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Bulletin of the American Physical Society

Self-consistent solution for the Hubbard model at the two-particle and one-particle level using the parquet formalism.

Herbert F. Fotso, Shuxiang Yang, Jun Liu, Mark Jarrell, Eduardo F. D'Azevedo, Thomas Maier, Karen A. Tomko, Richard T. Scalettar

Published Mar 17, 2009• 0 citations
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We present a numerical solution of the parquet approximation, a conserving diagrammatic approach which is self-consistent at both the single-particle and the two-particle levels. The fully irreducible vertex is approximated by the bare interaction thus producing the simplest approximation that one can perform with the set of equations involved in the formalism. The method is applied to the Hubbard model on a half-filled 4x4 cluster. Results are compared to those obtained from determinant quantum Monte Carlo (DQMC), FLuctuation EXchange (FLEX), and self-consistent second-order approximation methods. This comparison shows a satisfactory agreement with DQMC and a significant improvement over the FLEX or the self-consistent second-order approximation.
We present a general framework for investigating the stability of solutions being a parquet-type extension of the Baym-Kadanoff construction of conserving approximations. To obtain a consistent description of one- and two-particle quantities, needed for the stability analysis, we use explicit equations for one- and two-particle Green functions simultaneously. We formulate a complete set of stability criteria and show that each instability, singularity in a two-particle function, is connected with a symmetry-breaking order parameter, either of density or anomalous type. We explicitly study the Hubbard model at intermediate coupling and demonstrate that approximations with static vertices become unstable with respect to spin flips, before the Mott-Hubbard metal-insulator transition in the paramagnetic phase or the Kondo strong-coupling regime can be reached. Further on, we use the parquet approximation and turn it into a workable scheme with dynamical vertex corrections. We obtain a theory with two-particle self-consistence, the complexity of which is comparable with fluctuation-exchange-type approximations. We show that it is the simplest consistent theory free of spurious phases being able to correctly describe qualitatively, quantum critical points and the transition from weak to strong coupling in correlated electron systems.