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Fortschritte der Physik

A Rigorous Formulation of the Boson Method in Superconductivity

H. Matsumoto, Hiroomi Umezawa

Published Jan 1, 1976• 57 citations
Publisher

Abstract

Abstract A rigorous formulation of the boson method in superconductivity is presented. Assuming the local gauge invariance and presence of a phase‐dependent order parameter and utilizing the path‐integral formalism, we derive a set of macroscopic equations which control all the superconducting states. These equations are model‐independent. They contain certain functions and parameters which should be calculated for a given model. Using this rigorous formulation, we study what kind of approximations are involved in the usual formulation of the boson method and how these approximations can be improved.

DOI
10.1002/prop.19760240702

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Abstract A rigorous formulation of the boson method in superconductivity is presented. Assuming the local gauge invariance and presence of a phase‐dependent order parameter and utilizing the path‐integral formalism, we derive a set of macroscopic equations which control all the superconducting states. These equations are model‐independent. They contain certain functions and parameters which should be calculated for a given model. Using this rigorous formulation, we study what kind of approximations are involved in the usual formulation of the boson method and how these approximations can be improved.
Abstract On the basis of an extended attractive Hubbard model with nearest‐neighbour interaction of opposite spins two functional methods are adopted for deriving a self‐consistent equation for the superconducting order parameter. The functional‐integral formalism is exploited in saddle‐point approximation. The functional‐derivative technique using Schwinger source fields leads via the self‐energy in Hartree‐Fock‐Bogolubov approximation to a nonlinear gap equation. The superconducting transition temperature is investigated for the case of s‐wave pairing.
We present a new bosonization procedure for two-dimensional fermion models in the path integral formulation. We introduce external sources for the fermion currents instead of the auxiliary boson fields used in the conventional procedure. The calculations become much simpler because the external sources do not interact with themselves, while the auxiliary boson fields do, and we can find the equivalent boson models directy. All the formulations are done in Minkowski space-time. The extended Bukhvostov-Lipatov model and the massive Thirring model are discussed here as typical two-dimensional fermion models with current-current-type interactions. The Rothe and Stamatescu model is also discussed, though this is slightly out of the context of bosonization, and the relation of the fermion current in the Heisenberg picture with that in the interaction picture is derived almost straightforwardly. Usually in the operator formalism, this relation is derived using the splitting procedure, which requires detailed calculations. It is expected that we can derive the boson model equivalent to the fermion model with SU(N) symmetry more easily and without any guess work.
Abstract We consider the coupling of a scalar field to linearised gravity and derive a relativistic gravitationally induced decoherence model using Ashtekar variables. The model is formulated at the gauge invariant level using suitable geometrical clocks in the relational formalism, broadening existing gauge invariant formulations of decoherence models. For the construction of the Dirac observables we extend the known observable map by a kind of dual map where the role of clocks and constraints is interchanged. We also discuss a second choice of geometrical clocks existing in the ADM literature. Then we apply a reduced phase space quantisation on Fock space and derive the final master equation choosing a Gibbs state for the gravitational environment and using the projection operator technique. The resulting master equation is not automatically of Lindblad type, a starting point sometimes assumed for phenomenological models, but still involves a residual time dependence at the level of the effective operators in the master equation due to the form of the correlation functions that we express in terms of thermal Wightman functions. Furthermore, we discuss why in the model analysed here the application of a second Markov approximation in order to obtain a set of time independent effective system operators is less straightforward than in some of the quantum mechanical models.
In a previous paper, a quantum-mechanical formulation involving both mean fields and independent-particle fields in many-fermion systems was proposed using the path-integral technique. Then the semiclassical calculation of the energy spectra was performed, and the quantization rule was derived by applying a stationary phase approximation on the path integral. In this paper, a boson-fermion expansion is derived from our formulation using Dirac quantization. As an illustration, the Lipkin model is utilized.