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arXiv:cond-mat/0604500 · Physical Review Letters

ϵ Expansion for a Fermi Gas at Infinite Scattering Length

Yusuke Nishida, Dam Thanh Son

Published Aug 4, 2006• 182 citations
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Abstract

We show that there exists a systematic expansion around four spatial dimensions for Fermi gas in the unitarity regime. We perform the calculations to leading and next-to-leading orders in the expansion over $ϵ=4\ensuremath{-}d$, where $d$ is the dimensionality of space. We find the ratio of chemical potential and Fermi energy to be $\ensuremath{\mu}/{\ensuremath{\epsilon}}_{F}=\frac{1}{2}{ϵ}^{3/2}+\frac{1}{16}{ϵ}^{5/2}\mathrm{ln}ϵ\ensuremath{-}0.0246{ϵ}^{5/2}+\ensuremath{\cdots}$ and the ratio of the gap in the fermion quasiparticle spectrum and the chemical potential to be $\ensuremath{\Delta}/\ensuremath{\mu}=2{ϵ}^{\ensuremath{-}1}\ensuremath{-}0.691+\ensuremath{\cdots}$. The minimum of the fermion dispersion curve is located at $|\mathbit{p}|=(2m{\ensuremath{\epsilon}}_{0}{)}^{1/2}$, where ${\ensuremath{\epsilon}}_{0}/\ensuremath{\mu}=2+O(ϵ)$. Extrapolation to $d=3$ gives results consistent with Monte Carlo simulations.

DOI
10.1103/physrevlett.97.050403

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We show that there exists a systematic expansion around four spatial dimensions for Fermi gas in the unitarity regime. We perform the calculations to leading and next-to-leading orders in the expansion over ϵ=4\ensuremath−dϵ=4\ensuremath{-}dϵ=4\ensuremath−d, where ddd is the dimensionality of space. We find the ratio of chemical potential and Fermi energy to be \ensuremathμ/\ensuremathϵF=12ϵ3/2+116ϵ5/2lnϵ\ensuremath−0.0246ϵ5/2+\ensuremath⋯\ensuremath{\mu}/{\ensuremath{\epsilon}}_{F}=\frac{1}{2}{ϵ}^{3/2}+\frac{1}{16}{ϵ}^{5/2}\mathrm{ln}ϵ\ensuremath{-}0.0246{ϵ}^{5/2}+\ensuremath{\cdots}\ensuremathμ/\ensuremathϵF​=21​ϵ3/2+161​ϵ5/2lnϵ\ensuremath−0.0246ϵ5/2+\ensuremath⋯ and the ratio of the gap in the fermion quasiparticle spectrum and the chemical potential to be \ensuremathΔ/\ensuremathμ=2ϵ\ensuremath−1\ensuremath−0.691+\ensuremath⋯\ensuremath{\Delta}/\ensuremath{\mu}=2{ϵ}^{\ensuremath{-}1}\ensuremath{-}0.691+\ensuremath{\cdots}\ensuremathΔ/\ensuremathμ=2ϵ\ensuremath−1\ensuremath−0.691+\ensuremath⋯. The minimum of the fermion dispersion curve is located at ∣\mathbitp∣=(2m\ensuremathϵ0)1/2|\mathbit{p}|=(2m{\ensuremath{\epsilon}}_{0}{)}^{1/2}∣\mathbitp∣=(2m\ensuremathϵ0​)1/2, where \ensuremathϵ0/\ensuremathμ=2+O(ϵ){\ensuremath{\epsilon}}_{0}/\ensuremath{\mu}=2+O(ϵ)\ensuremathϵ0​/\ensuremathμ=2+O(ϵ). Extrapolation to d=3d=3d=3 gives results consistent with Monte Carlo simulations.
We extend previous work on applying the ϵϵϵ expansion to universal properties of a cold, dilute Fermi gas in the unitary regime of infinite scattering length. We compute the ratio $\ensuremath{\xi}=\ensuremath{\mu}∕{ϵ}_{\mathrm{F}}$ of chemical potential to ideal gas Fermi energy to next-to-next-to-leading-order (NNLO) in $ϵ=4\ensuremath{-}d$, where ddd is the number of spatial dimensions. We also explore the nature of corrections from the order after NNLO.
We construct systematic expansions around four and two spatial dimensions for a Fermi gas near the unitarity limit. Near four spatial dimensions such a Fermi gas can be understood as a weakly-interacting system of fermionic and bosonic degrees of freedom. To the leading and next-to-leading orders in the expansion over ε=4-d, with d being the dimensionality of space, we determine the thermodynamic functions, the fermion quasiparticle spectrum, the critical polarizations, and the critical temperature as functions of the binding energy of the two-body state. We also show that the unitary Fermi gas near two spatial dimensions reduces to a weakly-interacting Fermi gas and calculate the thermodynamic functions and the fermion quasiparticle spectrum in the expansion over \barε=d-2. We discuss the matching of the two systematic expansions around four and two spatial dimensions in order to extract physical observables at d=3. We find good agreement of the results with those from recent Monte Carlo simulations.
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.
We construct systematic expansions around four and two spatial dimensions for a Fermi gas near the unitarity limit. Near four spatial dimensions such a Fermi gas can be understood as a weakly interacting system of fermionic and bosonic degrees of freedom. To the leading and next-to-leading orders in the expansion over $ϵ=4\ensuremath{-}d$, with ddd being the dimensionality of space, we calculate the thermodynamic quantities and the fermion quasiparticle spectrum, both at unitarity and around the unitarity point. Then the phase structure of the polarized Fermi gas in the unitary regime is studied in the ϵϵϵ expansion. At unitarity the unpolarized superfluid state and a fully polarized normal state are separated by a first order phase transition. However, on the BEC side of the unitarity point, in a certain range of the two-body binding energy, these two phases are separated in the phase diagram by more exotic phases, including gapless superfluid phases with one or two Fermi surfaces and a superfluid phase with a spatially varying condensate. We also show that the unitary Fermi gas near two spatial dimensions is weakly interacting, and calculate the thermodynamic quantities and the fermion quasiparticle spectrum in the expansion over $\overline{ϵ}=d\ensuremath{-}2$.
We consider a polarized Fermi gas in the unitarity limit. Results are calculated analytically up to next-to-leading order in an expansion about d=4d=4d=4 spatial dimensions. We find a first order transition from superfluid to normal phase. The critical chemical potential asymmetry for this phase transition is $\ensuremath{\delta}{\ensuremath{\mu}}_{c}=\frac{2\ensuremath{\mu}}{\ensuremath{\epsilon}}(1\ensuremath{-}0.467\ensuremath{\epsilon})$, where $\ensuremath{\epsilon}=4\ensuremath{-}d$ is the expansion parameter and $\ensuremath{\mu}$ is the average chemical potential of the two fermion species. Stability of the superfluid phase in the presence of supercurrents is also studied.