Microscopic foundations of the Meißner effect: thermodynamic aspects (Q2853432)

The main subject of the paper is the full Meißner effect including the existence of currents concentrating near the bulk which annihilate the total magnetic induction inside the superconductor. The presented microscopic theory is based on the strong-coupling BCS-Hubbard model with a self-generated magnetic induction, which is driven by a spatial inhomogeneous external magnetic induction. First, the quantum many-body problem at fixed magnetic induction based on Hamiltonian is set up including the terms representing the strong coupling limit of the kinetic energy, the screened Coulomb repulsion as in the Hubbard model, and the BCS interaction written in the \(x\)-space. The important result obtained consists in that the macroscopic system can create any smooth current density by paying an infinitesimal energy price in the thermodynamic limit. Then, the Biot-Savart operator used is explained to define magnetic inductions from currents. The self-generated magnetic induction is discussed taking into account that the energy is carried by the total magnetic induction, by adding a magnetic term to the free-energy density. Minimizers of this new magnetic free-energy functional do carry currents in general. Then, the thermodynamics of the quantum system under consideration with a self-generated magnetic induction in relation to the existence of the Meißner effect are analyzed. The authors are focused on the Meißner effect, i.e., the existence of superconducting states with self-generated magnetic inductions, which vanish inside the unit box, while being created by currents supported on the box boundary at the corresponding conditions. First, the theorem on sufficient but not necessary conditions for the Meißner effect is considered. By adding a sufficiently small hopping term to the strong-coupling BCS-Hubbard model, the authors obtain a more realistic model, which has essentially the same correlation functions, by the Grassmann interaction and Brydges-Kennedy tree expansion methods together with determinant bounds. The assertions, obtained for models of the Meißner effect with hopping terms, the result from a before developed method, which gives access to domains of the phase diagram usually difficult to reach via other mathematical methods. In the analysis of the full Meißner effect from first principles of quantum mechanics, the results are provided concerning the free-energy taking into account contributions of the magnetic energy due to currents. By adding a magnetic term to the usual free-energy density, the obtained results show that the minimizers of this new magnetic free-energy can create surface currents which annihilate the total magnetic induction inside the bulk in the thermodynamic limit. The corresponding Euler-Lagrange equations for these minimizers seem to indicate that an effective magnetic susceptibility equal to \(-1\) is not the mechanism behind the Meißner effect.