Constrained-path auxiliary-field quantum Monte Carlo for coupled electrons and phonons

Abstract

We present an extension of constrained-path auxiliary-field quantum Monte Carlo (CP-AFQMC) for the treatment of correlated electronic systems coupled to phonons. The algorithm follows the standard CP-AFQMC approach for description of the electronic degrees of freedom while phonons are described in first quantization and propagated via a diffusion Monte Carlo approach. Our method is tested on the one- and two-dimensional Holstein and Hubbard-Holstein models. With a simple semiclassical trial wave function, our approach is remarkably accurate for $\ensuremath{\omega}/(2dt\ensuremath{\lambda})<1$ for all parameters in the Holstein model considered in this study where $d$ is the dimensionality, $\ensuremath{\omega}$ is the phonon frequency, $t$ is the electronic hopping strength, and $\ensuremath{\lambda}$ is the dimensionless electron-phonon coupling strength. In addition, we empirically show that the autocorrelation timescales as $1/\ensuremath{\omega}$ for $\ensuremath{\omega}/t\ensuremath{\lesssim}1$, which is an improvement over the $1/{\ensuremath{\omega}}^{2}$ scaling of the conventional determinant quantum Monte Carlo algorithm. In the Hubbard-Holstein model, the accuracy of our algorithm is found to be consistent with that of standard CP-AFQMC for the Hubbard model when the Hubbard $U$ term dominates the physics of the model, and is nearly exact when the ground state is dominated by the electron-phonon coupling scale $\ensuremath{\lambda}$. The ap- proach developed in this work should be valuable for understanding the complex physics arising from the interplay between electrons and phonons in both model lattice problems and ab initio systems.