Path-Integral Monte Carlo Determination of the Fourth-Order Virial Coefficient for a Unitary Two-Component Fermi Gas with Zero-Range Interactions

Abstract

The unitary equal-mass Fermi gas with zero-range interactions constitutes a paradigmatic model system that is relevant to atomic, condensed matter, nuclear, particle, and astro physics. This work determines the fourth-order virial coefficient b4 of such a strongly-interacting Fermi gas using a customized ab initio path integral Monte Carlo (PIMC) algorithm. In contrast to earlier theoretical results, which disagreed on the sign and magnitude of b4, our b4 agrees within error bars with the experimentally determined value, thereby resolving an ongoing literature debate. Utilizing a trap regulator, our PIMC approach determines the fourth-order virial coefficient by directly sampling the partition function. An on-the-fly anti-symmetrization avoids the Thomas collapse and, combined with the use of the exact two-body zero-range propagator, establishes an efficient general means to treat small Fermi systems with zero-range interactions. [8,9], which can nowadays be produced routinely in table-top experiments, are ideal for studying strongly-interacting systems since (i) the van der Waals interaction is short-ranged, which means that it can be approximated by a contact potential that introduces a single length scale, i.e., the s-wave scattering length a s ; and (ii) a s can be tuned at will utilizing Feshbach resonance techniques [10]. When a s diverges, i.e., becomes infinitely large, the two-body contact potential does not define a length scale [11]. Just like the non-interacting Fermi gas, the properties of the unitary Fermi gas (Fermi gas with infinite a s ) are determined by two length scales, the de Broglie wavelength λ and interparticle spacingr [12].At high temperature, λ is much smaller thanr and the grand canonical thermodynamic potential Ω can be expanded in terms of the fugacity [13,14]. The n thorder expansion or virial coefficient b n is determined by the partition functions of clusters containing n or fewer fermions. Since all thermodynamic properties at high temperature can be derived from the virial coefficients b n [15], the b n 's are essential to understanding the normal state of strongly-interacting Fermi gases.While the second-and third-order virial coefficients are well understood [13,[15][16][17][18][19][20][21], none of the theoretical calculations for b 4 [22][23][24][25] agree with the experimental data [19,26]. This letter rectifies this situation: our theoretically determined b 4 agrees with the experimentally determined value. Our approach uses a trap regulator [27,28] and employs the path integral Monte Carlo

Third- and fourth-order virial coefficients of harmonically trapped fermions in a semiclassical approximation

“…2 our results in 3D at unitarity as a function of βω, superimposed with the data from Ref. [7]. While we do not expect, a priori, good quantitative agreement in this strong coupling regime, we find at least qualitative agreement for both ∆b 3 and ∆b 4 , and surprisingly good agreement at the quantitative level for ∆b 4 .…”

“…Comparison of our LO-SCLA results for ∆b3 (top) and ∆b4 (bottom) as a function of βω. The high-temperature fits and PIMC results are from Ref [7][8]…”

Universality of the unitary Fermi gas: a few-body perspective

“…and we have used the Bertsch parameter in Eq. (22). Similarly, we find the upper bound for a uniform system…”

Virial coefficients of one-dimensional and two-dimensional Fermi gases by stochastic methods and a semiclassical lattice approximation

“…Obtaining b 3 and beyond, however, typically requires numerical methods (see e.g. [9][10][11]). Although the b n are a proxy for other quantities, their calculation has become an attractive challenge per se, especially in cases such as the unitary limit [12] (the universal limit of zero interaction range and infinite scattering length), where the b n represent universal constants of quantum many-body physics.…”

“…Although the b n are a proxy for other quantities, their calculation has become an attractive challenge per se, especially in cases such as the unitary limit [12] (the universal limit of zero interaction range and infinite scattering length), where the b n represent universal constants of quantum many-body physics. For that reason, the calculation of the b n has been vigorously pursued by several groups [11,[13][14][15][16][17][18].…”