The BCS Critical Temperature at High Density

Abstract

We investigate the BCS critical temperature $$T_c$$ 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 $$T_c$$ 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.

The BCS Energy Gap at High Density

“…To show the universality we prove in Theorem 3 an asymptotic formula for the energy gap Ξ in the high-density limit, similar to the corresponding formula for the critical temperature given in [12]. This formula depends strongly on the strength of the interaction potential V on the Fermi sphere {p 2 = µ} and we refer to [12] for a discussion. Together with the formula for the critical temperature [12] we prove the universality (3) in Corollary 4.…”

“…This formula depends strongly on the strength of the interaction potential V on the Fermi sphere {p 2 = µ} and we refer to [12] for a discussion. Together with the formula for the critical temperature [12] we prove the universality (3) in Corollary 4. All proofs are given in Section 2.…”

“…The evaluation of V pointwise (and in particular on a codim−1 submanifold) is well defined since V ∈ L 1 (R 3 ). The condition that V ∈ L 1 (R 3 ) could potentially be relaxed, see [4] and Remark 9 in [12]. The lowest eigenvalue of V µ , which we denote by e µ = inf spec V µ will be of particular importance.…”

“…In order to obtain an asymptotic formula for the energy gap that is valid up to second order (see [8,12]), we define the operator W (κ)…”

“…for any fixed κ ≥ 0 (cf. [12]). Here φ(p) = (2π) −3/2 S 2 V (p− √ µq)u(q)dω(q), and (|p|, ω(p)) ∈ (0, ∞) × S 2 denote spherical coordinates for p ∈ R 3 .…”