Langevin Simulations of a Long Range Electron Phonon Model

TL;DR

This paper introduces a Langevin-based Quantum Monte Carlo method for simulating electron-phonon systems, demonstrating improved efficiency and enabling exploration of phase diagrams and charge order phenomena in lattice models.

Contribution

The study develops a Langevin QMC approach with Fourier Acceleration for electron-phonon models, offering faster convergence and larger system size capabilities compared to traditional methods.

Findings

Langevin QMC outperforms determinant QMC for lattices larger than 8x8.

The method effectively maps phase diagrams, identifying charge density wave regions.

It enables studies of charge order in 3D Holstein models.

Abstract

We present a Quantum Monte Carlo (QMC) study, based on the Langevin equation, of a Hamiltonian describing electrons coupled to phonon degrees of freedom. The bosonic part of the action helps control the variation of the field in imaginary time. As a consequence, the iterative conjugate gradient solution of the fermionic action, which depends on the boson coordinates, converges more rapidly than in the case of electron-electron interactions, such as the Hubbard Hamiltonian. Fourier Acceleration is shown to be a crucial ingredient in reducing the equilibration and autocorrelation times. After describing and benchmarking the method, we present results for the phase diagram focusing on the range of the electron-phonon interaction. We delineate the regions of charge density wave formation from those in which the fermion density is inhomogeneous, caused by phase separation. We show that the Langevin approach is more efficient than the Determinant QMC method for lattice sizes $N \gtrsim 8 \times 8$ and that it therefore opens a potential path to problems including, for example, charge order in the 3D Holstein model.