The BCS Energy Gap at High Density

Abstract

We study the BCS energy gap $$\Xi $$ Ξ in the high–density limit and derive an asymptotic formula, which strongly depends on the strength of the interaction potential V on the Fermi surface. In combination with the recent result by one of us (Math. Phys. Anal. Geom. 25, 3, 2022) on the critical temperature $$T_c$$ T c at high densities, we prove the universality of the ratio of the energy gap and the critical temperature.

Universality in low-dimensional BCS theory

“…) and as such measuring the strength of the interaction V on the (rescaled) Fermi surface (see [HS08b;Hen22;HL22]). To assure that V (d) µ and W (d) µ will be well-defined and compact, we assume the following.…”

“…where γ ≈ 0.577 is the Euler-Mascheroni constant, in each of the three physically very different limits mentioned above. This result follows as a limiting equality by combining asymptotic formulas for the critical temperature T c (see [FHNS07;HS08a;HS08b;Hen22]) and the energy gap Ξ (see [HS08b;HL22;Lau21]) in the three different regimes. Although these scenarios (weak coupling, low density, and high density) are physically rather different, they all have in common that 'superconductivity is weak' and one can hence derive an asymptotic formula for T c and Ξ as they depart from being zero (in the extreme cases λ = 0, µ = 0, µ = ∞, respectively).…”

“…has the interpretation of an energy gap associated with the approximate BCS Hamiltonian and as such represents a second key quantity of the theory. Analogously to the critical temperature, the asymptotic behavior of this energy gap, again in three spatial dimensions, has been studied in the same three different limiting regimes: In a weak coupling limit [HS08b], in a low density limit [Lau21], and in a high density limit [HL22].…”

“…Essentially, only the technical details in Lemma 3.6 and the calculation of m • High density: For the high-density limit in [HL22], we needed some additional a priori bounds on ∆ before we could employ the Birman-Schwinger argument. Apart from that, in [HL22] the comparison of ∆( √ µ) and Ξ are done right after these a priori bounds. Additionally, since one starts with finding a priori bounds on ∆, one does not need the first-order analysis in Step 3.…”

“…Additionally, since one starts with finding a priori bounds on ∆, one does not need the first-order analysis in Step 3. One may think of the structure in [HL22] as being ordered in the above steps as follows: 4, 7, 1, 2, 4 (again), 5, 6.…”