Hubbard-like models in high spatial dimensions

Abstract

The present paper studies the properties of Hubbard-like models in high spatial dimensionsD. In a first par the limit of infinite dimension and its main features-i.e.i) the mapping onto a generalized atomic model with an additional auxiliary field andii) the validity of the local approximation for the self-energy-are worked out in a systematic (1/D)-expansion. Since the hopping matrix elements have to be properly scaled with the dimensionD, the (1/D)-expansion is also an expansion in the hopping amplitude. Thus for small hopping theD→∞-limit may serve as a proper approximation for finite-dimensional systems. The second part of the paper adopts the hybridisation-perturbation theory of the single impurity Anderson model in order to construct a perturbation theory for the auxiliary field of the generalized atom which can also be interpreted as an expansion in the hopping amplitude. The non-crossing approximation (NCA) is used to study the antiferromagnetic phase transtion of theD→∞-Hubbard model in the case of half filling: the critical temperature, the antiferromagnetic order parameter and the free energy of the lattice system are calculated. The NCA-results are in quite good agreement with recent results from the imaginary-time discretisation method.