Hamiltonian Monte Carlo enhanced by Exact Diagonalization

TL;DR

This paper introduces a hybrid algorithm combining exact diagonalization and Hamiltonian Monte Carlo to efficiently simulate large 2D fermionic systems, overcoming limitations of each method individually.

Contribution

The authors develop H$^2$MC, a novel hybrid approach that outperforms traditional methods in scaling, sign problem mitigation, and autocorrelation reduction for simulating interacting fermionic systems.

Findings

H$^2$MC scales better than pure ED, enabling larger system simulations.

H$^2$MC reduces the sign problem compared to standard HMC.

H$^2$MC decreases autocorrelation times, improving simulation efficiency.

Abstract

Strongly correlated fermionic systems are of great interest in condensed matter physics and numerical methods are indispensable tools for their study. However, existing approaches such as exact diagonalization (ED) and stochastic quantum Monte Carlo methods each suffer from fundamental limitations: ED is hindered by exponential scaling in system size, while Monte Carlo methods are plagued by sign problems and long autocorrelation times. These limitations restrict the accessible parameter space and developing algorithms that efficiently alleviate them remains a central challenge in computational physics. In this work, we propose a hybrid algorithm that combines ED and Hamiltonian Monte Carlo (HMC) to simulate 2D arrays of coupled quantum wires, modeled as interacting fermionic Hubbard chains. We demonstrate how our hybrid implementation of HMC, which we dub H$^2$MC, outperforms either method alone across several key simulation facets. When compared to pure ED, H$^2$MC has a much more favorable computational scaling, which allows us to push simulations to much larger 2D arrays. H$^2$MC also greatly alleviates the sign problem and reduces autocorrelation times when compared to pure HMC formulations utilizing either real or imaginary auxiliary fields. Our formalism demonstrates how complementary strengths of seemingly disparate methods can be leveraged to enable feasible simulations in an extended parameter space.