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Physical Review B

Periodic cluster perturbation theory

Minh-Tien Tran

Published Oct 25, 2006• 3 citations
Publisher

Abstract

A periodic cluster perturbation theory is used to investigate a fermion system. The theory is based on the dynamical cluster approximation with relaxing of self-consistency. Within the periodic cluster perturbation theory the self-energy is approximated by the self-energy of an isolated cluster with periodic boundary conditions. The small parameter of the theory is of order of $1∕{L}_{c}$, where ${L}_{c}$ is the linear size of the cluster. It is found that within the periodic cluster perturbation theory, quantities averaged over the cluster converge rapidly with corrections $O(1∕{L}_{c}^{2})$, whereas within the cluster perturbation theory, they converge with corrections $O(1∕{L}_{c})$.

DOI
10.1103/physrevb.74.155121

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A periodic cluster perturbation theory is used to investigate a fermion system. The theory is based on the dynamical cluster approximation with relaxing of self-consistency. Within the periodic cluster perturbation theory the self-energy is approximated by the self-energy of an isolated cluster with periodic boundary conditions. The small parameter of the theory is of order of 1∕Lc1∕{L}_{c}1∕Lc​, where Lc{L}_{c}Lc​ is the linear size of the cluster. It is found that within the periodic cluster perturbation theory, quantities averaged over the cluster converge rapidly with corrections O(1∕Lc2)O(1∕{L}_{c}^{2})O(1∕Lc2​), whereas within the cluster perturbation theory, they converge with corrections O(1∕Lc)O(1∕{L}_{c})O(1∕Lc​).
Quantum periodic cluster methods for strongly correlated electron systems are reformulated and developed. The reformulation and development are based on a canonical transformation which periodizes the fermions in the cluster space. The dynamical cluster approximation and the cellular dynamical mean field theory are related each other through the canonical transformation. A cluster perturbation theory with periodic boundary conditions is developed. It is found that the periodic cluster perturbation theory converges rapidly with corrections O(1/Lc2)\mathcal{O}(1/L_c^2)O(1/Lc2​), where LcL_cLc​ is the linear size of the clusters, whereas the ordinary cluster perturbation theory converges with corrections O(1/Lc)\mathcal{O}(1/L_c)O(1/Lc​).
Quantum periodic cluster methods for strongly-correlated electron systems are reformulated and developed. The real lattice is divided into a superlattice of clusters. The reformulation and development are based on a canonical transformation, which periodizes the fermions in the cluster space. The dynamical cluster approximation and the cellular dynamical mean field theory are related to each other through the canonical transformation. A cluster perturbation theory with periodic boundary conditions is developed. The periodic cluster perturbation theory is found to converge rapidly with corrections O(1/L(c)(2)), where L(c) is the linear size of the clusters, whereas the ordinary cluster perturbation theory converges with corrections O(1/L(c)).
Recent years have seen the development of two types of nonlocal extensions to the single-site dynamical mean field theory. On one hand, cluster approximations, such as the dynamical cluster approximation, recover short-range momentum-dependent correlations nonperturbatively. On the other hand, diagrammatic extensions, such as the dual-fermion theory, recover long-ranged corrections perturbatively. The correct treatment of both strong short-ranged and weak long-ranged correlations within the same framework is therefore expected to lead to a quick convergence of results, and offers the potential of obtaining smooth self-energies in nonperturbative regimes of phase space. In this paper, we present an exact cluster dual-fermion method based on an expansion around the dynamical cluster approximation. Unlike previous formulations, our method does not employ a coarse-graining approximation to the interaction, which we show to be the leading source of error at high temperature, and converges to the exact result independently of the size of the underlying cluster. We illustrate the power of the method with results for the second-order cluster dual-fermion approximation to the single-particle self-energies and double occupancies.
We have designed a multiscale approach for strongly correlated systems by combining the dynamical cluster approximation (DCA) and the recently introduced dual fermion formalism. This approach employs an exact mapping from a real lattice to a DCA cluster of linear size Lc{L}_{c}Lc​ embedded in a dual fermion lattice. Short-length-scale physics is addressed by the DCA cluster calculation, while longer-length-scale physics is addressed diagrammatically using dual fermions. The bare and dressed dual fermionic Green functions scale as O(1/Lc)\mathcal{O}(1/{L}_{c})O(1/Lc​), so perturbation theory on the dual lattice converges very quickly, e.g., the dual Fermion self-energy calculated with simple second-order perturbation theory is of order O(1/Lc3)\mathcal{O}(1/{L}_{c}^{3})O(1/Lc3​) with third-order and three-body corrections down by an additional factor of O(1/Lc)\mathcal{O}(1/{L}_{c})O(1/Lc​).