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Dynamical Mean Field and Dynamical Cluster Approximations

Mark Jarrell, Alexandru Macridin

Published Jan 1, 2011• 0 citations
OpenAlex

Abstract

We present a pedagogical discussion of the dynamical mean field and dynamical cluster approximations.

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arXiv:cond-mat/0504778•Physical Review B•Sep 6, 2005•23 citations
We present a pedagogical discussions of the dynamical mean field (DMFA) and dynamical cluster (DCA) approximations and associated Monte Carlo and entropy‐based methods of Bayesian data analysis. The DMFA and DCA methods are developed as coarse‐graining approximations and the relationship between the cluster and lattice problems are detailed. The Hirsch‐Fye and continuous time Quantum Monte Carlo (QMC) algorithms are used to solve the cluster problem. The algorithms are discussed, together with methods for efficient measurements and the modifications required by the self‐consistency of the DMFA/DCA. Then, several principles of Bayesian data analysis are presented. When coupled with information theory, this analysis produces a precise and systematic way to analytically continue Matsubara‐time QMC results to real frequencies. We show how to use Bayesian inference to qualify the solution of the continuation and optimize the inputs. Besides developing the Bayesian formalism, we also present a detailed description of the data qualification, sketch an efficient algorithm to solve for the optimal spectra, give cautionary notes where appropriate, and present two detailed case studies to demonstrate the method.
The variational cluster approximation (VCA) proposed by M. Potthoff et al. [Phys. Rev. Lett. 91, 206402 (2003)] is extended to electron or spin systems with nonlocal interactions. By introducing more than one source field in the action and employing the Legendre transformation, we derive a generalized self-energy functional with stationary properties. Applying this functional to a proper reference system, we construct the extended VCA (EVCA). In the limit of continuous degrees of freedom for the reference system, EVCA can recover the cluster extension of the extended dynamical mean-field theory (EDMFT). For a system with correlated hopping, the EVCA recovers the cluster extension of the dynamical mean-field theory for correlated hopping. Using a discrete reference system composed of decoupled three-site single impurities, we test the theory for the extended Hubbard model. Quantitatively good results as compared with EDMFT are obtained. We also propose VCA (EVCA) based on clusters with periodic boundary conditions. It has the (extended) dynamical cluster approximation as the continuous limit. A number of related issues are discussed.