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Progress of Theoretical Physics

Self-Consistency in the Quasi-Chemical Equilibrium Theory of Superconductivity

Colin J. Thompson, Takeo Matsubara

Published Apr 1, 1963• 1 citation
Publisher

Abstract

By expressing the density matrix, for the special case of one pair state only and simple pairing, in exponential form, we reproduce the results of earlier papers and obtain an expression for the entropy which is formally identical with that of Bardeen, Cooper and Schrieffer and that of Bogoliubov. The expression for the entropy is shown to be exact in the limit of infinite volume. A variational calculation is then performed and the resulting set of equations is identical with that of B.C.S., thus establishing the relation between the Quasi-Chemical Equilibrium, the B.C.S., and the Bogoliubov theories of superconductivity at non-zero temperature.

DOI
10.1143/ptp.29.494

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Progress of Theoretical Physics•Nov 1, 1965•1 citation
By expressing the density matrix, for the special case of one pair state only and simple pairing, in exponential form, we reproduce the results of earlier papers and obtain an expression for the entropy which is formally identical with that of Bardeen, Cooper and Schrieffer and that of Bogoliubov. The expression for the entropy is shown to be exact in the limit of infinite volume. A variational calculation is then performed and the resulting set of equations is identical with that of B.C.S., thus establishing the relation between the Quasi-Chemical Equilibrium, the B.C.S., and the Bogoliubov theories of superconductivity at non-zero temperature.
The theory of superconductivity is formulated starting from the configuration-space form of the Bardeen-Cooper-Schrieffer (BCS) wave function. Self-consistent equations for treating nonu- niform systems are obtained. These equations have the advantage over the Bogoliubov-de Gennes equations that an effective condensate wave function and quasiparticle eigenstates can separately be obtained. The intrinsic orbital angular momentum density of the condensate is naturally iden- tified in this formalism due to a manifest two-particle bound-state nature in the one-particle density matrix.
We investigate the thermodynamic properties of an interacting Bose gas with a condensate within the energy-functional formulation of the Hartree-Fock-Bogoliubov (HFB) approach. For a contact interaction, we derive a self-consistent solution to the HFB equations that intrinsically eliminates divergence. This solution characterizes the equilibrium state featuring a condensate of correlated pairs of particles. We analyze the temperature dependence of key thermodynamic quantities such as condensate density, chemical potential, entropy, pressure, specific heat capacity at constant volume, and isothermal compressibility and compare them with predictions from the Popov approximation (PA). We predict that the transition temperature shifts to higher values due interactions, with the HFB approach yielding a larger shift than the PA. Analysis of the compressibility indicates that a pure pair condensate is unstable, and the stable equilibrium corresponds to only a mixture of single-particle and pair condensates.
The quasi-chemical equilibrium, or pair correlation, approximation to statistical mechanics is written in terms of second-quantization formalism. This involves an entension of the original theory to include an Ansatz for off-diagonal elements of the density matrix. The formal expressions involve “labelling operators” acting in a purely formal Hilbert space. These labelling operators obey Bose commutation rules for correlations between even numbers particles (e.g. pair correlations), Fermi commutation rules for correlations between odd numbers of particles. The labelling operators provide an algebraic way of formulating the restriction to certain types of “graphs”. We also show how to include higher-order correlations into the Ansatz. The main results of the original theory can be derived more rapidly with the new formalism. However, certain correction terms obtained earlier are shown to be in error, necessitating a reinvestigation of the nature of the condensation phenomenon. This re-investigation is carried out in two following papers, and proves to lead to no essential modifications of the previous results. The correlation matrices which enter into the Ansatz need to be related to the Hamiltonian of the system, and to the thermodynamic variables which define its thermodynamic state. A variational formulation is developed for this purpose, but no explicit calculations are carried out in this paper.
The new generalized Bose–Einstein condensation (GBEC) quantum-statistical theory starts from a noninteracting ternary boson-fermion (BF) gas of two-hole Cooper pairs (2hCPs) along with the usual two-electron Cooper pairs (2eCPs) plus unpaired electrons. Here we obtain the entropy and heat capacity and confirm once again that GBEC contains as a special case the Bardeen–Cooper–Schrieffer (BCS) theory. The energy gap is first calculated and compared with that of BCS theory for different values of a new dimensionless coupling parameter n/n[Formula: see text] where n is the total electron number density and n[Formula: see text] that of unpaired electrons at zero absolute temperature. Then, from the entropy, the heat capacity is calculated. Results compare well with elemental-superconductor data suggesting that 2hCPs are indispensable to describe superconductors (SCs).
The spectrum and eigenstates of Bardeen-Cooper-Schrieffer's reduced Hamiltonian in the theory of superconductivity are investigated in a mathematically rigorous manner. The Hamiltonian is defined first for a finite number of electrons contained in a finite volume. The interaction Hamiltonian is assumed to take a simple form. The Hamiltonian after the Bogoliubov transformation (transformed Hamiltonian) has a limit in the weak topology of Hilbert space when the volume tends to infinity with the number density of electrons kept constant. In the limit the interaction part vanishes and the free part gives a continuous excitation spectrum of Bogolons (transformed electrons). It should be noticed that the operator norm of the former does not tend to zero and has no strong limit as well as the total Hamilton. An equivalent boson which consists of two Bogolons with opposite momenta and spins is introduced. In the Fock space for the bosons the Hamiltonian has a strong limit for infinite volume. The limit operator has the same excitation spectrum of Bogolons as the free Hamiltonian and the existence of the lower limit of the spectrum is shown for some special class of the interactions. For large volume the limit operator is a good approximation to be transformed Hamiltonian. The ground state of the free part of the transformed Hamiltonian (i.e. the ground state of Bardeen-Cooper-Schrieffer's calculation) is not the exact eigenstate of the transformed total Hamiltonian and the difference does not vanish when the volume tends to infinity.