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arXiv:math-ph/0103004 · Journal of Physics A Mathematical and General

Metastability in the BCS model

J. Lauwers, André F. Verbeure

Published Jun 27, 2001• 3 citations
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Abstract

We discuss metastable states in the mean-field version of the strong coupling BCS-model and study the evolution of a superconducting equilibrium state subjected to a dynamical semi-group with Lindblad generator in detailed balance w.r.t. another equilibrium state. The intermediate states are explicitly constructed and their stability properties are derived. The notion of metastability in this genuine quantum system, is expressed by means of energy-entropy balance inequalities and canonical coordinates of observables.

DOI
10.1088/0305-4470/34/27/304

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We discuss metastable states in the mean-field version of the strong coupling BCS-model and study the evolution of a superconducting equilibrium state subjected to a dynamical semi-group with Lindblad generator in detailed balance w.r.t. another equilibrium state. The intermediate states are explicitly constructed and their stability properties are derived. The notion of metastability in this genuine quantum system, is expressed by means of energy-entropy balance inequalities and canonical coordinates of observables.
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Stabilizing superconductivity beyond its equilibrium regime has been one of the quests of nonequilibrium state engineering. We report the presence of nonthermal pairing gap in BCS superconductors coupled to two thermal baths, where one of the baths can be at a temperature well above the equilibrium TcT_cTc​. Superconducting state is enabled by a highly non-equilibrium steady state stabilized by weak coupling to baths that break detailed balance condition. The steady-state is well described by a generalized Gibbs ensemble parametrized with chemical potentials for associated Bogoliubov quasiparticle. A strong, non-perturbative effect is reported for both sss-wave and ddd-wave superconductors, and its possible realization in cavity-based experiments is discussed.
Metastability in many-body quantum systems and its associated exponentially-long timescales have been the subject of considerable recent interest. Here, we focus on a class of driven-dissipative many-body open quantum systems described by a Lindbladian having hidden time-reversal symmetry (a form of quantum detailed balance). Examples include boundary-driven interacting spin chains, bosonic lattice models and driven-dissipative collective spin models. We suggest that for such systems, slow timescales in the vicinity of a dissipative first-order phase transition can be analytically predicted using a special purification of the non-equilibrium steady state. We show the accuracy of our conjecture through detailed studies of a dissipative transverse-field Ising model with collective and local decay, and a driven-dissipative nonlinear cavity model. Our results allow quantitative insights into metastability and slow dynamics for a range of systems, including cases where semiclassical or path-integral instanton approaches are intractable.
I analyze the low temperature limit of the BCS theory of s-wave single-band superconductors, when the attraction band may be asymmetric with respect to the chemical potential. I discuss equilibrium systems, taking consistently into account the variation of the energy and of the total number of particles with the populations of the quasiparticle energy levels. I show that the equation for the energy gap has two solutions, one of which is stable and the other one is metastable. When the chemical potential is the center of the attraction band (the standard BCS assumption), the energy gap in the stable solution is Δ0Δ_0Δ0​, whereas in the metastable one is Δ0/3Δ_0/3Δ0​/3. If the chemical potential is not in the center of the attraction band, then a quasiparticle imbalance appears. If the absolute value of the difference between the chemical potential and center of the attraction band is bigger than 2Δ02Δ_02Δ0​, then the superconducting energy gap cannot be formed. If the number of particles is conserved and the attraction band is asymmetric, then the stable solution is unphysical and only metastable solutions are realized.
The aim of the current paper is to illustrate, in a simple example, our recent, very general, rigorous results [1, 2] on the dynamical properties of fermions and quantum-spin systems with long-range, or mean-field, interactions, in infinite volume. We consider here the strong-coupling BCS-Hubbard model studied in [3, 4], because this example is very pedagogical and, at the same time, physically relevant for it highlights the impact of the (screened) Coulomb repulsion on ( sss -wave) superconductivity.