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Journal of Mathematical Physics

Common Interpretation of Phase Transitions in Various Models

Gérard G. Emch

Published Jan 1, 1967• 9 citations
Publisher

Abstract

A discussion is presented of the formal analogy which exists between the Bardeen-Cooper-Schrieffer (BCS) model for superconductivity and the several variations of the model suggested by Kac for the exact study of phase transitions. Some of the algebraic techniques, pioneered by Haag for his formulation of axiomatic field theory and his application of this theory to the BCS model, are shown to extend to all these cases. Some arguments are given in the beginning of the paper to point out the necessity of a new interpretation of the existence of various phases in the same physical system. The results obtained so far for equilibrium situations confirm the consistency of the proposed tentative description of the phenomena considered.

DOI
10.1063/1.1705092

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A discussion is presented of the formal analogy which exists between the Bardeen-Cooper-Schrieffer (BCS) model for superconductivity and the several variations of the model suggested by Kac for the exact study of phase transitions. Some of the algebraic techniques, pioneered by Haag for his formulation of axiomatic field theory and his application of this theory to the BCS model, are shown to extend to all these cases. Some arguments are given in the beginning of the paper to point out the necessity of a new interpretation of the existence of various phases in the same physical system. The results obtained so far for equilibrium situations confirm the consistency of the proposed tentative description of the phenomena considered.
A method of general applicability to the solution of second-quantized field theories at finite temperature is illustrated using the BCS (Bardeen–Cooper–Schrieffer) model of superconductivity. Finite-temperature field theory is treated using the thermo field-theory formalism of Umezawa and collaborators. The solution of the field theory uses an expansion in thermal modes analogous to the Haag expansion in asymptotic fields used in the N-quantum approximation at zero temperature. The lowest approximation gives the usual gap equation.
Lattice models in theoretical physics A major part of theoretical physics involves the construction and systematic analysis of mathematical models as a description of the physical world. For microscopic phenomena, in particular, the reference to models becomes quite explicit. The phenomena are often complex, and any theory that attempts to include every detail soon becomes intractable. It is more fruitful to develop models which ignore irrelevant details but, hopefully, capture the essential physics of the phenomena of interest. Thus we have the Heisenberg model of magnetic order, the Bardeen–Cooper– Schrieffer (BCS) model of superconductivity, and so on. It is important to note, at the outset, that the models we shall be discussing describe strongly interacting, and therefore highly correlated, systems of particles. These are difficult and interesting problems. Where interactions are absent, or weak, elementary treatments are possible and the resulting phenomena are generally unspectacular. Why lattice models? In solid-state phenomena there is usually an underlying lattice structure, and the symmetry properties of this lattice play an important role in the analysis. Even the process of electrical conduction in metals or semiconductors can be equally well described in terms of localized quantum states or in terms of the more usual continuum picture. In quantum field theory, which is formulated in a space–time continuum, a lattice is often introduced for computational purposes. One can think of this in two ways: as an approximation to the continuum, which is then recovered as a limit at the end of the calculations, or as a necessary means of regulating the theory (i.e. controlling divergences) in calculating the Feynman path integral.
The ground-state wave function and some of the excited states of the BCS reduced Hamiltonian are found. In the limit of large volume, the boundary and continuity conditions on the exact wave function lead directly to the equations which Bardeen, Cooper, and Schrieffer found by a variational technique. It is also shown in what sense the BCS trial wave function may be considered asymptotically exact in this limit. Finite-volume corrections are included in an appendix, and explicit calculations are carried out for a one-step model of the kinetic energy which has possible applications to the problem of the finite nucleus.
The equilibrium statistical mechanics of the Bardeen-Cooper-Schrieffer model of superconductivity, as well as that of a wide class of similar models, can be evaluated exactly by the ``thermodynamically equivalent Hamiltonian'' method of Bogoliubov, Zubarev, Tserkovnikov, and Wentzel. It has been pointed out by Wentzel that this method can be extended to certain weakly nonequilibrium situations. It is shown here that the method allows an exact evaluation of the nonequilibrium statistical mechanics of the following situations: (a) temporal evolution of the statistical expectation value of an observable O(r) whose initial deviation from equilibrium is spatially localized, but not necessarily small; (b) temporal evolution of the statistical expectation value of an observable O due to a perturbation V which is spatially localized, but not necessarily small.
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.