Parallelized Traveling Cluster Approximation to Study Numerically Spin-Fermion Models on Large Lattices

Lattice spin-fermion models are important to study correlated systems where quantum dynamics allows for a separation between slow and fast degrees of freedom. The fast degrees of freedom are treated quantum mechanically while the slow variables, generically refereed to as the "spins", are treated classically. At present, exact diagonalization coupled with classical Monte Carlo (ED+MC) is extensively used to solve numerically a general class of lattice spin-fermion problems. In this common setup, the classical variables (spins) are treated via the standard MC method while the fermion problem is solved by exact diagonalization. The "Traveling Cluster Approximation" (TCA) is a real space variant of the ED+MC method that allows to solve spin-fermion problems on lattice sizes with up to $10^3$ sites. In this publication, we present a novel reorganization of the TCA algorithm in a manner that can be efficiently parallelized. This allows us to solve generic spin-fermion models easily on $10^4$ lattice sites and with some effort on $10^5$ lattice sites, representing the record lattice sizes studied for this family of models.