Self-consistent mean field approximations

Ahmet Elçi

Friedrich Krien, Erik G. C. P. van Loon, Hartmut Hafermann +3 more

Extensions of dynamical mean-field theory (DMFT) make use of quantum impurity models as nonperturbative and exactly solvable reference systems which are essential to treat the strong electronic correlations. Through the introduction of retarded interactions on the impurity, these approximations can be made two-particle self-consistent. This is of interest for the Hubbard model because it allows to suppress the antiferromagnetic phase transition in two dimensions in accordance with the Mermin-Wagner theorem, and to include the effects of bosonic fluctuations. For a physically sound description of the latter, the approximation should be conserving. In this paper, we show that the mutual requirements of two-particle self-consistency and conservation lead to fundamental problems. For an approximation that is two-particle self-consistent in the charge and longitudinal spin channels, the double occupancy of the lattice and the impurity is no longer consistent when computed from single-particle properties. For the case of self-consistency in the charge and longitudinal as well as transversal spin channels, these requirements are even mutually exclusive so that no conserving approximation can exist. We illustrate these findings for a two-particle self-consistent and conserving DMFT approximation.

Lizeng Zhang, Michael Ma, Fu‐Chun Zhang

Using a self-consistent mean-field theory, where the anyon gauge field is approximated by its expectation value, we obtain a linear excitation spectrum for noninteracting anyon systems. The sound velocity is found to be \ensuremath{\surd}3 /2 of the value obtained by the random-phase approximation. The uniform liquid state of the anyon gas is shown to be locally stable against solid formation.

V. Janiš, Peter Zalom, Vladislav Pokorný +1 more

A two-particle self-consistency is rarely part of mean-field theories. It is, however, essential for avoiding spurious critical transitions and unphysical behavior. We present a general scheme for constructing analytically controllable approximations with self-consistent equations for the two-particle vertices based on the parquet equations. We explain in detail how to reduce the full set of parquet equations so as not to miss quantum criticality in strong coupling. We further introduce a decoupling of convolutions of the dynamical variables in the Bethe-Salpeter equations to make them analytically solvable. We connect the self-energy with the two-particle vertices to satisfy the Ward identity and the Schwinger-Dyson equation and discuss the role of the one-particle self-consistency in making the approximations reliable in the whole spectrum of the input parameters. Finally, we demonstrate the general construction on the simplest static approximation that we apply to the Kondo behavior of the single-impurity Anderson model.