Normalizing flows have recently demonstrated the ability to learn the Boltzmann distribution of the Hubbard model, opening new avenues for generative modeling in condensed matter physics. In this work, we investigate the steps required to extend such simulations to larger lattice sizes and lower temperatures, with a focus on enhancing stability and efficiency. Additionally, we present the scaling behavior of stochastic normalizing flows and non-equilibrium Markov chain Monte Carlo methods for this fermionic system.
Normalizing flows have recently demonstrated the ability to learn the Boltzmann distribution of the Hubbard model, opening new avenues for generative modeling in condensed matter physics. In this work, we investigate the steps required to extend such simulations to larger lattice sizes and lower temperatures, with a focus on enhancing stability and efficiency. Additionally, we present the scaling behavior of stochastic normalizing flows and non-equilibrium Markov chain Monte Carlo methods for this fermionic system.
Normalizing flows have recently demonstrated the ability to learn the Boltzmann distribution of the Hubbard model, opening new avenues for generative modeling in condensed matter physics. In this work, we investigate the steps required to extend such simulations to larger lattice sizes and lower temperatures, with a focus on enhancing stability and efficiency. Additionally, we present the scaling behavior of stochastic normalizing flows and non-equilibrium Markov chain Monte Carlo methods for this fermionic system.
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.
Generative models, particularly normalizing flows, have shown exceptional performance in learning probability distributions across various domains of physics, including statistical mechanics, collider physics, and lattice field theory. In the context of lattice field theory, normalizing flows have been successfully applied to accurately learn the Boltzmann distribution, enabling a range of tasks such as direct estimation of thermodynamic observables and sampling independent and identically distributed (i.i.d.) configurations. In this work, we present a proof-of-concept demonstration that normalizing flows can be used to learn the Boltzmann distribution for the Hubbard model. This model is widely employed to study the electronic structure of graphene and other carbon nanomaterials. State-of-the-art numerical simulations of the Hubbard model, such as those based on Hybrid Monte Carlo (HMC) methods, often suffer from ergodicity issues, potentially leading to biased estimates of physical observables. Our numerical experiments demonstrate that leveraging i.i.d. sampling from the normalizing flow effectively addresses these issues.
We present the first proof of principle that normalizing flows can accurately learn the Boltzmann distribution of the fermionic Hubbard model—a key framework for describing the electronic structure of graphene and related materials. State-of-the-art methods like hybrid Monte Carlo often suffer from ergodicity issues near the time-continuum limit, leading to biased estimates. Leveraging symmetry-aware architectures as well as independent and identically distributed sampling, our approach resolves these issues and achieves significant speedups over traditional methods.
We introduce a novel technique for scalable sampling of spin-system states with continuous symmetries using diffusion models. By applying our approach to the XY model, a fundamental continuous-spin model in condensed matter physics, we show that our technique addresses the shortfalls of the Markov chain Monte Carlo (MCMC) in generalization to varying system sizes. More specifically, we show that training a temperature-conditioned diffusion model on smaller-size XY model lattices enables the generation of accurate samples in larger lattice sizes. By tracking physically important observables of the model, such as spin correlations, our experiments demonstrate that diffusion sampling followed by a few MCMC steps reduces the thermalization time by an order of magnitude relative to the standard MCMC with random initialization. Our study provides valuable insight as to how generative models can be used to study continuous-state condensed matter systems at scale.
We give an overview of the Gaussian phase-space methods for fermions, focusing on the example of the Hubbard model, which describes ultracold fermions in a lattice. The phase-space methods do not suffer the sort of sign problem encountered by other quantum Monte Carlo methods, and provide an efficient means of calculating exact zero-temperature Hubbard results. However at finite temperature, there are known issues to do with the boundedness of the underlying distribution. We show how ‘stochastic gauges’ can be used to control the distribution tails and thereby improve the accuracy of the simulations at low temperatures.