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arXiv:2608.03894

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

Jie Zhang, Xiang Li, Yu Wang

Published Aug 4, 2026• 0 citations
arXivHTMLPDF

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.

DOI
10.48550/arxiv.2608.03894

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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 N2.49N^{2.49}N2.49, substantially below the O(N3)\mathcal{O}(N^3)O(N3) computational cost of conventional DQMC simulations.
Determinant Quantum Monte Carlo (DQMC) provides numerically exact solutions for strongly correlated fermionic systems but faces significant computational challenges with increasing system size. While submatrix updates were originally developed for Hirsch-Fye QMC with onsite interactions at finite temperatures, their comprehensive application in DQMC has remained unexplored despite noted algorithmic similarities. We present the first comprehensive application of submatrix updates in DQMC, significantly extending beyond the original scope by enabling simulations with extended interactions and at zero temperature. Building upon conventional fast updates and delay updates, our generalized implementation achieves an order-of-magnitude improvement in computational efficiency, enabling simulations of the half-filled Hubbard model on lattices up to 8,000 sites - a scale previously challenging with standard DQMC implementations. This enhanced computational capability allows us to accurately determine the finite-temperature phase diagram of the 3D Hubbard model at half-filling. Our findings not only shed light on the phase transitions within these complex systems but also pave the way for more effective simulations of strongly correlated electrons, potentially guiding experimental efforts in cold atom simulations of the 3D Hubbard model.
We present a systematic study of the quantum phase transition from a Dirac semimetal to an antiferromagnetic Mott insulator in the honeycomb-lattice Hubbard model, using finite-temperature auxiliary-field quantum Monte Carlo (AFQMC) simulations. Given the dynamical critical exponent z=1z=1z=1 at criticality, we scale the inverse temperature βββ with the linear system size LLL, i.e., βt=Lβt=Lβt=L (with ttt as the hopping amplitude), to approach the thermodynamic and zero-temperature limits simultaneously. On the algorithmic front, we generalize the fast Fourier transform in path propagation to multi-sublattice systems and employ the delayed update technique in configuration updates to accelerate the AFQMC simulations. These advances enable us to access unprecedented system sizes of up to 352835283528 sites (with L=42L=42L=42 and βt=42βt = 42βt=42). We additionally use twist-averaged boundary conditions to reduce the finite-size effects in computed observables. On the physical side, we determine the Dirac quantum criticality via finite-size scaling analysis of the spin correlations and off-diagonal single-particle Green's function, and locate the critical point at Uc/t=3.731(4)U_c/t=3.731(4)Uc​/t=3.731(4) with the critical exponents ν=1.097(7)ν=1.097(7)ν=1.097(7), ηϕ=0.72(2)η_ϕ=0.72(2)ηϕ​=0.72(2) and ηψ=0.155(6)η_ψ=0.155(6)ηψ​=0.155(6). We further corroborate the critical point by examining several relevant quantities, including the quasiparticle weight, Fermi-liquid parameter, charge compressibility, UUU-derivative of double occupancy, and fidelity susceptibility. Our work establishes an independent benchmark for the chiral Heisenberg Gross-Neveu-Yukawa criticality in the honeycomb Hubbard model, offering a complementary perspective to previous ground-state simulations.
Determinant quantum Monte Carlo (DQMC) is a widely used unbiased numerical method for simulating strongly correlated electron systems. However, the update process in DQMC is often a bottleneck for its efficiency. To address this issue, we propose a generalized delay update scheme that can handle both on-site and extended interactions. Our delay update scheme can be implemented in both zero-temperature and finite-temperature versions of DQMC. We apply the delay update scheme to various strongly correlated electron models and evaluate its efficiency under different conditions. Our results demonstrate that the proposed delay update scheme significantly improves the efficiency of DQMC simulations, enabling it to simulate larger system sizes.
We introduce methodologies for highly scalable quantum Monte Carlo simulations of electron-phonon models, and we report benchmark results for the Holstein model on the square lattice. The determinant quantum Monte Carlo (DQMC) method is a widely used tool for simulating simple electron-phonon models at finite temperatures, but it incurs a computational cost that scales cubically with system size. Alternatively, near-linear scaling with system size can be achieved with the hybrid Monte Carlo (HMC) method and an integral representation of the Fermion determinant. Here, we introduce a collection of methodologies that make such simulations even faster. To combat "stiffness" arising from the bosonic action, we review how Fourier acceleration can be combined with time-step splitting. To overcome phonon sampling barriers associated with strongly bound bipolaron formation, we design global Monte Carlo updates that approximately respect particle-hole symmetry. To accelerate the iterative linear solver, we introduce a preconditioner that becomes exact in the adiabatic limit of infinite atomic mass. Finally, we demonstrate how stochastic measurements can be accelerated using fast Fourier transforms. These methods are all complementary and, combined, may produce multiple orders of magnitude speedup, depending on model details.
Flat-band systems provide a promising platform for exploring exotic correlated phases owing to quenched kinetic energy and enhanced interaction effects. In this work, we investigate the flat-band superfluid transition in the attractive kagome Hubbard model using large-scale determinant quantum Monte Carlo simulations accelerated by fast Fourier transforms and delay updates. Through systematic calculations of the transition temperature, spin susceptibility, and density compressibility, we establish the finite-temperature phase diagram at the flat-band filling ρ=5/3ρ=5/3ρ=5/3 and find a clear linear scaling of the superfluid transition temperature with the interaction strength, Tc=0.0314∣U∣T_{\mathrm{c}}=0.0314|U|Tc​=0.0314∣U∣. Our results reveal a wide separation of temperature scales between local pair formation and global phase coherence, which stabilizes a broad intermediate supercompressible bosonic fluid region of preformed pairs that bridges the normal Fermi gas and the flat-band superfluid. We further demonstrate that both the transition temperature and density compressibility are significantly enhanced in the flat band compared to van Hove fillings. Our findings highlight the essential role of flat-band-enhanced pair-density fluctuations in separating the pairing and phase-coherence scales, offering clear experimental signatures for probing flat-band superfluidity and superconductivity in cold-atom and kagome-material platforms.
Using auxiliary-field quantum Monte Carlo simulations, we study the zero-temperature attractive Hubbard model on the kagome lattice at various relevant electronic fillings. The low-energy physics at 2/32/32/3-density is influenced by the Dirac point at the Fermi energy, wherein we unveil a superfluid transition at a critical attractive interaction Uc/t=−4.58(3)U_c/t=-4.58(3)Uc​/t=−4.58(3), which belongs to the chiral-XY universality class. This U(1) symmetry-breaking is not accompanied by charge order, even for substantially large interaction strengths, in a regime where a description in terms of hardcore bosons becomes increasingly suitable. An investigation of the latter shows that charge ordering at 1/31/31/3-filling only occurs at interaction strengths much larger than those corresponding to the mapping to the original fermionic model. Additionally, for densities exhibiting van Hove singularities in the non-interacting density of states, our results show that any attractive interaction gives rise to superfluidity, yet again without charge ordering, contrary to recent studies employing mean-field theory.