The temperature dependence of an isolated quantum vortex, embedded in an otherwise homogeneous fermionic superfluid of infinite extent, is determined via the Bogoliubov--de Gennes (BdG) equations across the BCS-BEC crossover. Emphasis is given to the BCS side of this crossover, where it is physically relevant to extend this study up to the critical temperature for the loss of the superfluid phase, such that the size of the vortex increases without bound. To this end, two techniques are introduced. The first one solves the BdG equations with ``free boundary conditions,'' which allows one to determine with high accuracy how the vortex profile matches its asymptotic value at a large distance from the center, thus avoiding a common practice of constraining the vortex in a cylinder with infinite walls. The second one improves on the regularization procedure of the self-consistent gap equation when the interparticle interaction is of the contact type, and permits us to considerably reduce the time needed for its numerical integration by drawing elements from the derivation of the Gross-Pitaevskii equation for composite bosons starting from the BdG equations.
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.
Making use of the Bogoliubov-de Gennes equation, we study the quasi-particle spectrum and the vortex core structure of a single vortex in quasi 2D s-wave superconductors for small
pFξ0, where
pF is the Fermi momentum and
ξ0=vF/Δ0 is the coherence length(
ℏ=1). In particular we find that the number of bound states decreases rapidly for decreasing
pFξ0. Also for
pFξ0∼1, the Kramer-Pesch effect stops around
T/Tc≃0.3.
We propose a fast and efficient approach for solving the Bogoliubov-de Gennes (BdG) equations in superconductivity, with a numerical matrix-size reduction procedure proposed by Sakurai and Sugiura [J. Comput. Appl. Math. 159, 119 (2003)]. The resultant small-size Hamiltonian contains the information of the original BdG Hamiltonian in a given energy domain. In other words, the present approach leads to a numerical construction of a low-energy effective theory in superconductivity. The combination with the polynomial expansion method allows a self-consistent calculation of the BdG equations. Through numerical calculations of quasi-particle excitations in a vortex lattice, thermal conductivity, and nuclear magnetic relaxation rate, we show that our approach is suitable for evaluating physical quantities in a large-size superconductor and a nano-scale superconducting device, with the mean-field superconducting theory.
Vortex mass in Fermi superfluids and superconductors and its influence on quantum tunneling of vortices are discussed. The vortex mass is essentially enhanced due to the fermion zero modes in the core of the vortex: the bound states of the Bogoliubov quasiparticles localized in the core. These bound states form the normal component, which is nonzero even in the low-temperature limit. In the collisionless regime ω 0 τ≫1 the normal component trapped by the vortex is unbound from the normal component in a bulk superfluid/superconductor and adds to the inertial mass of the moving vortex. In a d-wave superconductor the vortex mass has an additional factor of (B c2/B)1/2 due to the gap nodes.