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Next-to-next-to-leading-order ε expansion for a Fermi gas at infinite scattering length

J.E. Drut, P. Arnold, Dam Thanh Son

Published Nov 9, 2022• 0 citations
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Abstract

We extend previous work on applying the expansion to universal properties of a cold, dilute Fermi gas in the unitary regime of infinite scattering length. We compute the ratio ξ=μ/F of chemical potential to ideal gas Fermi energy to next-to-next-to-leading-order (NNLO) in =4-d, where d is the number of spatial dimensions. We also explore the nature of corrections from the order after NNLO.

DOI
10.17615/xyts-xr50

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We extend previous work on applying the expansion to universal properties of a cold, dilute Fermi gas in the unitary regime of infinite scattering length. We compute the ratio ξ=μ/F of chemical potential to ideal gas Fermi energy to next-to-next-to-leading-order (NNLO) in =4-d, where d is the number of spatial dimensions. We also explore the nature of corrections from the order after NNLO.
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 update the ground-state energy ratio of unitary Fermi gas to noninteracting Fermi gas $(\ensuremath{\xi})$ from the ϵϵϵ expansion by including the next-to-next-to-leading-order (NNLO) term near two spatial dimensions. Interpolations of the NNLO ϵϵϵ expansions around four and two spatial dimensions with the use of Pad\'e approximants give $\ensuremath{\xi}\ensuremath{\approx}0.360\ifmmode\pm\else\textpm\fi{}0.020$ in three dimensions with the uncertainty due to different interpolation functions. This value is consistent with the previous interpolations of the NLO ϵϵϵ expansions $\ensuremath{\xi}\ensuremath{\approx}0.377\ifmmode\pm\else\textpm\fi{}0.014$ in spite of the large NNLO corrections.
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 ddd 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=3d=3d=3 gives results consistent with 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-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.
We obtain a rigorous solution of universal Bose gases near resonance and offer an answer to one of the long-standing challenges of quantum gases at large scattering lengths, where the standard dilute theory breaks down. The solution was obtained by using an $\ensuremath{\epsilon}$ expansion near four spatial dimensions. In dimension $d=4\ensuremath{-}\ensuremath{\epsilon}$, the chemical potential of Bose gases near resonance is shown to approach the universal value ${\ensuremath{\epsilon}}^{\frac{2}{4\ensuremath{-}\ensuremath{\epsilon}}}{\ensuremath{\epsilon}}_{F}\sqrt{\frac{2}{3}}(1+0.474\ensuremath{\epsilon}\ensuremath{-}i1.217\ensuremath{\epsilon}+\ensuremath{\cdots})$, where ${\ensuremath{\epsilon}}_{F}$ is the Fermi energy defined for a Fermi gas of density nnn, and the condensation fraction is equal to $\frac{2}{3}(1+0.0877\ensuremath{\epsilon}+\ensuremath{\cdots})$. We also discuss the implications on ultracold gases in physical dimensions.