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Theoretical and Mathematical Physics

Dynamical description of the phase transition to the superconducting state

L. A. Gosteva, Mikhail Yu. Nalimov, Artem Sergeevich Yashugin

Published Nov 1, 2024• 0 citations
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Abstract

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.

DOI
10.1134/s0040577924110126

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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.
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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.
Within a mean-field approach, the stationary states of the kinetic spin-1 Blume-Capel model in the presence of a time-dependent oscillating external magnetic field is studied. The Glauber-type stochastic dynamics is used to describe the time evolution of the system and obtain the mean-field dynamic equation of motion. The dynamic phase-transition points are calculated and phase diagrams are presented in the temperature and crystal-field interaction plane. According to the values of the magnetic field amplitude, three fundamental types of phase diagrams are found: One exhibits a dynamic tricritical point, while the other two exhibit a dynamic zero-temperature critical point.
In standard nonrelativistic quantum mechanics the expectation of the energy is a conserved quantity. It is possible to extend the dynamical law associated with the evolution of a quantum state consistently to include a nonlinear stochastic component, while respecting the conservation law. According to the dynamics thus obtained, referred to as the energy-based stochastic Schrödinger equation, an arbitrary initial state collapses spontaneously to one of the energy eigenstates, thus describing the phenomenon of quantum state reduction. In this paper, two such models are investigated: one that achieves state reduction in infinite time and the other in finite time. The properties of the associated energy expectation process and the energy variance process are worked out in detail. By use of a novel application of a nonlinear filtering method, closed-form solutions—algebraic in character and involving no integration—are obtained of both these models. In each case, the solution is expressed in terms of a random variable representing the terminal energy of the system and an independent noise process. With these solutions at hand it is possible to simulate explicitly the dynamics of the quantum states of complicated physical systems.