Applying the Worldvolume Hybrid Monte Carlo method to the Hubbard model away from half filling

TL;DR

This paper applies the Worldvolume Hybrid Monte Carlo method to the doped two-dimensional Hubbard model, effectively mitigating the sign problem and enabling calculations where traditional methods fail.

Contribution

It demonstrates the effectiveness of WV-HMC in simulating the Hubbard model away from half filling, overcoming sign problem challenges in practical lattice sizes.

Findings

WV-HMC remains effective where standard methods fail.

Computed density and energy densities on 6x6 and 8x8 lattices.

Results show WV-HMC mitigates the sign problem in challenging regimes.

Abstract

The Worldvolume Hybrid Monte Carlo (WV-HMC) method [arXiv:2012.08468] is an efficient algorithm for addressing the numerical sign problem at moderate computational cost. It mitigates the sign problem while avoiding the ergodicity issues inherent in approaches based on Lefschetz thimbles. In this study, we apply WV-HMC to the two-dimensional Hubbard model doped away from half filling, which is known to suffer from a severe sign problem. We compute the number density and the energy density on lattices of size $6 \times 6$ and $8 \times 8$ at temperature $T/t = 1/6.4 \simeq 0.156$ and interaction strength $U/t = 8.0$, using Trotter number $N_t = 20$ (Trotter step $\epsilon = 0.32$). Our results demonstrate that WV-HMC remains effective even in parameter regimes where standard (non-thimble) determinant quantum Monte Carlo methods fail. In this work, fermion matrix inversions are performed using direct solvers, leading to a computational cost of $O(N^3)$, where $N$ denotes the number of degrees of freedom and is proportional to the spacetime lattice volume. An alternative algorithm employing pseudofermions and iterative solvers, which reduces the cost to $O(N^2)$ at the expense of careful parameter tuning, will be discussed in a separate publication.