A travelling cluster approximation for lattice fermions strongly coupled to classical degrees of freedom

We suggest and implement a new Monte Carlo strategy for correlated models involving fermions strongly coupled to classical degrees of freedom, with accurate handling of quenched disorder as well. Current methods iteratively diagonalise the full Hamiltonian for a system of N sites with computation time τN ∼N4. This limits achievable sizes to N ∼100. In our method the energy cost of a Monte Carlo update is computed from the Hamiltonian of a cluster, of size Nc, constructed around the reference site, and embedded in the larger system. As MC steps sweep over the system, the cluster Hamiltonian also moves, being reconstructed at each site where an update is attempted. In this method τN,Nc ∼NNc3. Our results are obviously exact when Nc=N, and converge quickly to this asymptote with increasing Nc, particularly in the presence of disorder. We provide detailed benchmarks on the Holstein model and the double exchange model. The `locality' of the energy cost, as evidenced by our results, suggests that several important but inaccessible problems can now be handled with control. This method forms the basis of our studies in Europhys. Lett. 68, 564 (2004), Phys. Rev. Lett. 94, 136601 (2005), and Phys. Rev. Lett. 96, 016602 (2006)