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arXiv:cond-mat/0406082

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

Sanjeev Kumar, Pinaki Majumdar

Published Jun 3, 2004• 0 citations
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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.

DOI
10.48550/arxiv.cond-mat/0406082

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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.
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We introduce regular series expansion for weakly and moderately correlated fermionic systems based on the fluctuating local field approach. The method relies on explicitly accounting for the leading fluctuating mode(s) and is therefore suitable for medium-sized lattices. It can also be used as a solver for the cluster approximations for infinite-size lattices. We introduce a classical fluctuating field coupled to a fermionic collective mode(s). In this way, fluctuations in selected modes are treated in a nonperturbative way. Other degrees of freedom are accounted for in the diagram expansion performed for each value of the fluctuating field. The method is benchmarked for the U/t=1U/t=1U/t=1 and U/t=2U/t=2U/t=2 Hubbard lattices at half filling. Results for susceptibility in the antiferromagnetic channel along with the spectral function are compared with the numerically exact quantum Monte Carlo data. Calculations up to the third order of the series expansion are performed and show a uniform convergence to the reference result for the susceptibility. This convergence is observed well below the dynamical mean-field theory (DMFT) Ne\'el temperature and makes our practically simple method applicable in a much wider temperature range than DMFT-based diagrammatic schemes.
Classic numerical approaches to lattice models such as the Hubbard model are usually based on a solution of the model on a periodic lattice with a small number of sites. For instance, Exact Diagonalizations (ED) calculations are performed on periodic systems with no more than ∼ 20 sites and Quantum Monte Carlo (QMC) calculations are limited in practice on systems with . 100 sites. Some extrapolation is then needed to make statements about the thermodynamic (e.g. infinite-size) limit. One advantage of such approaches is their relative simplicity and lack of ambiguity. A disadvantage is that broken symmetry states need careful analysis to be identified, as they are fully revealed only in the thermodynamic limit. Quantum cluster methods are a set of closely related approaches that instead consider a finite cluster of sites embedded in the infinite lattice. The embedding is done by adding to the cluster additional fields or bath degrees of freedom such as to best represent the effect of the surrounding infinite lattice. Variational or self-consistency principles are used to set the values of these extra parameters. In these approaches, broken symmetry states can appear even for the smallest clusters used, somewhat like ordinary mean-field theory. However, unlike mean field theory, these approaches are dynamical and retain the full effect of strong correlations. These methods are usually known by their acronyms: