The Travelling Cluster Approximation for Strong Correlation Models of Lattice Fermions Coupled to Classical Fields

TL;DR

The paper introduces a new Monte Carlo method called the Travelling Cluster Approximation for efficiently simulating strongly correlated lattice fermion models, significantly reducing computational costs while maintaining accuracy.

Contribution

It presents a novel cluster-based Monte Carlo approach that improves scalability and accuracy in simulating fermion models coupled to classical fields, especially under strong disorder.

Findings

Method reduces computational complexity from N^4 to N N_c^3.

Results converge quickly to exact solutions as cluster size increases.

Preliminary results demonstrate effectiveness on Holstein and Double Exchange models.

Abstract

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 tau_N proportional to N^4. This limits achievable sizes to N \sim 100. In our method the energy cost of a Monte Carlo update is computed from the Hamiltonian of a cluster, of size N_c, 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 tau_{N,N_c} is proportional to NN_c^3. Our results are obviously exact when N_c=N, and converge quickly to this asymptote with increasing N_c. The accuracy improves in systems where the effective disorder seen by the fermions is large. We provide results of preliminary calculations 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.