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arXiv:1110.1400 · Physical Review A

Thermodynamics of the two-component Fermi gas with unequal masses at unitarity

K. M. Daily, D. Blume

Published Jan 5, 2012• 31 citations
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Abstract

We consider mass-imbalanced two-component Fermi gases for which the unequal-mass atoms interact via a zero-range model potential with a diverging $s$-wave scattering length ${a}_{s}$, that is, with $1/{a}_{s}=0$. The high-temperature thermodynamics of the harmonically trapped and homogeneous systems are examined using a virial expansion approach up to third order in the fugacity. We find that the universal part of the third-order virial coefficient associated with two light atoms and one heavy atom is negative, while that associated with two heavy and one light atom changes sign from negative to positive as the mass ratio $\ensuremath{\kappa}$ increases and diverges when Efimov physics sets in at $\ensuremath{\kappa}=13.61$. By examining the Helmholtz free energy, we find that the equilibrium polarization of the trapped and homogeneous systems is 0 for $\ensuremath{\kappa}=1$, but finite for $\ensuremath{\kappa}\ensuremath{\ne}1$ (with a majority of heavy particles). Compared to the equilibrium polarization of the noninteracting system, the equilibrium polarization at unitarity is increased for the trapped system and decreased for the homogeneous system. We find that unequal-mass Fermi gases are stable for all polarizations.

