We consider a class of Hubbard-Stratonovich transformations suitable for treating Hubbard interactions in the context of quantum Monte Carlo simulations. A tunable parameter p allows us to continuously vary from a discrete Ising auxiliary field (p=∞) to a compact auxiliary field that couples to electrons sinusoidally (p=0). In tests on the single-band square and triangular Hubbard models, we find that the severity of the sign problem decreases systematically with increasing p. Selecting p finite, however, enables continuous sampling methods such as the Langevin or Hamiltonian Monte Carlo methods. We explore the tradeoffs between various simulation methods through numerical benchmarks.
We consider a class of Hubbard-Stratonovich transformations suitable for treating Hubbard interactions in the context of quantum Monte Carlo simulations. A tunable parameter p allows us to continuously vary from a discrete Ising auxiliary field (p=∞) to a compact auxiliary field that couples to electrons sinusoidally (p=0). In tests on the single-band square and triangular Hubbard models, we find that the severity of the sign problem decreases systematically with increasing p. Selecting p finite, however, enables continuous sampling methods such as the Langevin or Hamiltonian Monte Carlo methods. We explore the tradeoffs between various simulation methods through numerical benchmarks.
Within the ground-state auxiliary-field quantum Monte Carlo technique, we introduce discrete Hubbard-Stratonovich transformations (HSTs) that are also suitable for spatially inhomogeneous trial functions. The discrete auxiliary fields introduced here are coupled to local spin or charge operators fluctuating around their Hartree-Fock values. The formalism can be considered a generalization of the discrete HSTs by J. E. Hirsch [Phys. Rev. B 28, 4059 (1983)] or a compactification of the shifted-contour auxiliary-field Monte Carlo formalism by N. Rom et al. [Chem. Phys. Lett. 270, 382 (1997)]. An improvement of the acceptance ratio is found for a real auxiliary field, while an improvement of the average sign is found for a purely imaginary auxiliary field. Efficiencies of the different HSTs are tested in the single-band Hubbard model at and away from half filling by studying the staggered magnetization and energy expectation values, respectively.
Ground-state properties of the Hubbard model on a two-dimensional square lattice are studied by the auxiliary-field quantum Monte Carlo method. Accurate results for energy, double occupancy, effective hopping, magnetization, and momentum distribution are calculated for interaction strengths of U/t from 2 to 8, for a range of densities including half-filling and n=0.3,0.5,0.6,0.75, and 0.875. At half-filling, the results are numerically exact. Away from half-filling, the constrained path Monte Carlo method is employed to control the sign problem. Our results are obtained with several advances in the computational algorithm, which are described in detail. We discuss the advantages of generalized Hartree-Fock trial wave functions and its connection to pairing wave functions, as well as the interplay with different forms of Hubbard-Stratonovich decompositions. We study the use of different twist angle sets when applying the twist averaged boundary conditions. We propose the use of quasirandom sequences, which improves the convergence to the thermodynamic limit over pseudorandom and other sequences. With it and a careful finite size scaling analysis, we are able to obtain accurate values of ground-state properties in the thermodynamic limit. Detailed results for finite-sized systems up to $16\ifmmode\times\else\texttimes\fi{}16$ are also provided for benchmark purposes.
Understanding strongly correlated electron systems remains one of the central challenges in condensed matter physics. Finding ab initio descriptions of non-perturbative systems is of great importance for modeling carbon-based materials such as graphene or fullerenes. In this regime, computational methods become indispensable tools for performing simulations, making predictions, and ultimately comparing to experiments. The Hubbard model offers a minimal description of the dynamics of electrons in such materials. As exact methods are limited to small system sizes by the exponentially growing Hilbert space, larger-scale first-principle studies rely on numerical approaches. Alongside powerful Hamiltonian-based methods, the auxiliary-field description of the model offers a natural setting for Lagrangian-based Monte Carlo sampling. However, substantial challenges, such as lack of ergodicity and the numerical sign problem, arise in these approaches, leading to long autocorrelation times, biased results, and large statistical uncertainties. In recent years, generative models, especially normalizing flows, have been applied extensively to various theories to address shortcomings of traditional Monte Carlo-based algorithms, such as critical slowing down or topological freezing. While these approaches have shown promise in alleviating these issues, a poorly understood and unfavorable scaling of these methods has stalled their progress, and whether they can reach state-of-the-art system sizes in lattice quantum chromodynamics remains to be seen. This motivates the application of these methods to condensed matter systems, where physically relevant system sizes are reachable at substantially smaller scales. In this work, we study for the first time the application of generative models to the finite-temperature auxiliary-field Hubbard model. We begin by employing an equivariant normalizing flow that respects multiple symmetries inherent to the Boltzmann distribution, demonstrating a substantial improvement in sampling efficiency over Hybrid Monte Carlo and standard non-equivariant flows. We further develop Symmetry-Enforcing Stochastic Modulation (SESaMo), a more scalable algorithm capable of modeling both exact and broken symmetries in normalizing flows, and show that it achieves significantly higher effective sample sizes than traditional equivariant techniques at large system sizes. Equipped with SESaMo, we compute physical observables at half-filling in excellent agreement with exact diagonalization at substantially lower computational cost than Hybrid Monte Carlo. We then turn to the doped Hubbard model, where the theory exhibits a numerical sign problem. We introduce an annealing scheme to simulate the Hubbard model at finite chemical potential in the spin basis. Making use of the less severe sign problem in this description, the annealing scheme enables ergodic sampling of the multi-modal distribution, yielding more accurate results with substantially reduced uncertainties compared to state-of-the-art optimized Hybrid Monte Carlo. Lastly, we employ a masked autoencoder for distribution estimation to learn the Boltzmann distribution of the discrete auxiliary-field Hubbard model. Preliminary results show a significantly higher effective sample size compared to normalizing flows with continuous degrees of freedom, while also exhibiting a milder sign problem than the continuous spin basis.
