Proceeding from a microscopic model Hamiltonian for a system of Fermi particles with attraction, we have derived an effective Lagrangian, which allows analysis of the superconducting properties of the system for arbitrary fermion concentrations. Exact solution of the equations for the gap and the chemical potential enables us to examine the crossover from the regime of local pairs and superfluidity to the Cooper pairing and the BCS-like superconductivity. It is shown that the physical parameter that demarcates the regions of exotic and normal behavior is the fermion bound state energy. The expressions derived suggest that in the region of relatively low concentrations the parameters of a 2D metal correspond to a type-II superconductor.
An attempt is made to analyze the state-of-the-art in the description of the superconducting phase in 2D metals with an arbitrary carrier density. It is shown that a decrease in the density leads to a nontrivial equation for chemical potential μ, in particular, to the inequality μ ≠ εF including μ < 0 (εF is the Fermi energy). In this case, the region of μ ≃ 0 corresponds to a transition (crossover) from Cooper pairing (with strongly overlapping pairs) and from the description of superconductivity in terms of the BCS–Bogoliubov theory to local (nonoverlapping) pairs and their Bose–Einstein condensation. Some examples of such a crossover are discussed.
The superconducting properties of the 2D fermion system with local and different types of the indirect boson-exchange attractions in the cases of s-wave and d-wave pairing are reviewed and analysed at T=0. In particular, the possibility of the crossover from the Bose-Einstein condensation regime to the BCS-like superconductivity with the carrier density and coupling changing in the case of different pairing channels is discussed. Gaussian fluctuations of the order parameter are taken into account and the carrier density dependence of the gap is studied in this case. The role of the interaction form in the physical behavior of the systems is also discussed.
The superconducting properties of a two-dimensional metallic system with arbitrary carrier density and both local and various types of attractive indirect boson-exchange interaction in the cases of s- and d-wave pairing are analyzed and reviewed at T=0. In particular, the possibility of a crossover from the Bose–Einstein condensation regime to Bardeen–Cooper–Schrieffer-like superconductivity with growing carrier density and changing coupling in the case of different pairing channels is discussed. Gaussian fluctuations of the order parameter are taken into account, and the carrier density dependence of the gap magnitude is studied. The role of the form of the interparticle attractive interaction in the physical behavior of the system is also considered.
One- and two-dimensional attractive Hubbard models in the presence of magnetic field h are studied in a wide range of on-site interaction strength U, temperature T and electron concentration n by means of the generalized self-consistent field (GSCF) approach with renormalized chemical potential. Pairing, the smooth crossover from the BCS regime into the Bose condensation state and the inhomogeneous superconductivity with the non-uniform order parameter are investigated. The excitation spectrum, the momentum distribution function and thermodynamic properties, including the concentration of double occupied sites, the kinetic energy, and the chemical potential, are calculated. The numerical calculations are found in a good agreement with the Bethe-ansatz results and quantum Monte-Carlo calculations in a wide range of system parameters U, n and h. The GSCF theory correctly displays the separation of the energy gap from the order parameter (crossover) as n or |U| decrease. The concentration of double occupied sites and the bound electron pairs n pair are calculated and we found a simple relationship between the order parameter and n pair , valid for the arbitrary U and all n in any dimension. The developed GSCF approach in magnetic field suggests a smooth transition into the inhomogeneous superconducting state.
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∼ℏωDe−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.