Virial expansion for a strongly correlated Fermi gas

Using a high temperature virial expansion, we present a controllable study of the thermodynamics of strongly correlated Fermi gases near the BEC-BCS crossover region. We propose a practical way to determine the expansion coefficients for both harmonically trapped and homogeneous cases, and calculate the third order coefficient $b_{3}(T)$ at finite temperatures $T$. At resonance, a $T$-independent coefficient $b_{3,\infty}^{\hom}\approx-0.29095295$ is determined in free space. These results are compared with a recent thermodynamic measurement of $^{6}$Li atoms, at temperatures below the degeneracy temperature, and with Monte Carlo simulations.

💡 Research Summary

The paper presents a systematic study of the thermodynamics of a strongly interacting two‑component Fermi gas across the BEC‑BCS crossover using a high‑temperature virial expansion. Starting from the grand canonical potential Ω = −kBT ln Z, the authors expand the partition function in powers of the fugacity z = exp(μ/kBT), which is small at high temperature. The expansion coefficients (virial coefficients) b_n are expressed in terms of n‑body cluster partition functions Q_n. To obtain b₂ and b₃, the authors solve the two‑ and three‑body problems exactly in a three‑dimensional isotropic harmonic trap, employing the Bethe‑Peierls boundary condition to model a broad Feshbach resonance.

For the two‑body sector, the relative motion spectrum is given by a transcendental equation involving the scattering length a. In the unitary limit (|a|→∞) the solutions are ν_n = n − ½, leading to an analytic expression for the second virial coefficient b₂,∞. The result shows a small temperature‑dependent correction proportional to (ℏω/kBT)², reflecting the finite trap frequency.

The three‑body problem is tackled by introducing Jacobi coordinates (r, ρ) and constructing the wavefunction as a symmetrized product of a two‑body relative state and a single‑particle harmonic state, with the exchange operator P₁₃ enforcing fermionic antisymmetry. The interaction enters through a matrix C_{nm} that couples different two‑body channels. Solving the resulting eigenvalue problem numerically for up to 10⁴ energy levels yields the third virial coefficient. In the unitary limit the authors find b₃,∞ − b₃^{(1)} = −0.06833960 + 0.038867 (ℏω/kBT)² + …, which translates to a universal homogeneous value b₃,∞^{hom} ≈ −0.29095295. This sign and magnitude differ from an earlier field‑theoretic calculation (which gave a positive value) but agree with recent quantum Monte‑Carlo data.

A key observation is that the harmonic confinement strongly suppresses higher‑order virial coefficients: the trapped coefficients are related to the homogeneous ones by b_n^{trap}=n^{-3/2} b_n^{hom}. Consequently, even at temperatures below the Fermi temperature T_F the virial series converges rapidly for a trapped gas. Using the calculated b₂ and b₃, the authors compute the energy E and entropy S as functions of temperature and compare the interaction energy E_int = E − E_{IG} with experimental measurements on ⁶Li at a broad Feshbach resonance. The theoretical curve matches the data down to T≈0.5 T_F, far below the conventional Boltzmann regime where virial expansions are thought to be valid. In free space, the same coefficients predict that the virial expansion becomes reliable only for T ≳ 2 T_F, consistent with Monte‑Carlo benchmarks.

The paper concludes that the virial expansion, when combined with exact few‑body solutions, provides a controllable and accurate description of strongly correlated Fermi gases across the crossover, even in the degenerate regime for trapped systems. The methodology is extensible to higher‑order coefficients (b₄, b₅, …) and offers a valuable benchmark for future experiments and quantum Monte‑Carlo simulations of strongly interacting fermionic matter.