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Journal of the Korean Physical Society

Quantum Periodic Cluster Methods for Strongly-Correlated Electron Systems

Minh-Tien Tran

Published Dec 15, 2008• 0 citations
Publisher

Abstract

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)).

DOI
10.3938/jkps.53.3619

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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)).
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​).
This article reviews quantum cluster theories, a set of approximations for infinite lattice models which treat correlations within the cluster explicitly, and correlations at longer length scales either perturbatively or within a mean-field approximation. These methods become exact when the cluster size diverges, and most recover the corresponding mean-field approximation when the cluster size becomes 1. Although quantum cluster theories were originally developed to treat disordered systems, they have more recently been applied to the study of ordered and disordered correlated systems, which will be the focus of this review. After a brief historical review, the authors provide detailed derivations of three cluster formalisms: the cluster perturbation theory, the dynamical cluster approximation, and the cellular dynamical mean-field theory. They compare their advantages and review their applications to common models of correlated electron systems.
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​).
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​).