping, containing a single nonzero entry at (δ)2,1. K is Fourier transformed into the block-diagonal matrix Λ{2×2} hon : K =(F ⊗ F ⊗ I2)[(I ⊗ I + I ⊗ Λs + ...

    work page

  4. [3]

    B. R. Ortiz, L. C. Gomes, J. R. Morey, M. Winiarski, M. Bor- delon, J. S. Mangum, I. W. H. Oswald, J. A. Rodriguez-Rivera, J. R. Neilson, S. D. Wilson, E. Ertekin, T. M. McQueen, and E. S. Toberer, New kagome prototype materials: discovery of KV3Sb5, RbV3Sb5, and CsV3Sb5, Phys. Rev. Mater. 3, 094407 (2019)

    work page 2019

  5. [4]

    These re - sults suggest that the proposed triangle-rule CDW orientat ion in individual Monte Carlo configurations is likely a finite-s ize effect

    For all fillings considered, D(Γ) extrapolates to zero in the thermodynamic limit, indicating the absence of a stable triangle-rule CDW ori- entation in individual Monte Carlo configurations. These re - sults suggest that the proposed triangle-rule CDW orientat ion in individual Monte Carlo configurations is likely a finite-s ize effect. Likewise, D(K) also ex...

    work page

  6. [5]

    Schäfer, T

    F. Schäfer, T. Fukuhara, S. Sugawa, Y . Takasu, and Y . Takahashi, Tools for quantum simulation with ultracold atoms in optica l lattices, Nat. Rev. Phys. 2, 411 (2020)

    work page 2020

  7. [6]

    First,[ ( F †) ⊗d ⊗ In ] vj Fourier transforms the input vector vj into a new vector v′ j through FFT

    can be interpreted as three consecutive steps. First,[ ( F †) ⊗d ⊗ In ] vj Fourier transforms the input vector vj into a new vector v′ j through FFT. Since the FFT is indepen- dently applied to the n sublattice components, each contain- ing Ld entries, the computational complexity of this step is O(nLd log ( Ld) ). Second, v′ j is multiplied by the block-...

    work page

  8. [7]

    L. Y e, M. Kang, J. Liu, F. von Cube, C. R. Wicker, T. Suzuki, C. Jozwiak, A. Bostwick, E. Rotenberg, D. C. Bell, L. Fu, R. Comin, and J. G. Checkelsky, Massive Dirac fermions in a ferromagnetic kagome metal, Nature(London) 555, 638 (2018)

    work page 2018

Show all 62 references
  1. [8]

    E. Liu, Y . Sun, N. Kumar, L. Muechler, A. Sun, L. Jiao, S.- Y . Yang, D. Liu, A. Liang, Q. Xu, J. Kroder, V . Süß, H. Borrmann, C. Shekhar, Z. Wang, C. Xi, W. Wang, W. Schnelle, S. Wirth, Y . Chen, S. T. B. Goennenwein, and C. Felser, Giant anomalous Hall effect in a ferromagn...

    2018

  2. [9]

    X. Feng, K. Jiang, Z. Wang, and J. Hu, Chiral flux phase in th e Kagome superconductor A V3Sb5, Sci. Bull. 66, 1384 (2021)

    2021

  3. [10]

    G.-B. Jo, J. Guzman, C. K. Thomas, P. Hosur, A. Vishwanath , and D. M. Stamper-Kurn, Ultracold Atoms in a Tunable Optical Kagome Lattice, Phys. Rev. Lett. 108, 045305 (2012)

    2012

  4. [11]

    L. Nie, K. Sun, W. Ma, D. Song, L. Zheng, Z. Liang, P. Wu, F. Yu, J. Li, M. Shan, D. Zhao, S. Li, B. Kang, Z. Wu, Y . Zhou, K. Liu, Z. Xiang, J. Ying, Z. Wang, T. Wu, and X. Chen, Charge- density-wave-driven electronic nematicity in a kagome sup er- conductor, Nature(London) 6...

    2022

  5. [12]

    Employing our accelerated DQMC scheme, we in- vestigate the SF critical behavior at the challenging Dirac fill- ing on system sizes from L = 12 to L = 24

    [18]. Employing our accelerated DQMC scheme, we in- vestigate the SF critical behavior at the challenging Dirac fill- ing on system sizes from L = 12 to L = 24 . These sub- stantially large system sizes enable reliable finite-size scalings required for an accurate determination ...

