Another operator-theoretical proof for the second-order phase transition in the BCS-Bogoliubov model of superconductivity

Abstract

In the preceding papers, imposing certain complicated and strong conditions, the present author showed that the solution to the BCS-Bogoliubov gap equation in superconductivity is twice differentiable only on the neighborhoods of absolute zero temperature and the transition temperature so as to show that the phase transition is of the second order from the viewpoint of operator theory. Instead, we impose a certain simple and weak condition in this paper, and show that there is a unique nonnegative solution and that the solution is indeed twice differentiable on a closed interval from a certain positive temperature to the transition temperature as well as pointing out several properties of the solution. We then give another operator-theoretical proof for the second-order phase transition in the BCS-Bogoliubov model. Since the thermodynamic potential has the squared solution in its form, we deal with the squared BCS-Bogoliubov gap equation. Here, the potential in the BCS-Bogoliubov gap equation is a function and need not be a constant.

arXiv comment18 pages

References in corpus (24)

• Generalized Hartree-Fock Theory and the Hubbard Model
• Critical Temperature and Energy Gap for the BCS Equation
• The BCS Functional for General Pair Interactions
• Variational Principle of Bogoliubov and Generalized Mean Fields in Many-Particle Interacting Systems
• The BCS functional of superconductivity and its mathematical properties
• Bogoliubov's Vision: Quasiaverages and Broken Symmetry to Quantum Protectorate and Emergence
• The BCS Critical Temperature for Potentials with Negative Scattering Length
• The external field dependence of the BCS critical temperature
• The solution to the BCS gap equation and the second-order phase transition in superconductivity
• The role of the chemical potential in the BCS theory
• Is the solution to the BCS gap equation continuous in the temperature ?
• The BCS gap equation for spin-polarized fermions
• Renormalization Group Methods: Landau-Fermi Liquid and BCS Superconductor
• An operator-theoretical treatment of the Maskawa-Nakajima equation in the massless abelian gluon model
• Persistence of translational symmetry in the BCS model with radial pair interaction
• Smoothness and monotone decreasingness of the solution to the BCS-Bogoliubov gap equation for superconductivity
• An operator-theoretical proof for the second-order phase transition in the BCS-Bogoliubov model of superconductivity (final version)
• General pairing mechanisms in the BCS-theory of superconductivity
• An operator-theoretical study of the specific heat and the critical magnetic field in the BCS-Bogoliubov model of superconductivity
• Another operator-theoretical proof for the second-order phase transition in the BCS-Bogoliubov model of superconductivity
• Superconducting phase in the BCS model with imaginary magnetic field
• Superconducting phase in the BCS model with imaginary magnetic field. II. Multi-scale infrared analysis
• An operator-theoretical study on the BCS-Bogoliubov model of superconductivity near absolute zero temperature
• Superconducting phase in the BCS model with imaginary magnetic field. III. Non-vanishing free dispersion relations

Cited by in corpus (3)

• Another operator-theoretical proof for the second-order phase transition in the BCS-Bogoliubov model of superconductivity
• An operator-theoretical study on the BCS-Bogoliubov model of superconductivity near absolute zero temperature
• The BCS-Bogoliubov gap equation with external magnetic field and the first-order phase transition