^{1,a)}and R. Seiringer

^{2,b)}

### Abstract

We review recent results concerning the mathematical properties of the Bardeen–Cooper–Schrieffer (BCS) functional of superconductivity, which were obtained in a series of papers, partly in collaboration with R. Frank, E. Hamza, S. Naboko, and J. P. Solovej. Our discussion includes, in particular, an investigation of the critical temperature for a general class of interaction potentials, as well as a study of its dependence on external fields. We shall explain how the Ginzburg–Landau model can be derived from the BCS theory in a suitable parameter regime.

This review originates from lecture notes put together for lectures on the subject given by C.H. at the summer school “Current Topics in Mathematical Physics” at the CIRM in Marseille in September 2013, as well as by R.S. at the workshop “Spectral Theory and Dynamics of Quantum Systems” at Blaubeuren in February 2014, and during a summer course at McGill University in July 2014. We are grateful to Gerhard Bräunlich and Andi Deuchert for many useful remarks and suggestions on various versions of this manuscript.

I. INTRODUCTION A. The BCS energy functional B. Brief summary of mathematical results 1. Translation-invariant case 2. The case of weak and slowly varying fields II. MATHEMATICAL BACKGROUND AND “DERIVATION” OF THE BCS FUNCTIONAL A. Quantum many-body systems 1. Fock space 2. Hamiltonian and Gibbs states 3. Quasi-free or BCS states 4. Representation of the energy 〈ℍ〉

_{ρ}in terms of

*γ*and

*α*B. Bcs energy functional 1. Reduction to

*SU*(2)-invariant states 2. Omitting direct and exchange energies C. Collective behavior and long range order III. BCS FUNCTIONAL RESTRICTED TO TRANSLATION-INVARIANT STATES A. Minimization of the BCS functional B. Critical temperature 1. Radial potentials C. Birman–Schwinger argument 1. Weak coupling limit 2. Low density limit 3. Zero-range limit 4. Zero temperature and energy gap IV. DERIVATION OF THE GINZBURG–LANDAU FUNCTIONAL A. The Ginzburg–Landau model B. Main results 1. The coefficients

*λ*C. Sketch of the proof of theorem 4.1 D. Useful identity and relative entropy inequality E. Proof of the key steps 1. Step 1 2. Step 3 3. Step 2 F. Absence of external fields

_{i}^{2}(ℝ

^{d}) below, it will always mean that .

Data & Media loading...

Article metrics loading...

### Abstract

We review recent results concerning the mathematical properties of the Bardeen–Cooper–Schrieffer (BCS) functional of superconductivity, which were obtained in a series of papers, partly in collaboration with R. Frank, E. Hamza, S. Naboko, and J. P. Solovej. Our discussion includes, in particular, an investigation of the critical temperature for a general class of interaction potentials, as well as a study of its dependence on external fields. We shall explain how the Ginzburg–Landau model can be derived from the BCS theory in a suitable parameter regime.

Full text loading...

Commenting has been disabled for this content