DOI
10.1103/physreva.85.013609

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We consider mass-imbalanced two-component Fermi gases for which the unequal-mass atoms interact via a zero-range model potential with a diverging sss-wave scattering length as{a}_{s}as​, that is, with 1/as=01/{a}_{s}=01/as​=0. The high-temperature thermodynamics of the harmonically trapped and homogeneous systems are examined using a virial expansion approach up to third order in the fugacity. We find that the universal part of the third-order virial coefficient associated with two light atoms and one heavy atom is negative, while that associated with two heavy and one light atom changes sign from negative to positive as the mass ratio \ensuremathκ\ensuremath{\kappa}\ensuremathκ increases and diverges when Efimov physics sets in at \ensuremathκ=13.61\ensuremath{\kappa}=13.61\ensuremathκ=13.61. By examining the Helmholtz free energy, we find that the equilibrium polarization of the trapped and homogeneous systems is 0 for \ensuremathκ=1\ensuremath{\kappa}=1\ensuremathκ=1, but finite for \ensuremathκ\ensuremath≠1\ensuremath{\kappa}\ensuremath{\ne}1\ensuremathκ\ensuremath=1 (with a majority of heavy particles). Compared to the equilibrium polarization of the noninteracting system, the equilibrium polarization at unitarity is increased for the trapped system and decreased for the homogeneous system. We find that unequal-mass Fermi gases are stable for all polarizations.
We treat small trapped unequal-mass two-component Fermi gases at unitarity within a nonperturbative microscopic framework and investigate the system properties as functions of the mass ratio κ, and the numbers N1 and N2 of heavy and light fermions. While equal-mass Fermi gases with infinitely large interspecies s-wave scattering length a(s) are universal, we find that unequal-mass Fermi gases are, for sufficiently large κ and in the regime where Efimov physics is absent, not universal. In particular, the (N₁,N₂) = (2, 1) and (3, 1) systems exhibit three-body and four-body resonances at κ=12.314(2) and 10.4(2), respectively, as well as surprisingly large finite-range effects. These findings have profound implications for ongoing experimental efforts and quantum simulation proposals that utilize unequal-mass atomic Fermi gases.
Equal-mass two-component Fermi gases under spherically symmetric external harmonic confinement with large sss-wave scattering length are considered. Using the stochastic variational approach, we determine the lowest 286 and 164 relative eigenenergies of the (2,2)(2,2)(2,2) and (3,1)(3,1)(3,1) systems at unitarity as a function of the range r0{r}_{0}r0​ of the underlying two-body potential and extrapolate to the ${r}_{0}\ensuremath{\rightarrow}0$ limit. Our calculations include all states with vanishing and finite angular momentum LLL (and natural and unnatural parity $\ensuremath{\Pi}$) with relative energy up to 10.5 $\ensuremath{\hbar}\ensuremath{\Omega}$, where $\ensuremath{\Omega}$ denotes the angular trapping frequency of the external confinement. Our extrapolated zero-range energies are estimated to have uncertainties of 0.1% or smaller. The (2,2)(2,2)(2,2) and (3,1)(3,1)(3,1) energies are used to determine the fourth-order virial coefficient of the trapped unitary two-component Fermi gas in the low-temperature regime. Our results are compared with recent predictions for the fourth-order virial coefficient of the homogeneous system. We also calculate small portions of the energy spectra of the (3,2)(3,2)(3,2) and (4,1)(4,1)(4,1) systems at unitarity.
We derive the expressions for the virial coefficients up to the third order for two-species fermion mixtures, and investigate the fugacity of the species and the interaction energy density around the sss-wave Feshbach resonance in 6Li^{6}\mathrm{Li}6Li-40K^{40}\mathrm{K}40K mixtures. The theoretical results show that the magnitude of the virial coefficients depends on the mass ratio in a mass-imbalanced system. We find that the contribution of two-body bound states to high-order virial coefficients is very small at temperatures of microkelvin order. With the decrease of T/TFT/{T}_{F}T/TF​, the positions of the maxima of fugacity and interaction energy density shift away from the resonance position in the third-order virial expansion calculation. The mass ratio determines the relevance between the fugacity and number density of heavy species. Our conclusion is also effective for other mass-imbalanced systems.
Thermodynamic properties of a Fermi system close to the unitarity limit, where the 2-body scattering length a approaches +/-infinity, are studied in the high temperature Boltzmann regime. For dilute systems the virial expansion coefficients in the Boltzmann regime are expected, from general arguments, to be universal. A model independent finite temperature T calculation of the third virial coefficient b3(T) is presented. At the unitarity limit, b3infinity approximately 1.11 is a universal number. The energy density up to the third virial expansion is derived. These calculations are of interest in dilute neutron matter and could be tested in current atomic experiments on dilute Fermi gases near the Feshbach resonance.
We theoretically investigate the high-temperature thermodynamics of a strongly interacting trapped Fermi gas near either sss-wave or ppp-wave Feshbach resonances, using a second-order quantum virial expansion. The second virial coefficient is calculated based on the energy spectrum of two interacting fermions in a harmonic trap. We consider both isotropic and anisotropic harmonic potentials. For the two-fermion interaction, either sss-wave or ppp-wave, we use a pseudopotential parametrized by a scattering length and an effective range. This turns out to be the simplest way of encoding the energy dependence of the low-energy scattering amplitude or phase shift. This treatment of the pseudopotential can be easily generalized to higher partial-wave interactions. We discuss how the second virial coefficient and thermodynamics are affected by the existence of these finite-range interaction effects. The virial expansion result for a strongly interacting sss-wave Fermi gas has already been proved very useful. In the case of ppp-wave interactions, our results for the high-temperature equation of state are applicable to future high-precision thermodynamic measurements for a spin-polarized Fermi gas near a ppp-wave Feshbach resonance.
We consider a two-component atomic Fermi gas within a mean-field, single-channel model, where both the mass and population of each component are unequal. We show that the tricritical point at zero temperature evolves smoothly from the BEC to BCS side of the resonance as a function of mass ratio r. We find that the interior gap state proposed by Liu and Wilczek is always unstable to phase separation, while the breached pair state with one Fermi surface for the excess fermions exhibits differences in its density of states and pair correlation functions depending on which side of the resonance it lies. Finally, we show that, when r greater, similar 3.95, the finite-temperature phase diagram of trapped gases at unitarity becomes topologically distinct from the equal mass system.