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Europhysics Letters (EPL)

Functional-Integral Approaches and Diagrammatic Many-Body Theory of the Hubbard Model

Cláudio Andrade Macedo, M. D. Coutinho Filho

Published Feb 15, 1987• 4 citations
Publisher

Abstract

We are able to demonstrate that we can attain a proper comprehension of the functional-integral method as applied to the Hubbard and similar model systems. This is done in the context of diagrammatic perturbation theory. In particular, new as well as well-known exact relations among physical quantities of interest are derived in the context of the formalism.

DOI
10.1209/0295-5075/3/4/001

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We are able to demonstrate that we can attain a proper comprehension of the functional-integral method as applied to the Hubbard and similar model systems. This is done in the context of diagrammatic perturbation theory. In particular, new as well as well-known exact relations among physical quantities of interest are derived in the context of the formalism.
In this work the basic formalism of non-equilibrium Green’s functions is presented and then applied to study a Ward identity in linear response theory, namely the frequency sum-rule. It can be proven that the frequency sum-rule is satisfied when the quantities involved are calculated using perturbation theory within a conserving approximation for the self-energy. To illustrate this equality along with other properties of the response function, a numerical application that solves the Kadanoff-Baym equations for systems of Hubbard chains was used. The results showed that the frequency sum-rule was satisfied to the same extent by all the conserving approximations used as by the exact diagonalization numerical results. The density response function was analyzed diagrammatically for a series of conserving approximations for the self-energy and this demonstrated that even for a first order in perturbation theory approximation for the self-energy, the response function has a corresponding complex, third order in the perturbation diagrammatic structure.
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.