  6. [13]

    Mekata, Kagome: The Story of the Basketweave Lattice, Phys

    M. Mekata, Kagome: The Story of the Basketweave Lattice, Phys. Today 56(2), 12 (2003)

    2003

  7. [14]

    M. Kang, S. Fang, J.-K. Kim, B. R. Ortiz, S. H. Ryu, J. Kim, J . Y oo, G. Sangiovanni, D. Di Sante, B.-G. Park, C. Jozwiak, A. Bostwick, E. Rotenberg, E. Kaxiras, S. D. Wilson, J.-H. Park , and R. Comin, T wofold van Hove singularity and origin of charge order in topological k...

    2022

  8. [15]

    Jiang, J.-X

    Y .-X. Jiang, J.-X. Yin, M. M. Denner, N. Shumiya, B. R. Ort iz, G. Xu, Z. Guguchia, J. He, M. S. Hossain, X. Liu, J. Ruff, L. Kautzsch, S. S. Zhang, G. Chang, I. Belopolski, Q. Zhang, T. A. Cochran, D. Multer, M. Litskevich, Z.-J. Cheng, X. P. Yang, Z . Wang, R. Thomale, T. Ne...

    2021

  9. [16]

    Mielke III, D

    C. Mielke III, D. Das, J.-X. Yin, H. Liu, R. Gupta, Y .-X. J iang, M. Medarde, X. Wu, H. C. Lei, J. Chang, P. Dai, Q. Si, H. Miao, R. Thomale, T. Neupert, Y . Shi, R. Khasanov, M. Z. Hasan, H. Luetkens, and Z. Guguchia, Time-reversal symmetry-breaki ng charge order in a kagome...

    2022

  10. [17]

    H. D. Scammell, J. Ingham, T. Li, and O. P. Sushkov, Chira l excitonic order from twofold van Hove singularities in kago me metals, Nat. Commun. 14, 605 (2023)

    2023

  11. [19]

    F. H. Yu, D. H. Ma, W. Z. Zhuo, S. Q. Liu, X. K. Wen, B. Lei, J. J. Ying, and X. H. Chen, Unusual competition of supercon- ductivity and charge-density-wave state in a compressed to po- logical kagome metal, Nat. Commun. 12, 3645 (2021)

    2021

  12. [20]

    X. Wu, T. Schwemmer, T. Müller, A. Consiglio, G. Sangiovanni, D. Di Sante, Y . Iqbal, W. Hanke, A. P. Schnyder, M. M. Denner, M. H. Fischer, T. Neupert, and R. Thomale, Nature of Un- conventional Pairing in the Kagome Superconductors AV3Sb5 (A = K , Rb, Cs), Phys. Rev. Lett. 1...

    2021

  13. [21]

    H. Tan, Y . Liu, Z. Wang, and B. Yan, Charge Density Waves and Electronic Properties of Superconducting Kagome Metal s, Phys. Rev. Lett. 127, 046401 (2021)

    2021

  14. [22]

    J. Zhao, W. Wu, Y . Wang, and S. A. Yang, Electronic cor- relations in the normal state of the kagome superconductor KV3Sb5, Phys. Rev. B 103, L241117 (2021)

    2021

  15. [23]

    Sun and X

    F. Sun and X. Y . Xu, Delay update in determinant quantum Monte Carlo, Phys. Rev. B 109, 235140 (2024)

    2024

  16. [24]

    X. Zhu, W. Han, S. Feng, and H. Guo, Quantum Monte Carlo study of the attractive kagome-lattice Hubbard model, Phys. Rev. Res. 5, 023037 (2023)

    2023

  17. [25]

    Blankenbecler, D

    R. Blankenbecler, D. J. Scalapino, and R. L. Sugar, Mont e Carlo calculations of coupled boson-fermion systems. I, Phys. Rev. D 24, 2278 (1981)

    1981

  18. [26]

    J. E. Hirsch, Discrete Hubbard-Stratonovich transfor mation for fermion lattice models, Phys. Rev. B 28, 4059(R) (1983)

    1983

  19. [27]

    We calculate the total computational time at different syste m sizes L, as shown in Fig

    Therefore, it is important to characterize the computa- tional performance in the experimentally relevant size regime. We calculate the total computational time at different syste m sizes L, as shown in Fig. 2 (b). The measured total com- putational time remains consistently be...

  20. [28]

    Alvarez, M

    G. Alvarez, M. S. Summers, D. E. Maxwell, M. Eisenbach, J . S. Meredith, J. M. Larkin, J. Levesque, T. A. Maier, P. R. C. Kent , E. F. D’ Azevedo, and T. C. Schulthess, New algorithm to enable 400+ TFlop/s sustained performance in simulations of disor - der effects in high- Tc ...

