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.
Superconducting transition temperature is studied in the strong-coupling region of the Holstein-Hubbard model that includes both electron-phonon and Coulomb interactions.By combining the dynamical mean field theory with the continuous-time quantum Monte Carlo method, transition temperature is derived numerically.Without Coulomb interaction, our calculation reproduces analytic expressions of the transition temperature for weak and strong coupling limits of electron-phonon interaction.When electron-phonon coupling is strong enough, Coulomb repulsion enhances superconductivity, in contrast to the weak coupling case.The enhancement is due to coherent motion of superconducting pairs that reduce the Coulomb repulsive energy of almost localized electrons.
Recent studies of pairing and charge order in materials such as FeSe, SrTiO
3, and 2H-NbSe
2 have suggested that momentum dependence of the electron-phonon coupling plays an important role in their properties. Initial attempts to study Hamiltonians which either do not include or else truncate the range of Coulomb repulsion have noted that the resulting spatial non-locality of the electron-phonon interaction leads to a dominant tendency to phase separation. Here we present Quantum Monte Carlo results for such models in which we incorporate both on-site and intersite electron-electron interactions. We show that these can stabilize phases in which the density is homogeneous and determine the associated phase boundaries. As a consequence, the physics of momentum dependent electron-phonon coupling can be determined outside of the trivial phase separated regime.