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Fluctuation and Noise Letters

THE OPEN BCS MODEL, ITS STOCHASTIC LIMIT AND SOME GENERALIZATIONS

Fabio Bagarello

Published Jun 1, 2005• 0 citations
Publisher

Abstract

In this paper we use the stochastic limit approach as a tool to discuss the open BCS model of low temperature superconductivity. We also briefly discuss the role of a second reservoir interacting with the first one (but not with the system) in the computation of the critical temperature corresponding to the transition from a normal to a superconducting phase.

DOI
10.1142/s0219477505002756

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In this paper we use the stochastic limit approach as a tool to discuss the open BCS model of low temperature superconductivity. We also briefly discuss the role of a second reservoir interacting with the first one (but not with the system) in the computation of the critical temperature corresponding to the transition from a normal to a superconducting phase.
We study the thermodynamics of a superconductor with strongly bound Cooper pairs by solving the self-consistent T-matrix equations for the attractive Hubbard model within a local approximation. We determine the transition temperature as a function of the interaction strength at quarter filling, and compute the temperature dependence of the superconducting order parameter, the specific heat, the spin susceptibility, and the density of local pairs in the normal and superconducting phase. Pronounced deviations from BCS behavior are found already for moderate interactions, where superconductivity sets in at a critical temperature above the coherence temperature for Fermi liquid behavior.
In papers devoted to superfluidity, the generally accepted statement that the dynamics of the corresponding phase transition is described by the stochastic model F or E can be frequently found. Nevertheless, the dynamical critical index has not been found even in the leading order of the perturbation theory. It is also unknown which model, E or F, in fact corresponds to this system. We use two different approaches to study this problem. First, we study the dynamics of the critical behavior in a neighborhood of the λ point using the renormalization group method based on a quantum microscopic model in the formalism of the time-dependent Green’s functions at a finite temperature. Second, we study the stochastic model F to find whether it is stable under compressibility effects. Both approaches lead to the same very unexpected result: the dynamics of the phase transition to the superfluid state are described by the stochastic model A with a known dynamical critical index.
Superconductivity beyond the Gorkov decoupling is examined with an emphasis on the thermodynamic properties. In the Gorkov decoupling, the interaction term is factorized into the pairing correlation. Therefore, in contrast to the strong pair-pair correlation in coordinate space, the correlation between the Cooper pairs in momentum space is neglected. In this paper, in addition to the conventional pairing interaction, we consider a more general type of attractive interaction that cannot be factorized into the product of the pairing correlation. Accordingly, it becomes necessary to include the neglected correlation. We assume a variational functional where the general type of interaction is taken into account, and determine the state self-consistently at finite temperature. The following results are obtained: (1) The state obtained is more stable at low temperature, compared with the BCS state with the same parameters. (2) As the temperature is increased, our state approaches the BCS state. (3) At the same temperature as in the BCS theory, our system undergoes a phase transition from the superconducting phase to the normal phase.
We present dynamical equations that are valid in the vicinity of a phase transition to the superconducting state. In writing the equations, we take the possible influence of the magnetic interaction between charge carriers and temperature fluctuations into account. We discuss the type of the phase transition under study. We argue in favor of the applicability of stochastic dynamical model A according to the standard classification (the stochastic model with one two-component field without a conservation law) to describe the dynamics of this phase transition.