Nonlocal dynamical correlations of strongly interacting electron systems

Abstract

We introduce an extension of the dynamical mean field approximation (DMFA) which retains the causal properties and generality of the DMFA, but allows for systematic inclusion of non-local corrections. Our technique maps the problem to a self-consistently embedded cluster. The DMFA (exact result) is recovered as the cluster size goes to one (infinity). As a demonstration, we study the Falicov-Kimball model using a variety of cluster sizes. We show that the sum rules are preserved, the spectra are positive definite, and the non-local correlations suppress the CDW transition temperature.Introduction. Strongly interacting electron systems have been on the forefront of theoretical and experimental interest for several decades. This interest has intensified with the discovery of a variety of Heavy Fermion and related non Fermi liquid systems and the high-T c superconductors. In all these systems strong electronic interactions play a dominant role in the selection of at least the low temperature phase. The simplest theoretical models of strongly correlated electrons, the Hubbard model (HM) and the periodic Anderson model (PAM), have remained unsolved in more than one dimension despite a multitude of sophisticated techniques introduced since the inception of the models.With the ground breaking work by Metzner and Vollhardt [1] it was realized that these models become significantly simpler in the limit of infinite dimensions, D = ∞. Namely, provided that the kinetic energy is properly rescaled as 1/ √ D, they retain only local, though nontrivial dynamics: The self energy is constant in momentum space, though it has a complicated frequency dependence. Consequently, the HM and PAM map onto a generalized single impurity Anderson model. The thermodynamics and phase diagram have been obtained numerically by quantum Monte Carlo (QMC) and other methods. [2][3][4] The name dynamical mean field approximation (DMFA) has been coined for approximations in which a purely local self energy (and vertex function) is assumed in the context of a finite dimensional electron system. While it has been shown that this approximation captures many key features of strongly correlated systems even in a finite dimensional context, the DMFA, which leads to an effective single site theory, has some obvious limitations. For example, the DMFA can not describe phases with explicitly nonlocal order parameters, such as d-wave superconductivity, nor can it describe

Theory of electronic transport in random alloys with short-range order: Korringa-Kohn-Rostoker nonlocal coherent potential approximation

“…42,43 This is a coarse graining of ␦Ĝ consistently in real and reciprocal space and a mapping to a self-consistently embedded impurity cluster problem with appropriate boundary conditions imposed. 48 The full translational symmetry of the underlying lattice is preserved.…”

“…43 The final step is to find the SPO for an impurity cluster ␥ IJ describing a particular configuration ␥ C of atoms, which is embedded into the NLCPA medium. By demanding that the average is equal to the SPO of the NLCPA enables the effective t matrix and structure constant corrections to be determined, i.e.,…”

“…This nonlocal CPA ͑NLCPA͒ theory is based on reciprocal space coarse-graining ideas introduced by Jarrell and Krishnamurthy, 42 originating from the dynamical cluster approximation ͑DCA͒. [43][44][45] The KKR-NLCPA 36,37 introduces an effective ͑translationally invariant͒ disorder term ␦G, which represents an effective propagator that accounts for all nonlocal scattering correlations on the electronic propagation due to disorder configurations and modifies the structure constants accordingly. By coarse-graining reciprocal space, one naturally introduces real space periodically repeating clusters.…”

Study of off-diagonal disorder using the typical medium dynamical cluster approximation

“…These include cluster extensions such as the molecular coherent potential approximation (MCPA) 27,28 , the dynamical cluster approximation (DCA) [29][30][31] , etc. Self-consistent mean field studies of off-diagonal disorder have been conducted by a number of authors 28,[32][33][34] .…”

“…However, he criticized the MCPA for violating translational invariance and other critical properties of a valid quantum cluster theory 15,35 . In order to take into account such non-local effects on offdiagonal disorder models while maintaining translational invariance, we extend the BEB formalism using the DCA scheme [29][30][31] .…”