Fast and scalable quantum Monte Carlo simulations of electron-phonon models

TL;DR

This paper presents advanced methodologies to significantly accelerate quantum Monte Carlo simulations of electron-phonon models, enabling more efficient and scalable analysis of complex systems like the Holstein model on square lattices.

Contribution

It introduces a suite of techniques combining Fourier acceleration, global updates, preconditioning, and FFTs to enhance the speed and scalability of hybrid Monte Carlo simulations for electron-phonon models.

Findings

Methods achieve near-linear scaling with system size.

Significant speedups are possible depending on model details.

Enhanced simulation efficiency for the Holstein model.

Abstract

We introduce methodologies for highly scalable quantum Monte Carlo simulations of electron-phonon models, and report benchmark results for the Holstein model on the square lattice. The determinant quantum Monte Carlo (DQMC) method is a widely used tool for simulating simple electron-phonon models at finite temperatures, but incurs a computational cost that scales cubically with system size. Alternatively, near-linear scaling with system size can be achieved with the hybrid Monte Carlo (HMC) method and an integral representation of the Fermion determinant. Here, we introduce a collection of methodologies that make such simulations even faster. To combat "stiffness" arising from the bosonic action, we review how Fourier acceleration can be combined with time-step splitting. To overcome phonon sampling barriers associated with strongly-bound bipolaron formation, we design global Monte Carlo updates that approximately respect particle-hole symmetry. To accelerate the iterative linear solver, we introduce a preconditioner that becomes exact in the adiabatic limit of infinite atomic mass. Finally, we demonstrate how stochastic measurements can be accelerated using fast Fourier transforms. These methods are all complementary and, combined, may produce multiple orders of magnitude speedup, depending on model details.