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arXiv:1310.4726 · Physical Review A

Higher virial coefficients of a unitary Fermi gas

R. K. Bhaduri, W. van Dijk, M. V. N. Murthy

Published Oct 30, 2013• 4 citations
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Abstract

Third- and higher-order quantum virial coefficients require the solution of the corresponding quantum many-body problem. Nevertheless, in an earlier paper [Phys. Rev. Lett. 108, 260402 (2012)], we proposed that the higher-order cluster integrals of a dilute unitary fermionic gas may be approximated in terms of the two-body cluster together with an appropriate suppression factor. Although not exact, this ansatz gave a fair agreement up to fugacity $z\ensuremath{\approx}6$ with the experimentally obtained equation of state. The objective of the present Brief Report is to give some physical arguments in favor of this ansatz.

DOI
10.1103/physreva.88.045602

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Third- and higher-order quantum virial coefficients require the solution of the corresponding quantum many-body problem. Nevertheless, in an earlier paper [Phys. Rev. Lett. 108, 260402 (2012)], we proposed that the higher-order cluster integrals of a dilute unitary fermionic gas may be approximated in terms of the two-body cluster together with an appropriate suppression factor. Although not exact, this ansatz gave a fair agreement up to fugacity z\ensuremath≈6z\ensuremath{\approx}6z\ensuremath≈6 with the experimentally obtained equation of state. The objective of the present Brief Report is to give some physical arguments in favor of this ansatz.
It is suggested that for a Fermi gas at unitarity, the two-body bond plays a special role. We propose an equation of state using an ansatz relating the interaction part of the l-body cluster to its two-body counterpart. This allows a parameter-free comparison with the recently measured equation of state by the ENS group. The agreement between the two over a range of fugacity (z<5 for a homogeneous gas, and z<10 for the trapped gas) leads us to perform the calculations of more sensitive quantities measured recently by the MIT group.
Virial expansion is widely used in cold atoms to analyze high temperature strongly correlated many-body systems. As the n-th order virial expansion coefficient can be accurately obtained by exactly solving up to n-body problems, the virial expansion offers a few-body approach to study strongly correlated many-body problems. In particular, the virial expansion has successfully been applied to unitary Fermi gas. We review recent progress of the virial expansion studies in the unitary Fermi gas, in particular the fourth order virial coefficient.
Abstract Using universal properties and a basic statistical mechanical approach, we propose a general equation of state for unitary Fermi gases. The universal equation of state is written as a series solution to a self consistent integral equation where the general solution is a linear combination of Fermi functions. First, by truncating our series solution to four terms with already known exact theoretical inputs at limiting cases, namely the first three virial coefficients and using the Bertsch parameter as a free parameter, we find a good agreement with experimental measurements in the entire temperature region in the normal state. This analytical equation of state agrees with experimental data up to the fugacity z = 18, which is a vast improvement over the other analytical equations of state available where the agreements is only up to z ≈ 7 . Second, by truncating our series solution to four terms again using first four virial coefficients, we find the Bertsch parameter ξ = 0.35 , which is in good agreement with the direct experimental measurement of ξ = 0.37 . This second form of equation of state shows a good agreement with self-consistent T -matrix calculations in the normal phase.
We combine a Tan's universal relation with a basic statistical mechanical approach to derive a general equation of state for unitary Fermi gases. The universal equation of state is written as a series solution to a self consistent integral equation where the general solution is a linear combination of Fermi functions. By truncating our series solution to four terms with already known exact theoretical inputs at limiting cases, namely the first three virial coefficients and the Bertsch parameter, we find a good agreement with experimental measurements in the entire temperature region in the normal state. Our analytical equation of state agrees with experimental data up to the fugacity z=18z = 18z=18, which is a vast improvement over the other analytical equations of state available where the agreements is \emph{only} up to z≈7z \approx 7z≈7.
In the current era of precision quantum many-body physics, one of the most scrutinized systems is the unitary limit of the nonrelativistic spin-1/2 Fermi gas, due to its simplicity and relevance for atomic, condensed matter, and nuclear physics. The thermodynamics of this strongly correlated system is determined by universal functions which, at high temperatures, are governed by universal virial coefficients b_{n} that capture the effects of the n-body system on the many-body dynamics. Currently, b_{2} and b_{3} are well understood, but the situation is less clear for b_{4}, and no predictions have been made for b_{5}. To answer these open questions, we implement a nonperturbative analytic approach based on the Trotter-Suzuki factorization of the imaginary-time evolution operator, using progressively finer temporal lattice spacings. By means of these factorizations and automated algebra codes, we obtain the interaction-induced change Δb_{n} from weak coupling to unitarity. At unitarity, we find that Δb_{3}=-0.356(4) in agreement with previous results, Δb_{4}=0.062(2), which is in agreement with all previous theoretical estimates but at odds with experimental determinations, and Δb_{5}=0.078(6), which is a prediction. We show the impact of those answers on the density equation of state and Tan contact, and trace their origin back to their polarized and unpolarized components.
We briefly review current research progress in the field of strongly correlated atomic Fermi gases. A deep understanding of strongly correlated quantum gases is plagued by the lack of either exact solutions or a controllable small interaction parameter. In this work, we present a systematic virial expansion method to study the high-temperature thermodynamics of a strongly correlated Fermi gas,where the controllable parameter is given by the fugacity,exp (μkBT),with μ being the chemical potential. A practical way is proposed to obtain the expansion coefficients for both homogeneous and harmonically trapped Fermi gases. We calculate, for the first time accurately, the third-order virial coefficient. The resulting equation of states at high temperatures is compared with a recent thermodynamic measurement and with Monte Carlo simulations.