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ε展開を用いたユニタリー・フェルミ気体の研究(<シリーズ>超伝導・超流動研究の接点)

Yusuke Nishida

Published Mar 5, 2011• 0 citations
OpenAlex

Abstract

本稿では、2007年日本物理学会若手奨励賞(理論核物理)の受賞対象となった論文[1]"ε expansion for a Fermi gas at infinite scattering length,"Yusuke Nishida and Dam Thanh Son,Phys.Rev.Lett.97(2006),050403について、関連論文[2,3]の結果も交えながら解説します。

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BEC-BCS crossover in the ϵ expansion

Jiunn-Wei Chen, Eiji Nakano

The $ϵ$ expansion (expansion around four spatial dimensions) developed by Nishida and Son for a cold Fermi gas with infinite scattering length is extended to finite scattering length to study the Bose-Einstein condensate (BEC) to BCS crossover. A resummation of higher-order logarithms and a suitable extension of fermion coupling in $d$ dimensions are developed in order to apply the theory in the BCS regime. The ratio between the chemical potential and the Fermi energy, $\ensuremath{\mu}∕{\ensuremath{\epsilon}}_{F}$, is computed to next-to-leading order in the $ϵ$ expansion as a function of $\ensuremath{\eta}=1∕(a{k}_{F})$, where $a$ is the scattering length and ${k}_{F}$ is the Fermi momentum in a noninteracting system. Near the unitarity limit $\ensuremath{\mid}\ensuremath{\eta}\ensuremath{\mid}\ensuremath{\rightarrow}0$, we found $\ensuremath{\mu}∕{\ensuremath{\epsilon}}_{F}=0.475\ensuremath{-}0.707\ensuremath{\eta}\ensuremath{-}0.5{\ensuremath{\eta}}^{2}$. Near the BEC limit $\ensuremath{\eta}\ensuremath{\rightarrow}\ensuremath{\infty}$, $\ensuremath{\mu}∕{\ensuremath{\epsilon}}_{F}=0.062∕\ensuremath{\eta}\ensuremath{-}{\ensuremath{\eta}}^{2}$, while near the BCS limit $\ensuremath{\eta}\ensuremath{\rightarrow}\ensuremath{-}\ensuremath{\infty}$, $\ensuremath{\mu}∕{\ensuremath{\epsilon}}_{F}=1+0.707∕\ensuremath{\eta}$. Overall good agreement with quantum Monte Carlo results is found.

arXiv:cond-mat/0610011•Physical Review A•Apr 23, 2007•28 citations
Using $\ensuremath{\epsilon}$ expansion technique proposed by Nishida and Son [Phys. Rev. Lett. 97, 050403 (2006)] we derive an effective Lagrangian (Ginzburg-Landau-type functional) of the degenerate unitary Fermi gas to the next-to-leading order in $\ensuremath{\epsilon}$. It is demonstrated that for many realistic situations it is sufficient to retain leading order terms in the derivative expansion. The functional is used to study vortex structure in the symmetric gas, and interface between normal and superfluid phases in the polarized gas. Typical vortex size, r0kF=0.92{r}_{0}{k}_{F}=0.92r0​kF​=0.92, where kF{k}_{F}kF​ is Fermi momentum, agrees well with results of previous work [A. Bulgac and Y. Yu, Phys. Rev. Lett. 91, 190404 (2003)]. Surface free energy is about 3 times larger than the value previously quoted in the literature.
The ϵϵϵ expansion (expansion around four spatial dimensions) developed by Nishida and Son for a cold Fermi gas with infinite scattering length is extended to finite scattering length to study the Bose-Einstein condensate (BEC) to BCS crossover. A resummation of higher-order logarithms and a suitable extension of fermion coupling in ddd dimensions are developed in order to apply the theory in the BCS regime. The ratio between the chemical potential and the Fermi energy, $\ensuremath{\mu}∕{\ensuremath{\epsilon}}_{F}$, is computed to next-to-leading order in the ϵϵϵ expansion as a function of $\ensuremath{\eta}=1∕(a{k}_{F})$, where aaa is the scattering length and kF{k}_{F}kF​ is the Fermi momentum in a noninteracting system. Near the unitarity limit $\ensuremath{\mid}\ensuremath{\eta}\ensuremath{\mid}\ensuremath{\rightarrow}0$, we found $\ensuremath{\mu}∕{\ensuremath{\epsilon}}_{F}=0.475\ensuremath{-}0.707\ensuremath{\eta}\ensuremath{-}0.5{\ensuremath{\eta}}^{2}$. Near the BEC limit $\ensuremath{\eta}\ensuremath{\rightarrow}\ensuremath{\infty}$, $\ensuremath{\mu}∕{\ensuremath{\epsilon}}_{F}=0.062∕\ensuremath{\eta}\ensuremath{-}{\ensuremath{\eta}}^{2}$, while near the BCS limit $\ensuremath{\eta}\ensuremath{\rightarrow}\ensuremath{-}\ensuremath{\infty}$, $\ensuremath{\mu}∕{\ensuremath{\epsilon}}_{F}=1+0.707∕\ensuremath{\eta}$. Overall good agreement with quantum Monte Carlo results is found.