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arXiv:0801.4159 · Physical Review B

Critical temperature and energy gap for the BCS equation

Christian Hainzl, Robert Seiringer

Published May 28, 2008• 61 citations

Abstract

We derive upper and lower bounds on the critical temperature ${T}_{c}$ and the energy gap $\ensuremath{\Xi}$ (at zero temperature) for the BCS gap equation, describing spin-$\frac{1}{2}$ fermions interacting via a local two-body interaction potential $\ensuremath{\lambda}V(x)$. At weak coupling $\ensuremath{\lambda}⪡1$ and under appropriate assumptions on $V(x)$, 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 $A$, $B$, and $C$ depending on the interaction $V(x)$ and the chemical potential $\ensuremath{\mu}$. The ratio $A∕C$ turns out to be a universal constant, independent of both $V(x)$ 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 $V(x)$.

DOI: 10.1103/physrevb.77.184517

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We derive upper and lower bounds on the critical temperature

{T}_{c}

and the energy gap

\ensuremath{\Xi}

(at zero temperature) for the BCS gap equation, describing spin-

\frac{1}{2}

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)

, 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

A

,

B

, and

C

depending on the interaction

V(x)

and the chemical potential

\ensuremath{\mu}

. The ratio

A∕C

turns out to be a universal constant, independent of both

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)

.

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.

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.

Following an approach similar to that of Miyake or Randeria, Duan, and Shieh in two dimensions, we study a three-dimensional many-fermion gas at zero temperature interacting via some short-ranged two-body potential. To accommodate a possible singularity (e.g., the Coulomb repulsion) in the interaction, the potential is eliminated in favor of the two-body scattering t-matrix, the low-energy form of which is expressible in terms of the s-wave scattering length

{\mathit{a}}_{\mathit{s}}

. The BCS gap equation for s-wave pairing is then solved simultaneously with the number equation in order to self-consistently obtain the zero-temperature BCS gap \ensuremath{\Delta} as well as the chemical potential \ensuremath{\mu} as functions of the dimensionless coupling variable \ensuremath{\lambda}\ensuremath{\equiv}

{\mathit{k}}_{\mathit{Fa}\mathit{s}}

, where

{\mathit{k}}_{\mathit{F}}

is the Fermi momentum. Results are valid for arbitrary coupling strength, and in the weak coupling limit reproduce the standard BCS results. Finally, root-mean-square pair sizes are obtained as a function of \ensuremath{\lambda} and compared with experimental values.

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.

We study the BCS gap equation for a Fermi gas with unequal population of spin-up and spin-down states. For cosh (δμ/T) ⩽ 2, with T the temperature and δμ the chemical potential difference, the question of existence of non-trivial solutions can be reduced to spectral properties of a linear operator, similar to the unpolarized case studied previously in [Frank, R. L., Hainzl, C., Naboko, S., and Seiringer, R., J., Geom. Anal. 17, 559–567 (2007)10.1007/BF02937429; Hainzl, C., Hamza, E., Seiringer, R., and Solovej, J. P., Commun., Math. Phys. 281, 349–367 (2008)10.1007/s00220-008-0489-2; and Hainzl, C. and Seiringer, R., Phys. Rev. B 77, 184517–110 435 (2008)]10.1103/PhysRevB.77.184517. For cosh (δμ/T) > 2 the phase diagram is more complicated, however. We derive upper and lower bounds for the critical temperature, and study their behavior in the small coupling limit.