A recently introduced one-dimensional three-orbital Hubbard model displays orbital-selective Mott phases with exotic spin arrangements such as spin block states [J. Rincón et al., Phys. Rev. Lett. 112, 106405 (2014)PRLTAO0031-900710.1103/PhysRevLett.112.106405]. In this publication we show that the constrained-path quantum Monte Carlo (CPQMC) technique can accurately reproduce the phase diagram of this multiorbital one-dimensional model, paving the way to future CPQMC studies in systems with more challenging geometries, such as ladders and planes. The success of this approach relies on using the Hartree-Fock technique to prepare the trial states needed in CPQMC. We also study a simplified version of the model where the pair-hopping term is neglected and the Hund coupling is restricted to its Ising component. The corresponding phase diagrams are shown to be only mildly affected by the absence of these technically difficult-to-implement terms. This is confirmed by additional density matrix renormalization group and determinant quantum Monte Carlo calculations carried out for the same simplified model, with the latter displaying only mild fermion sign problems. We conclude that these methods are able to capture quantitatively the rich physics of the several orbital-selective Mott phases (OSMP) displayed by this model, thus enabling computational studies of the OSMP regime in higher dimensions, beyond static or dynamic mean-field approximations.
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.