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arXiv:2009.03701 · Letters in Mathematical Physics

The BCS energy gap at low density

Asbjørn Bækgaard Lauritsen

Published Feb 1, 2021• 4 citations
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Abstract

Abstract We show that the energy gap for the BCS gap equation is $$\begin{aligned} \varXi = \mu \left( 8 {\mathrm{e}}^{-2} + o(1)\right) \exp \left( \frac{\pi }{2\sqrt{\mu } a}\right) \end{aligned}$$ Ξ = μ 8 e - 2 + o ( 1 ) exp π 2 μ a in the low density limit $$\mu \rightarrow 0$$ μ → 0 . Together with the similar result for the critical temperature by Hainzl and Seiringer (Lett Math Phys 84: 99–107, 2008), this shows that, in the low density limit, the ratio of the energy gap and critical temperature is a universal constant independent of the interaction potential V. The results hold for a class of potentials with negative scattering length a and no bound states.

DOI
10.1007/s11005-021-01358-5

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It is a remarkable property of BCS theory that the ratio of the energy gap at zero temperature [Formula: see text] and the critical temperature [Formula: see text] is (approximately) given by a universal constant, independent of the microscopic details of the fermionic interaction. This universality has rigorously been proven quite recently in three spatial dimensions and three different limiting regimes: weak coupling, low density and high density. The goal of this short note is to extend the universal behavior to lower dimensions [Formula: see text] and give an exemplary proof in the weak coupling limit.
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