    2008

  21. [29]

    P. K. V . V . Nukala, T. A. Maier, M. S. Summers, G. Alvarez, and T. C. Schulthess, Fast update algorithm for the quantum Monte Carlo simulation of the Hubbard model, Phys. Rev. B80, 195111 (2009)

    2009

  22. [30]

    Sun and X

    F. Sun and X. Y . Xu, Boosting determinant quantum Monte Carlo with submatrix updates: Unveiling the phase diagram o f the 3D Hubbard model, SciPost Phys. 18, 055 (2025)

    2025

  23. [31]

    Du and Y .- Y

    H. Du and Y .- Y . He, Accelerating ground-state auxiliar y-field quantum Monte Carlo simulations by delayed update and block force-bias update, Phys. Rev. B 112, 235120 (2025)

    2025

  24. [32]

    Chuang, Y .-S

    C.-C. Chuang, Y .-S. Chiu, Q.-L. Kao, Z.-H. Chen, and C.- R. Lee, Accelerating block checkerboard method on GPU for performance enhancement of 2D and 3D quantum Monte Carlo simulations, in Proceedings of the 4th IEEE International Con- ference on Cloud Computing Technology and...

    2012

  25. [33]

    Lee, Minimal split checkerboard method for expon enti- ating sparse matrices and its applications in quantum stati stical mechanics, SIAM J

    C.-R. Lee, Minimal split checkerboard method for expon enti- ating sparse matrices and its applications in quantum stati stical mechanics, SIAM J. Sci. Comput. 35, C143 (2013)

    2013

  26. [34]

    Cohen-Stead, S

    B. Cohen-Stead, S. M. Costa, J. Neuhaus, A. T. Ly, Y . Zhan g, R. Scalettar, K. Barros, and S. Johnston, SmoQyDQMC.jl: A flexible implementation of determinant quantum Monte Carlo for Hubbard and electron-phonon interactions, SciPost Phys. Codebases 29 (2024)

    2024

  27. [35]

    Y .- Y . He, H. Shi, and S. Zhang, Reaching the Continuum Limit in Finite-Temperature Ab Initio Field-Theory Computations in Many-Fermion Systems, Phys. Rev. Lett. 123, 136402 (2019)

    2019

  28. [36]

    Celledoni and A

    E. Celledoni and A. Iserles, Approximating the exponen tial from a Lie algebra to a Lie group,Math. Comp. 69, 1457 (2000)

    2000

  29. [37]

    Z. Bai, W. Chen, R. Scalettar, and I. Yamazaki, Numerica l methods for quantum Monte Carlo simulations of the Hubbard model, in Multi-Scale Phenomena in Complex Fluids , edited by T. Y . Hou, C. Liu, and J.-G. Liu (Higher Education Press and World Scientific, 2009), pp. 1–110

    2009

  30. [38]

    Y .-F. Song, Y . Deng, and Y .- Y . He, Magnetic, thermodynamic, and dynamical properties of the three-dimensional fermion ic Hubbard model: A comprehensive Monte Carlo study, Phys. Rev. B 111, 035123 (2025)

    2025

  31. [39]

    Z. Meng, L. Wang, W. Han, F. Liu, K. Wen, C. Gao, P. Wang, C. Chin, and J. Zhang, Atomic Bose–Einstein condensate in twisted-bilayer optical lattices, Nature(London) 615, 231–236 (2023)

    2023

  32. [40]

    J. Yang, L. Liu, J. Mongkolkiattichai, and P. Schauss, S ite- Resolved Imaging of Ultracold Fermions in a Triangular-Lattice Quantum Gas Microscope, PRX Quantum 2, 020344 (2021)

    2021

  33. [41]

    Hartke, B

    T. Hartke, B. Oreg, C. Turnbaugh, N. Jia, and M. Zwierlei n, Direct observation of nonlocal fermion pairing in an attrac tive Fermi-Hubbard gas, Science 381, 82–86 (2023)

    2023

  34. [42]

    Parisen Toldin, M

    F. Parisen Toldin, M. Hohenadler, F. F. Assaad, and I. F. Her- but, Fermionic quantum criticality in honeycomb and π-flux Hubbard models: Finite-size scaling of renormalization-g roup- invariant observables from quantum Monte Carlo, Phys. Rev. B 91, 165108 (2015)

    2015

  35. [43]

    Binder, Finite size scaling analysis of Ising model b lock distribution functions, Z

    K. Binder, Finite size scaling analysis of Ising model b lock distribution functions, Z. Physik B – Condensed Matter 43, 119 (1981)

    1981

  36. [44]

    Pujari, T

    S. Pujari, T. C. Lang, G. Murthy, and R. K. Kaul, Interact ion- Induced Dirac Fermions from Quadratic Band Touching in Bi- layer Graphene, Phys. Rev. Lett. 117, 086404 (2016)

    2016

  37. [45]

    F. F. Assaad and H. G. Evertz, World-line and Determinan tal Quantum Monte Carlo Methods for Spins, Phonons and Elec- trons, in Computational Many-Particle Physics, Lecture Notes in Physics, Vol. 739, edited by H. Fehske, R. Schneider, and A. Weiße (Springer, Berlin, Heidelber...

