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

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.

References in corpus (15)

• The Kernel Polynomial Method
• The Hubbard Model
• The Hubbard model: A computational perspective
• Sign-problem-free quantum Monte Carlo of the onset of antiferromagnetism in metals
• Electron-Phonon Coupling from $\text{Ab Initio}$ Linear-Response Theory within the $GW$ Method: Correlation-Enhanced Interactions and Superconductivity in Ba${}_{1-x}$K${}_x$BiO${}_3$
• Hybridization effects and bond-disproportionation in the bismuth perovskites
• Valence-bond solid to antiferromagnet transition in the two-dimensional Su-Schrieffer-Heeger model by Langevin dynamics
• Superconductivity and charge density wave order in the 2D Holstein model
• Antiferromagnetism induced by electron-phonon-coupling
• Superconductivity, Charge-Density-Waves, and Bipolarons in the Holstein model
• Particle-hole asymmetry in the dynamical spin and charge structure factors of the corner-shared one-dimensional cuprates
• Effect of Strain on Charge Density Wave Order in the Holstein Model
• Quantum Monte Carlo study of an anharmonic Holstein model
• Deconvolving the components of the sign problem
• Polaron and bipolaron tendencies in a semiclassical model for hole-doped bismuthates

Cited by in corpus (7)

• A perspective on machine learning and data science for strongly correlated electron problems
• A hybrid Monte Carlo study of bond-stretching electron-phonon interactions and charge order in BaBiO${}_3$
• Charge order in the kagome lattice Holstein model: A hybrid Monte Carlo study
• Non-Hertz-Millis scaling of the antiferromagnetic quantum critical metal via scalable Hybrid Monte Carlo
• A flexible class of exact Hubbard-Stratonovich transformations
• Scalable hybrid quantum Monte Carlo simulation of U(1) gauge field coupled to fermions on GPU
• A lattice pairing-field approach to ultracold Fermi gases