Stable Determinant Monte Carlo Simulations at Large Inverse Temperature $\beta$

TL;DR

This paper presents methods to improve the stability and precision of determinant quantum Monte Carlo simulations at large inverse temperatures, enabling accurate simulations at room temperature for graphene.

Contribution

The authors introduce matrix decomposition techniques that address numerical instabilities in DQMC and HMC methods at high $eta$, maintaining computational efficiency.

Findings

Simulations achieved at $eta oughly 90$, corresponding to room temperature for graphene.

Numerical costs scale as $ ext{O}(N_x^3 N_t)$, similar to naive implementations.

Enhanced stability and precision in fermion determinant evaluations at low temperatures.

Abstract

At low temperatures $T$ where $1/T=\beta\gg1$ the na\"ive implementation of determinant quantum Monte Carlo (DQMC) methods suffers from loss of precision and numerical instabilities when evaluating the fermion determinant. This instability propagates into the calculation of observables that rely on the evaluation of the inverse of the fermion matrix, or the Greens function. For DQMC methods that rely on the Hamiltonian Monte Carlo (HMC) algorithm, an additional complication comes from evaluating the force terms required for integrating Hamilton's equations of motion, since here loss of precision and numerical instabilities are also prevalent. We show how to address all these issues using various choices of matrix decompositions, allowing us to simulate at $\beta\gtrsim 90$, which corresponds to room temperature for graphene structures. Furthermore, our implementation has numerical costs that scale similarly to the na\"ive implementation, namely as $\mathcal{O}(N_x^3N_t)$, where $N_x$ ($N_t$) is the number of spatial (temporal) sites.