Leveraging Machine Learning to Alleviate Hubbard Model Sign Problems

TL;DR

This paper introduces a neural network-based approach to mitigate the sign problem in lattice Monte Carlo simulations of the Hubbard model on non-bipartite lattices, significantly reducing computational costs and enabling calculations previously infeasible.

Contribution

The authors demonstrate that neural networks can efficiently parameterize manifolds for complex integration, alleviating the sign problem in non-bipartite Hubbard models at strong interactions.

Findings

Neural networks effectively reduce the sign problem in the Hubbard model.

The method outperforms standard reweighting techniques on challenging cases.

Exact benchmarks validate the approach's accuracy.

Abstract

Lattice Monte Carlo calculations of interacting systems on non-bipartite lattices exhibit an oscillatory imaginary phase known as the phase or sign problem, even at zero chemical potential. One method to alleviate the sign problem is to analytically continue the integration region of the state variables into the complex plane via holomorphic flow equations. For asymptotically large flow times the state variables approach manifolds of constant imaginary phase known as Lefschetz thimbles. However, flowing such variables and calculating the ensuing Jacobian is a computationally demanding procedure. In this paper we demonstrate that neural networks can be trained to parameterize suitable manifolds for this class of sign problem and drastically reduce the computational cost. We apply our method to the Hubbard model on the triangle and tetrahedron, both of which are non-bipartite. At strong interaction strengths and modest temperatures the tetrahedron suffers from a severe sign problem that cannot be overcome with standard reweighting techniques, while it quickly yields to our method. We benchmark our results with exact calculations and comment on future directions of this work.