Auxiliary field Quantum Monte Carlo simulations of interacting fermions require sampling over a Hubbard-Stratonovich field x introduced to decouple the interactions. The weight for a given configuration involves the products of the determinant of matrices Mσ(x), where σ labels the species, and hence is typically not positive definite. Indeed, the average sign ⟨S⟩ of the determinants goes to zero exponentially with increasing spatial size and decreasing temperature for most Hamiltonians of interest. This statement, however, does not explicitly separate two possible origins for the vanishing of ⟨S⟩. Does ⟨S⟩→0 because {\it randomly} chosen field configurations have det(M(x))<0, or does the `sign problem' arise as a result of {\it importance sampling} in the simulations? In the latter case, the process of weighting the configurations with ∣det(M(x))∣ might steer the simulation to a region of configuration space of x where positive and negative determinants are equally likely, even though randomly chosen x would preferentially have determinants with a single dominant sign. In this paper we address the relative importance of these two mechanisms for the vanishing of ⟨S⟩ in quantum simulations.
In the study of strongly correlated, many-electron systems, the Hubbard-Kanamori (HK) model has emerged as one of the prototypes for transition metal oxide physics. The model is multiband in nature and contains Hund's coupling terms, which have pronounced effects on metal-insulator transitions, high-temperature superconductivity, and other physical properties. In the following, we present a complete theoretical framework for treating the HK model using the ground-state auxiliary field quantum Monte Carlo (AFQMC) method and analyze its performance on few-band models whose parameters approximate those observed in ruthenates, rhodates, and other materials exhibiting Hund's physics. Unlike previous studies, the constrained path and phaseless approximations are used to respectively control the sign and phase problems, which enables high-accuracy modeling of the HK model's ground-state properties within parameter regimes of experimental interest. We demonstrate that, after careful consideration of the Hubbard-Stratonovich transformations and trial wave functions employed, relative errors in the energy of less than 1 can routinely be achieved for moderate to large values of the Hund's coupling constant. Crucially, our methodology also accurately predicts magnetic ordering and phase transitions. The results presented open the door to more predictive modeling of Hund's physics within a wide range of strongly correlated materials using AFQMC.
Understanding the properties of strongly correlated materials is a tough work since most of the times analytical tools cannot be applied due to the complicated nature of the problem. Here we apply quantum Monte Carlo techniques in order to tackle with one of the most important and unknown phenomena of many materials: the mechanism of superconductivity. We study the Hubbard-Holstein model which is one of the most simple theoretical models including strong correlation and electron-phonon coupling. By using twist-averaging boundary conditions we eliminate finite-size errors in the Monte Carlo simulations. With this useful tool we first solve the Hubbard-Holstein model by variational Monte Carlo by reporting the phase diagram of the model for different phonon frequencies at half filling. Then we investigate the phase separation away from half filling. Finally we try to attack the problem with an essentially unbiased method based on the auxiliary field quantum Monte Carlo technique and accelerated Langevin dynamics. By curing the sign problem via Cauchy integration in the complex plane where the auxiliary fields have been defined, we report the effect of the electron-phonon coupling on some observables such as magnetization, spin and charge structure factors. We show preliminary results that are already meaningful to understand the nature of the transition between magnetism and charge order in the model at half filling.