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arXiv:cond-mat/0608321 · Physical Review A

Unitary Fermi gas at finite temperature in the ϵ expansion

Yusuke Nishida

Published Jun 18, 2007• 42 citations
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Abstract

Thermodynamics of the unitary Fermi gas at finite temperature is investigated from the perspective of the expansion over $ϵ=4\ensuremath{-}d$ with $d$ being the dimensionality of space. We show that the thermodynamics is dominated by bosonic excitations in the low temperature region $T\ensuremath{\ll}{T}_{c}$. Analytic formulas for the thermodynamic functions as functions of the temperature are derived to the lowest order in $ϵ$ in this region. In the high temperature region where $T\ensuremath{\sim}{T}_{c}$, bosonic and fermionic quasiparticles are excited. We determine the critical temperature ${T}_{c}$ of the superfluid phase transition and the thermodynamic functions around ${T}_{c}$ to the leading and next-to-leading orders in $ϵ$.

DOI
10.1103/physreva.75.063618

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Thermodynamics of the unitary Fermi gas at finite temperature is investigated from the perspective of the expansion over ϵ=4\ensuremath−dϵ=4\ensuremath{-}dϵ=4\ensuremath−d with ddd being the dimensionality of space. We show that the thermodynamics is dominated by bosonic excitations in the low temperature region T\ensuremath≪TcT\ensuremath{\ll}{T}_{c}T\ensuremath≪Tc​. Analytic formulas for the thermodynamic functions as functions of the temperature are derived to the lowest order in ϵϵϵ in this region. In the high temperature region where T\ensuremath∼TcT\ensuremath{\sim}{T}_{c}T\ensuremath∼Tc​, bosonic and fermionic quasiparticles are excited. We determine the critical temperature Tc{T}_{c}Tc​ of the superfluid phase transition and the thermodynamic functions around Tc{T}_{c}Tc​ to the leading and next-to-leading orders in ϵϵϵ.
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 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 investigate the low-temperature thermodynamics of the unitary Fermi gas by introducing a model based on the zero-temperature spectra of both bosonic collective modes and fermonic single-particle excitations. We calculate the Helmholtz free energy and from it we obtain the entropy, the internal energy, and the chemical potential as a function of the temperature. By using these quantities and the Landau's expression for the superfluid density we determine analytically the superfluid fraction, the critical temperature, the first sound velocity, and the second sound velocity. We compare our analytical results with other theoretical predictions and experimental data of ultracold atoms and dilute neutron matter.
The finite-temperature thermodynamics of a unitary Fermi gas is studied in detail. The chemical potential, energy density, and entropy are given analytically with the quasilinear approximation. The ground-state energy agrees with previous theoretical and experimental results. Recently, the generalized exclusion statistics is applied to the discussion of the finite-temperature unitary Fermi-gas thermodynamics. A concrete comparison between the two different approaches is performed. Emphasis is made on the behavior of the entropy per particle. In physics, the slope of entropy gives the information on the effective fermion mass ${m}^{\ensuremath{\ast}}/m$ in the low-temperature strongly degenerate region. Compared with ${m}^{\ensuremath{\ast}}/m\ensuremath{\approx}0.70<1$ given in terms of the generalized exclusion statistics, our quasilinear approximation determines that ${m}^{\ensuremath{\ast}}/m\ensuremath{\approx}1.11>1$.
We investigate the superfluid transition of the unitary Fermi gas by means of the functional renormalization group, aiming at quantitative precision. We extract ${T}_{\mathrm{c}}/\ensuremath{\mu}=0.38(2)$ and $\ensuremath{\Delta}/\ensuremath{\mu}=1.04(15)$ for the critical temperature and the superfluid gap at zero temperature, respectively, within a systematic improvement of the truncation for the effective average action. The key ingredient in comparison to previous approaches consists in the use of regulators which cut off both frequencies and momenta. We incorporate renormalization effects on both the bosonic and the fermionic propagators, include higher-order bosonic scattering processes, and investigate the regulator and specification parameter dependence for an error estimate. The ratio $\ensuremath{\Delta}/{T}_{\mathrm{c}}=2.7(3)$ becomes less sensitive to the relative cutoff scale of bosons and fermions when improving the truncation. The techniques developed in this work are easily carried over to the cases of finite scattering length, lower dimensionality, and spin imbalance.
We investigate the superfluid phase transition in an SU(N)\mathrm{SU}(N)SU(N)-symmetric Fermi gas with NNN distinct spin states using the functional renormalization group. To capture pairing phenomena beyond mean-field theory, we introduce an auxiliary bosonic field and employ the leading order of the derivative expansion of the partially bosonized effective average action. By discretizing the effective potential on a grid and numerically integrating the flow equations, we resolve the thermodynamic behavior near the transition. Our results reveal a fluctuation-induced first-order phase transition for N≥4N \geq 4N≥4, which is absent at the mean-field level. In the unitary regime, we provide quantitative predictions for the critical temperature, as well as for the discontinuities in the superfluid gap and entropy density as functions of NNN. With increasing NNN, the critical temperature decreases, while the discontinuities become more pronounced, indicating a stronger first-order transition.