Simulating the Hubbard Model with Equivariant Normalizing Flows

TL;DR

This paper demonstrates that equivariant normalizing flows can effectively learn the Boltzmann distribution of the Hubbard model, providing a new approach to address ergodicity issues in numerical simulations of strongly correlated electron systems.

Contribution

It introduces the use of equivariant normalizing flows to learn the Hubbard model's distribution, offering a novel method to improve sampling and reduce biases in simulations.

Findings

Normalizing flows successfully learn the Hubbard model's Boltzmann distribution.

Flow-based sampling mitigates ergodicity issues in numerical simulations.

The approach enables unbiased estimation of physical observables.

Abstract

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.