    2008

  38. [46]

    H. Xu, X. Li, Z. Zhou, X. Wang, L. Wang, C. Wu, and Y . Wang, Trion states and quantum criticality of attractive SU(3) Di rac fermions, Phys. Rev. Res. 5, 023180 (2023)

    2023

  39. [47]

    Janke, Monte Carlo methods in classical statistical physics, in Computational Many-Particle Physics, edited by H

    W. Janke, Monte Carlo methods in classical statistical physics, in Computational Many-Particle Physics, edited by H. Fehske, R. Schneider, and A. Weiße (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008), pp. 79–140

    2008

  40. [48]

    H. Shao, W. Guo, and A. W. Sandvik, Quantum criticality w ith two length scales, Science 352, 213 (2016)

    2016

  41. [49]

    Senthil, A

    T. Senthil, A. Vishwanath, L. Balents, S. Sachdev, and M . P. A. Fisher, Deconfined quantum critical points, Science 303, 1490–1494 (2004)

    2004

  42. [50]

    Senthil, L

    T. Senthil, L. Balents, S. Sachdev, A. Vishwanath, and M . P. A. Fisher, Quantum criticality beyond the Landau–Ginzburg– Wilson paradigm, Phys. Rev. B 70, 144407 (2004)

    2004

  43. [51]

    A. W. Sandvik, Evidence for Deconfined Quantum Critical ity in a T wo-Dimensional Heisenberg Model with Four-Spin Inter- actions, Phys. Rev. Lett. 98, 227202 (2007)

    2007

  44. [52]

    D. Wang, L. Wang, and C. Wu, Slater and Mott insulating st ates in the SU(6) Hubbard model, Phys. Rev. B100, 115155 (2019)

    2019

  45. [53]

    Melchert, autoScale.py – A program for automatic fini te- size scaling analyses: A user’s guide, arXiv:0910.5403 (2009)

    O. Melchert, autoScale.py – A program for automatic fini te- size scaling analyses: A user’s guide, arXiv:0910.5403 (2009)

    2009 arXiv

  46. [54]

    Houdayer and A

    J. Houdayer and A. K. Hartmann, Low-temperature behavi or of two-dimensional Gaussian Ising spin glasses, Phys. Rev. B 70, 014418 (2004)

    2004

  47. [55]

    Bradley, B

    O. Bradley, B. Cohen-Stead, S. Johnston, K. Barros, and R. T. Scalettar, Charge order in the kagome lattice Holstein mode l: a hybrid Monte Carlo study, npj Quantum Materials 8, 21 (2023). 8

    2023

  48. [56]

    Z. Zhou, Z. Cai, C. Wu, and Y . Wang, Quantum Monte Carlo simulations of thermodynamic properties of SU(2N ) ultracold fermions in optical lattices, Phys. Rev. B 90, 235139 (2014)

    2014

  49. [57]

    Z. Zhou, D. Wang, Z. Y . Meng, Y . Wang, and C. Wu, Mott insulating states and quantum phase transitions of correla ted SU(2N ) Dirac fermions, Phys. Rev. B 93, 245157 (2016)

    2016

  50. [58]

    Z. Zhou, D. Wang, C. Wu, and Y . Wang, Finite-temperature valence-bond-solid transitions and thermodynamic proper ties of interacting SU(2N ) Dirac fermions, Phys. Rev. B95, 085128 (2017)

    2017

  51. [59]

    Z. Zhou, C. Wu, and Y . Wang, Mott transition in the π-flux SU(4) Hubbard model on a square lattice, Phys. Rev. B 97, 195122 (2018)

    2018

  52. [60]

    X. Li, H. Xu, and Y . Wang, Quantum Monte Carlo simula- tions of thermodynamic properties of attractive SU(3) Dirac fermions, Phys. Rev. B 108, 165102 (2023)

    2023

  53. [61]

    Li and Y

    X. Li and Y . Wang, Coexisting Néel and charge density wave or- ders in attractive three-color fermions,Phys. Rev. B110, 115105 (2024)

    2024

  54. [62]

    X. Li, Y . Li, Q. Fu, and Y . Wang, Trion ordering in the attractive three-color Hubbard model on a π-flux square lattice, Phys. Rev. A 112, 063319 (2025)

    2025

Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.