Abstract

We investigate the BCS critical temperature T_c in the high-density limit and derive an asymptotic formula, which strongly depends on the behavior of the interaction potential V on the Fermi-surface. Our results include a rigorous confirmation for the behavior of T_c at high densities proposed by Langmann et al. (Phys Rev Lett 122:157001, 2019) and identify precise conditions under which superconducting domes arise in BCS theory.

Highlights

  • The Bardeen–Cooper–Schrieffer (BCS) gap equation [2] (p) = − 1V ( p − q) (q) tanh E,μ(q) dq, (1) (2π )3/2 R3E,μ(q) with dispersion relation E,μ( p) = ( p2 − μ)2 + | ( p)|2, has played an important role in physics since its introduction

  • Whenever the temperature T is below a certain critical temperature Tc, the gap equation (1) admits non

  • One might think of the operator KT,μ( p) + V (x) as the Hessian in the BCS functional of superconductivity at a normal state, where the positivity corresponds to the “stability" of this normal state, which is directly related to the existence of a non-trivial solution of the BCS gap equation (1)

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Summary

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At some intermediate density, where ξ ∼ μ−1/2, a superconducting dome arises This simple argument is reflected in our results by the presence of the operator Vμ, defined in Eq (2), acting on functions on the (rescaled) Fermi surface. 2 are threefold: first, we confirm a proposed asymptotic formula from [16] for the critical temperature at high densities for s-wave superconductivity (to leading order) by proving a more general result for radially symmetric interaction potentials V (Theorem 2); second, we provide a counterexample, showing that the assumptions on V from [16] are not quite sufficient to conclude a non-monotonic behavior of Tc and need to be slightly strengthened (Proposition 4); third, we use these strengthened assumptions to improve the asymptotics obtained in Theorem 2 to second order with the aid of perturbation theory, and obtain an analogous formula to the ones proven in the weak-coupling and low-density limit (Theorem 7).

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Results
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Proofs
Proof of Theorem 2
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Proof of Proposition 4
Proof of Theorem 7
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Properties of Spherical Bessel Functions

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