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Physical review. B, Condensed matter

Hubbard model: Functional-integral approach and diagrammatic perturbation theory

Cláudio Andrade Macedo, Mauriaacio D. Coutinho-Filho

Published Jun 1, 1991• 13 citations
Publisher

Abstract

We present a careful analysis of the functional-integral approach as applied to the Hubbard model in the framework of diagrammatic perturbation theory. It is demonstrated that a proper comprehension of the formalism can be attained and that the functional-integral method is a very systematic technique, free of ambiguity. In particular, additional exact relations among physical quantities of interest are derived in the context of the formalism.

DOI
10.1103/physrevb.43.13515

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We present a careful analysis of the functional-integral approach as applied to the Hubbard model in the framework of diagrammatic perturbation theory. It is demonstrated that a proper comprehension of the formalism can be attained and that the functional-integral method is a very systematic technique, free of ambiguity. In particular, additional exact relations among physical quantities of interest are derived in the context of the formalism.
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In this work we introduce the dual boson diagrammatic Monte Carlo technique for strongly interacting electronic systems. This method combines the strength of dynamical mean-filed theory for nonperturbative description of local correlations with the systematic account of nonlocal corrections in the dual boson theory by the diagrammatic Monte Carlo approach. It allows us to get a numerically exact solution of the dual boson theory at the two-particle local vertex level for the extended Hubbard model. We show that it can be efficiently applied to description of single-particle observables in a wide range of interaction strengths. We compare our exact results for the self-energy with the ladder dual boson approach and determine a physical regime, where the description of collective electronic effects requires more accurate consideration beyond the ladder approximation. Additionally, we find that the order-by-order analysis of the perturbative diagrammatic series for the single-particle Green's function allows to estimate the transition point to the charge density wave phase.
The formalism of nonequilibrium perturbation theory was constructed by Schwinger and Keldysh and then was developed with the diagrammatical technique by Lifshitz and Pitaevskii. Until now there has been widespread application to various researches in physics, condensed matter, plasmas, atoms and molecules, nuclear matter etc.. In spite of this, the formalism has not been established as perturbation theory. For example, there is no perturbative method to derive arbitrary self-energy properly. In addition, the connection with other formalism, e.g., the Matsubara imaginary-time perturbative formalism is uncertain. Although there must be the relationship between self-energies in the perturbative formalism, such basic problems remain to be solved. The solution is given by the present work. The real-time perturbative expansion is performed on the basis of the adiabatic theorem. As the results, the requirements of self-energies as functions in time are demonstrated and the formulated self-energies meet the known relations. Besides, it gives exact agreement with functions derived by perturbative expansion in imaginary-time and analytical continuity. As a consequence, it implies that the present formalism can be generalized.