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Physical review. B, Condensed matter

Formulation of a self-consistent diagram method in thep-dmixing model

H. Matsumoto, M. Sasaki, M. Tachiki

Published May 1, 1991• 7 citations
Publisher

Abstract

A formulation of a self-consistent perturbation method is presented, which is convenient to deal with the strongly correlated p-d mixing model. A real-time formulation of thermofield dynamics is used. The method allows one to develop a finite-temperature perturbation theory for operators that do not allow the use of Wick's theorem. By introducing local decomposition of the operators, a diagram method that does not rely on Wick's theorem is constructed. For the p-d mixing model, self-consistent equations for the electron Green's functions are derived.

DOI
10.1103/physrevb.43.10247

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A formulation of a self-consistent perturbation method is presented, which is convenient to deal with the strongly correlated p-d mixing model. A real-time formulation of thermofield dynamics is used. The method allows one to develop a finite-temperature perturbation theory for operators that do not allow the use of Wick's theorem. By introducing local decomposition of the operators, a diagram method that does not rely on Wick's theorem is constructed. For the p-d mixing model, self-consistent equations for the electron Green's functions are derived.
Wick's theorem is combined with a cluster composition to derive a generalized statistical Wick's theorem for a non-equilibrium system with initial correlations. This enables a diagrammatic time-dependent perturbation theory which is a simple extension of the usual one to be developed for time-path causal statistical Green functions and their matrix equivalents. This provides a complete non-equilibrium theory. Reduction of diagrams and equations of motion for the Green functions are also discussed.
Perturbation theory for the spin-frac12frac12frac12 Heisenberg ferromagnet I. Wick's theorem and diagrammatic representation
Pauli operators are treated as Bose operators fulfilling q -number commutation relations. An analogue of Wick's theorem for Pauli operators is formulated suitable for perturbation theory. The lowest-order contributions to the perturbation series for the temperature dependent one-particle causal Green function are calculated, and a diagrammatic representation is introduced for them. It is not necessary to have a different meaning of the same graph depending on the order of temperatures contained in it. The connected graphs representing the perturbation series up to second order are drawn explicitly.
Perturbation theory in thermofield dynamics (a real time finite temperature quantum field theory) is formulated in the interaction representation and a diagrammatic expansion for the expectation value of time ordered products is derived solely on the basis of the thermal state condition and the algebraic properties of the operators. This new formulation has a much wider domain of applicability than those presented previously. The method is applicable even in the case of purely quantum mechanical systems. When certain bosonic operators, which commute with the unperturbed Hamiltonian, contribute to the renormalization of the thermal state vacuum the vacuum diagrams do not cancel in the usual way. It will be shown how these diagrams arise and how they may be computed.
A perturbation theory scheme in terms of electron hopping, which is based on the Wick theorem for Hubbard operators, is developed. Diagrammatic series contain single-site vertices connected by hopping lines and it is shown that for each vertex the problem splits into the subspaces with ``vacuum states'' determined by the diagonal Hubbard operators and only excitations around these vacuum states are allowed. The rules to construct diagrams are proposed. In the limit of infinite spatial dimensions the total auxiliary single-site problem exactly splits into subspaces that allows to build an analytical thermodynamically consistent approach for a Hubbard model. Some analytical results are given for the simple approximations when the two-pole (alloy-analogy solution) and four-pole (Hartree-Fock approximation) structure for Green's function is obtained. Two poles describe contribution from the Fermi-liquid component, which is dominant for small electron and hole concentrations (``overdoped case'' of high-Tc′s),{T}_{c}'\mathrm{s}),Tc′​s), whereas other two describe contribution from the non-Fermi liquid and are dominant close to half-filling (``underdoped case'').