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arXiv:cond-mat/0303136 · Physical Review Letters

Variational Cluster Approach to Correlated Electron Systems in Low Dimensions

Michael Potthoff, Markus Aichhorn, Christopher Dahnken

Published Nov 11, 2003• 332 citations
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Abstract

A self-energy-functional approach is applied to construct cluster approximations for correlated lattice models. It turns out that the cluster-perturbation theory [Phys. Rev. Lett. 84, 522 (2000)]] and the cellular dynamical mean-field theory [Phys. Rev. Lett. 87, 186401 (2001)]] are limiting cases of a more general cluster method. The results for the one-dimensional Hubbard model are discussed with regard to boundary conditions, bath degrees of freedom, and cluster size.

DOI
10.1103/physrevlett.91.206402

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A self-energy-functional approach is applied to construct cluster approximations for correlated lattice models. It turns out that the cluster-perturbation theory [Phys. Rev. Lett. 84, 522 (2000)]] and the cellular dynamical mean-field theory [Phys. Rev. Lett. 87, 186401 (2001)]] are limiting cases of a more general cluster method. The results for the one-dimensional Hubbard model are discussed with regard to boundary conditions, bath degrees of freedom, and cluster size.
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.
These lecture notes provide an introduction to quantum cluster methods for strongly correlated systems. Cluster Perturbation Theory (CPT), the Variational Cluster Approximation (VCA) and Cellular Dynamical Mean Field Theory (CDMFT) are described, as well as the exact diagonalization solver for the cluster. Potthoff's self-energy functional formalism is reviewed. Some numerical procedures, in particular regarding the exact diagonalization method and the frequency-momentum integrals needed in VCA, are discussed in detail.
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.
The variational cluster approach (VCA) is applied to the one-dimensional Hubbard model at zero temperature using clusters (chains) of up to ten sites with full diagonalization and the Lanczos method as cluster solver. Within the framework of the self-energy-functional theory (SFT), different cluster reference systems with and without bath degrees of freedom, in different topologies and with different sets of variational parameters, are considered. Static and one-particle dynamical quantities are calculated for half-filling as a function of UUU as well as for fixed UUU as a function of the chemical potential to study the interaction- and filling-dependent metal-insulator (Mott) transition. The recently developed QQQ-matrix technique is used to compute the SFT grand potential. For benchmarking purposes we compare the VCA results with exact results available from the Bethe ansatz, with essentially exact dynamical density-matrix renormalization-group data, with (cellular) dynamical mean-field theory and full diagonalization of isolated Hubbard chains. Several issues are discussed including convergence of the results with cluster size, the ability of cluster approaches to access the critical regime of the Mott transition, efficiency in the optimization of correlated-site vs bath-site parameters, and of multidimensional parameter optimization. We also study the role of bath sites for the description of excitation properties and as charge reservoirs for the description of filling dependencies. The VCA turns out to be a computationally cheap method which is competitive with established cluster approaches.
We explore the use of exact diagonalization methods for solving the self-consistent equations of the cellular dynamical mean field theory for the one-dimensional regular and extended Hubbard models. We investigate the nature of the Mott transition and convergence of the method as a function of cluster size as well as the optimal allocation of computational resources between bath and ``cluster-impurity'' sites, with a view to develop a renormalization group method in higher dimensions. We assess the performance of the method by comparing results for the Green's functions in both the spin density wave and charge density wave phases with accurate density matrix renormalization group (DMRG) calculations.
We study the Hubbard model using the cellular dynamical mean-field theory (CDMFT) with quantum Monte Carlo (QMC) simulations. We present the algorithmic details of CDMFT with the Hirsch-Fye QMC method for the solution of the self-consistently embedded quantum cluster problem. We use the one- and two-dimensional half filled Hubbard model to gauge the performance of CDMFT+QMC\mathrm{CDMFT}+\mathrm{QMC}CDMFT+QMC particularly for small clusters by comparing with the exact results and also with other quantum cluster methods. We calculate single-particle Green's functions and self-energies on small clusters to study their size dependence in one and two dimensions. It is shown that in one dimension, CDMFT with two sites in the cluster is already able to describe with high accuracy the evolution of the density as a function of the chemical potential and the compressibility divergence at the Mott transition, in good agreement with the exact Bethe ansatz result. With increasing UUU the result on small clusters rapidly approaches that of the infinite size cluster. Large scattering rate and a positive slope in the real part of the self-energy in one dimension suggest that the system is a non-Fermi liquid for all the parameters studied here. In two dimensions, at intermediate to strong coupling, even the smallest cluster $({N}_{c}=2\ifmmode\times\else\texttimes\fi{}2)$ accounts for more than 95% of the correlation effect of the infinite-size cluster in the single particle spectrum, suggesting that some of the important problems in strongly correlated electron systems may be studied highly accurately with a reasonable computational effort. Finally, as an application that is sensitive to details of correlations, we show that CDMFT+QMC\mathrm{CDMFT}+\mathrm{QMC}CDMFT+QMC can describe spin-charge separated Luttinger liquid physics in one dimension. The spinon and holon branches appear only for sufficiently large system sizes.