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arXiv:2106.02028 · Journal of Statistical Physics

The BCS Energy Gap at High Density

Joscha Henheik, Asbjørn Bækgaard Lauritsen

Published Jul 29, 2022• 5 citations
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Abstract

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.

DOI
10.1007/s10955-022-02965-9

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Abstract We study the BCS energy gap Ξ\Xi Ξ 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 TcT_cTc​ T c at high densities, we prove the universality of the ratio of the energy gap and the critical temperature.
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Abstract We show that the energy gap for the BCS gap equation is Ξ=μ(8e−2+o(1))exp⁡(π2μa)\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(8e−2+o(1))exp(2μ​aπ​)​ Ξ = μ 8 e - 2 + o ( 1 ) exp π 2 μ a in the low density limit μ→0\mu \rightarrow 0μ→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.
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We derive upper and lower bounds on the critical temperature Tc{T}_{c}Tc​ and the energy gap $\ensuremath{\Xi}$ (at zero temperature) for the BCS gap equation, describing spin-12\frac{1}{2}21​ fermions interacting via a local two-body interaction potential $\ensuremath{\lambda}V(x)$. At weak coupling $\ensuremath{\lambda}⪡1$ and under appropriate assumptions on V(x)V(x)V(x), our bounds show that ${T}_{c}\ensuremath{\sim}A\phantom{\rule{0.2em}{0ex}}\mathrm{exp}(\ensuremath{-}B∕\ensuremath{\lambda})$ and $\ensuremath{\Xi}\ensuremath{\sim}C\phantom{\rule{0.2em}{0ex}}\mathrm{exp}(\ensuremath{-}B∕\ensuremath{\lambda})$ for some explicit coefficients AAA, BBB, and CCC depending on the interaction V(x)V(x)V(x) and the chemical potential $\ensuremath{\mu}$. The ratio A∕CA∕CA∕C turns out to be a universal constant, independent of both V(x)V(x)V(x) and $\ensuremath{\mu}$. Our analysis is valid for any $\ensuremath{\mu}$; for small $\ensuremath{\mu}$, or low density, our formulas reduce to well-known expressions involving the scattering length of V(x)V(x)V(x).
We develop a general description of the superconductivity of lattice fermions based on the BCS theory. We propose a modeling of the density of states (DOS) of lattice fermions, where divergent and semi-metallic structures are described by asymptotic expansions around the Fermi energy. This modeling leads to a unified representation of the transition temperature TcT_cTc​ at half filling, which reproduces asymptotic forms of TcT_cTc​ derived in several lattices, such as the square, honeycomb, and Lieb lattices, for the weakly interacting limit. The derived asymptotic forms of TcT_cTc​ are categorized into four types, which is attributed to the different responses of the degenerate fermions depending on the DOS structures. The DOS with a delta-functional singularity induces the highest TcT_cTc​ in the weakly interacting region, where TcT_cTc​ is linearly proportional to the pairing interaction UUU. Three kinds of universal ratios defined in the BCS theory no longer reduce to constants independent of the system parameters but can be parameterized with a certain variable that characterizes the singular structures of the DOS. We find universal relationship among thermodynamic quantities that holds for all parameter regions. Further, we numerically demonstrate that in multi-band systems the correlation effects can induce an effective delta-functional singularity. This phenomenon generally appears in multi-energy (or multi-gap) systems and may provide a plausible guideline for material designs of high-TcT_cTc​ superconductor.