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arXiv:1403.2563 · Mathematical Results in Quantum Mechanics (QMath)

On the BCS gap equation for superfluid fermionic gases

Gerhard Bräunlich, Christian Hainzl, Robert Seiringer

Published Nov 5, 2014• 1 citation
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Abstract

We present a rigorous derivation of the BCS gap equation for superfluid fermionic gases with point interactions. Our starting point is the BCS energy functional, whose minimizer we investigate in the limit when the range of the interaction potential goes to zero.
DOI
10.1142/9789814618144_0007

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The Bardeen-Cooper-Schrieffer (BCS) functional has recently received renewed attention as a description of fermionic gases interacting with local pairwise interactions. We present here a rigorous analysis of the BCS functional for general pair interaction potentials. For both zero and positive temperature, we show that the existence of a non-trivial solution of the nonlinear BCS gap equation is equivalent to the existence of a negative eigenvalue of a certain linear operator. From this we conclude the existence of a critical temperature below which the BCS pairing wave function does not vanish identically. For attractive potentials, we prove that the critical temperature is non-zero and exponentially small in the strength of the potential.
We calculate Bardeen-Cooper-Schrieffer (BCS) state of a unitary Fermi gas of atoms interacting with the finite-ranged Jost-Kohn potential which has been recently shown to account for the resonant interactions [2019 {\rm J. Phys. B: At. Mol. Opt. Phys.} {\bf 52}, 165004]. Using exact scattering solution of the potential, we derive two-body T{\mathbf T}T-matrix element which is employed to construct the BCS Hamiltonian in momentum space. We present results on the energy- and range-dependence of the pairing gap and superfluid density and the range-dependence of the chemical potential for a wide variation of the scattering length including the unitarity regime. In the zero range limit our calculated gap at the Fermi energy is found to be nearly equal to that calculated with contact potential. The mean gap averaged over the full width at half maximum of the gap function in the zero range and unitarity limits is found to be 0.42EF0.42 E_F0.42EF​ which is quite close to the recent result of the quantum Monte Carlo simulation [2018 {\rm Phys. Rev.A} {\bf 97}, 013601]. The chemical potential in the zero range limit also agrees well with that for the contact potential.
Abstract We calculate Bardeen–Cooper–Schrieffer (BCS) state of a unitary Fermi gas of atoms interacting with the finite-ranged Jost-Kohn potential which has been recently shown to account for the resonant interactions (2019 J. Phys. B: At. Mol. Opt. Phys. 52 165004). Using exact scattering solution of the potential, we derive two-body T-matrix element which is employed to construct the BCS Hamiltonian in momentum space. We present results on the energy- and range-dependence of the pairing gap and superfluid density and the range-dependence of the chemical potential for a wide variation of the scattering length including the unitary regime. In the zero range limit our calculated gap at the Fermi energy is found to be nearly equal to that calculated in mean-field theory with contact potential. The mean gap averaged over the full width at half maximum of the gap function in the zero range and unitary limits is found to be 0.42E F which is quite close to the recent result of the quantum Monte-Carlo simulation (2018 Phys. Rev. A 97 013601). The chemical potential in the zero range limit also agrees well with that for the contact potential.