2019/02/06 by J. Le Magnen, Magnen, J., Jérémie Unterberger +1
Physics and Astronomy · #81T08 #81V70 #82D55 #Cold Atom Physics and Bose-Einstein Condensates #FOS: Physical sciences #High Energy Physics - Theory (hep-th) #Mathematical Physics (math-ph) #Physics of Superconductivity and Magnetism #Quantum and electron transport phenomena #Superconductivity (cond-mat.supr-con)
paper · pdf · doi:10.48550/arxiv.1902.02337
openalex publication_date 2019/02/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Starting from H. Fröhlich's second-quantized Hamiltonian for a d-dimensional electron gas in interaction with lattice phonons describing the quantum vibrations of a metal, we present a rigorous mathematical derivation of the superconducting state, following the principles laid out originally in 1957 by J. Bardeen, L. Cooper and J. Schrieffer. As in the series of papers written on the subject in the 90es, of which the present paper is a continuation, the representation of ions as a uniform charge background allows for a (1+d)-dimensional fermionic quantum-field theoretic reformulation of the model at equilibrium. For simplicity, we restrict in this article to d=2 dimensions and zero temperature, and disregard effects due to electromagnetic interactions. Under these assumptions, we prove transition from a Fermi liquid state to a superconducting state made up of Cooper pairs of electrons at an energy level Γϕ∼ ℏωD e-π/mλ equal to the mass gap, expressed in terms of the Debye frequency ωD, electron mass m and coupling constant λ. The dynamical U(1)-symmetry breaking produces at energies lower than the energy gap Γϕ a Goldstone boson, a non-massive particle described by an effective (2+1)-dimensional non-linear sigma-model, whose parameters and correlations are computed. The proof relies on a mixture of general concepts and tools (multi-scale cluster expansions, Ward identities), adapted to this quantum many-body problem with its extended infra-red singularity located on the Fermi circle, and a specific 1/N-expansion giving the leading diagrams at intermediate energies. Ladder diagrams are proved to provide the leading behavior in the infra-red limit, in agreement with mean-field theory predictions.