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arXiv:0808.3826 · Physical Review A

Ground-state energy of the unitary Fermi gas from the ϵ expansion

Yusuke Nishida

Published Jan 27, 2009• 21 citations
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Abstract

We update the ground-state energy ratio of unitary Fermi gas to noninteracting Fermi gas $(\ensuremath{\xi})$ from the $ϵ$ expansion by including the next-to-next-to-leading-order (NNLO) term near two spatial dimensions. Interpolations of the NNLO $ϵ$ 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 $ϵ$ expansions $\ensuremath{\xi}\ensuremath{\approx}0.377\ifmmode\pm\else\textpm\fi{}0.014$ in spite of the large NNLO corrections.

DOI
10.1103/physreva.79.013627

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We update the ground-state energy ratio of unitary Fermi gas to noninteracting Fermi gas (\ensuremathξ)(\ensuremath{\xi})(\ensuremathξ) 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 \ensuremathξ\ensuremath≈0.360\ifmmode±\else\textpm\fi0.020\ensuremath{\xi}\ensuremath{\approx}0.360\ifmmode\pm\else\textpm\fi{}0.020\ensuremathξ\ensuremath≈0.360\ifmmode±\else\textpm\fi0.020 in three dimensions with the uncertainty due to different interpolation functions. This value is consistent with the previous interpolations of the NLO ϵϵϵ expansions \ensuremathξ\ensuremath≈0.377\ifmmode±\else\textpm\fi0.014\ensuremath{\xi}\ensuremath{\approx}0.377\ifmmode\pm\else\textpm\fi{}0.014\ensuremathξ\ensuremath≈0.377\ifmmode±\else\textpm\fi0.014 in spite of the large NNLO corrections.
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.
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.
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 $\overline{ϵ}=d\ensuremath{-}2$.
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.