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arXiv:cond-mat/0404537 · Physical Review B

Reply to “Comment on ‘Cluster methods for strongly correlated electron systems’ ”

Giulio Biroli, Gabriel Kotliar

Published Jan 7, 2005• 21 citations
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Abstract

We reply to the Comment by Aryanpour, Maier, and Jarrell [Phys. Rev. B 71, 037101 (2005)] on our paper [Phys. Rev. B 65, 155112 (2002)]. We demonstrate, using general arguments and explicit examples, that whenever the correlation length is finite, local observables converge exponentially fast in the cluster size ${L}_{c}$ within cellular dynamical mean field theory. This is a faster rate of convergence than the $1∕{L}_{c}^{2}$ behavior of the dynamical cluster approximation, thus refuting the central assertion of their Comment.

DOI
10.1103/physrevb.71.037102

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We reply to the Comment by Aryanpour, Maier, and Jarrell [Phys. Rev. B 71, 037101 (2005)] on our paper [Phys. Rev. B 65, 155112 (2002)]. We demonstrate, using general arguments and explicit examples, that whenever the correlation length is finite, local observables converge exponentially fast in the cluster size Lc{L}_{c}Lc​ within cellular dynamical mean field theory. This is a faster rate of convergence than the 1∕Lc21∕{L}_{c}^{2}1∕Lc2​ behavior of the dynamical cluster approximation, thus refuting the central assertion of their Comment.
We comment on the article by Biroli and Kotliar [Phys. Rev. B 65, 155112 (2002)] in which they make a comparison between two cluster techniques, the cellular dynamical mean field theory (CDMFT) and the dynamical cluster approximation (DCA). Based upon an incorrect implementation of the DCA technique in their work, they conclude that the CDMFT is a faster converging technique than the DCA. We present the correct DCA prescription for the particular model Hamiltonian studied in their article and conclude that the DCA, once implemented correctly, is a faster converging technique for the quantities averaged over the cluster.
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.
We recently introduced the dynamical cluster approximation (DCA), a technique that includes short-ranged dynamical correlations in addition to the local dynamics of the dynamical mean-field approximation while preserving causality. The technique is based on an iterative self-consistency scheme on a finite-size periodic cluster. The dynamical mean-field approximation (exact result) is obtained by taking the cluster to a single site (the thermodynamic limit). Here, we provide details of our method, explicitly show that it is causal, systematic, $\ensuremath{\Phi}$ derivable, and that it becomes conserving as the cluster size increases. We demonstrate the DCA by applying it to a quantum Monte Carlo and exact enumeration study of the two-dimensional Falicov-Kimball model. The resulting spectral functions preserve causality, and the spectra and the charge-density-wave transition temperature converge quickly and systematically to the thermodynamic limit as the cluster size increases.