Conserving approximations in direct perturbation theory: new semianalytical impurity solvers and their application to general lattice problems

Abstract

Abstract. For the treatment of interacting electrons in crystal lattices approximations based on the picture of effective sites, coupled in a self-consistent fashion, have proven very useful. Particularly in the presence of strong local correlations, a local approach to the problem, combining a powerful method for the short ranged interactions with the lattice propagation part of the dynamics, determines the quality of results to a large extent. For a considerable time the non crossing approximation (NCA) in direct perturbation theory, an approach originally developed by Keiter for the Anderson impurity model, built a standard for the description of the local dynamics of interacting electrons. In the last couple of years exact methods like the numerical renormalization group (NRG) as pioneered by Wilson, have surpassed this approximation as regarding the description of the low energy regime. We present an improved approximation level of direct perturbation theory for finite Coulomb repulsion U , the crossing approximation one (CA1) and discuss its connections with other generalizations of NCA. CA1 incorporates all processes up to fourth order in the hybridization strength V in a self-consistent skeleton expansion, retaining the full energy dependence of the vertex functions. We reconstruct the local approach to the lattice problem from the point of view of cumulant perturbation theory in a very general way and discuss the proper use of impurity solvers for this purpose. Their reliability can be tested in applications to e.g. the Hubbard model and the Anderson-lattice model. We point out shortcomings of existing impurity solvers and improvements gained with CA1 in this context. This paper is dedicated to the memory of Hellmut Keiter.

Competition between quantum spin tunneling and Kondo effect

“…This leads to a set of coupled integral equations for the PP propagators and selfenergies that have to be solved self-consistently. Once the OCA equations are solved the real electron spectral function ρ α (ω) for the impurity levels entering equation (1) for calculating the conductance spectrum is obtained from convolutions of the PP spectral functions OCA captures both the weak and strong coupling Kondo regimes, and has shown to produce reliable spectra for the single-orbital Anderson model, as long as the temperatures are not too low compared to the Kondo temperature [35,36]. For the general multi-orbital situation considered here, benchmarking of spectra is difficult since Numerical Renormalization Group [37] is computationally too demanding to be applied.…”

Spectral properties of the two-impurity Anderson model with varying distance and various interactions

“…In contrast to what is commonly needed for (effective) single-impurity models, 17 It is important to realize here that the time-ordered processes, generated by expanding the resolvents in Eq. (14) in terms of the hybridizationV , generally extend over all sites of the cluster.…”

“…These techniques have been laid out in the literature, e.g., in connection with the Kondo effect or the mixed valence problem, 16 and have served to define approximation schemes of great usefulness and quality. 17,18 There have already been approaches to the TIAM with these techniques, 19 but these were restricted to infinite Coulomb interaction U → ∞, neglected vertex corrections, and ignored nondiagonal ionic propagators (see the following for details). The present scheme is an extension of the finite-U enhanced noncrossing approximation (ENCA), 20 which improves on the well-known noncrossing approximation 21 (NCA) by the inclusion of low-order vertex corrections.…”

Dynamical mean-field theory within the full-potential methods: Electronic structure ofCeIrIn5,CeCoIn5, andHaule1, Yee2, Kim3 2010Phys. Rev. B

“…In the past, most of calculations were limited to the lowest order self-consistent approximation, called the Non-crossing approximation (NCA). Recently, many generalization of the approach were studied 25,[54][55][56] , to overcome the difficulty of the NCA at low temperature, below the Kondo temperature. It is well known that the NCA approximation fails to recover the Fermi liquid fixed point at low temperature and low energy.…”