Flow-Based Generative Sampling for the Auxiliary-Field Hubbard Model

Abstract

Understanding strongly correlated electron systems remains one of the central challenges in condensed matter physics. Finding <em>ab initio</em> 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. <br/>

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. <br/>

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. <br/>

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. <br/>

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.