arXiv:1107.0405 · Journal of Mathematical Physics

The gap equation for spin-polarized fermions

Abraham Freiji, Christian Hainzl, Robert Seiringer

Published Jan 1, 2012• 13 citations

Abstract

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.

DOI: 10.1063/1.3670747

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arXiv:1107.0405 · Journal of Mathematical Physics

The gap equation for spin-polarized fermions

Abraham Freiji, Christian Hainzl, Robert Seiringer

Published Jan 1, 2012• 13 citations

arXiv HTML PDF

Abstract

DOI: 10.1063/1.3670747

Related papers

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Takashi Kashimura, Ryota Watanabe, Yoji Ohashi

We investigate strong-coupling corrections to single-particle excitations in the normal state of a spin-polarized unitary Fermi gas. Within the framework of an extended $T$ -matrix approximation, we calculate the single-particle density of states, as well as the single-particle spectral weight, to show that the so-called pseudogap phenomenon gradually disappears with increasing magnitude of an effective magnetic field. In the highly spin-polarized regime, the calculated spin polarization as a function of the effective magnetic field agrees well with the recent experiment on a ${}^{6}$ Li Fermi gas. Although this experiment has been considered to be incompatible with the existence of the pseudogap in an unpolarized Fermi gas, our result clarifies that the observed spin polarization in the highly spin-polarized regime and the pseudogap in the unpolarized limit can be explained in a consistent manner when one correctly includes effects of population imbalance on single-particle excitations. Since it is a crucial issue to clarify whether the pseudogap exists or not in the BCS-BEC crossover regime of an ultracold Fermi gas, our results would be useful for the understanding of this strongly interacting fermion system.We investigate strong-coupling corrections to single-particle excitations in the normal state of a spin-polarized unitary Fermi gas. Within the framework of an extended $T$-matrix approximation, we calculate the single-particle density of states, as well as the single-particle spectral weight, to show that the so-called pseudogap phenomenon gradually disappears with increasing magnitude of an effective magnetic field. In the highly spin-polarized regime, the calculated spin polarization as a function of the effective magnetic field agrees well with the recent experiment on a ${}^{6}$Li Fermi gas. Although this experiment has been considered to be incompatible with the existence of the pseudogap in an unpolarized Fermi gas, our result clarifies that the observed spin polarization in the highly spin-polarized regime and the pseudogap in the unpolarized limit can be explained in a consistent manner when one correctly includes effects of population imbalance on single-particle excitations. Since it is a crucial issue to clarify whether the pseudogap exists or not in the BCS-BEC crossover regime of an ultracold Fermi gas, our results would be useful for the understanding of this strongly interacting fermion system.

arXiv:1310.5490Physical Review AJan 22, 201415 citations

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 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.

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.

{\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.

We investigate strong-coupling corrections to single-particle excitations in the normal state of a spin-polarized unitary Fermi gas. Within the framework of an extended

T

-matrix approximation, we calculate the single-particle density of states, as well as the single-particle spectral weight, to show that the so-called pseudogap phenomenon gradually disappears with increasing magnitude of an effective magnetic field. In the highly spin-polarized regime, the calculated spin polarization as a function of the effective magnetic field agrees well with the recent experiment on a

{}^{6}

Li Fermi gas. Although this experiment has been considered to be incompatible with the existence of the pseudogap in an unpolarized Fermi gas, our result clarifies that the observed spin polarization in the highly spin-polarized regime and the pseudogap in the unpolarized limit can be explained in a consistent manner when one correctly includes effects of population imbalance on single-particle excitations. Since it is a crucial issue to clarify whether the pseudogap exists or not in the BCS-BEC crossover regime of an ultracold Fermi gas, our results would be useful for the understanding of this strongly interacting fermion system.