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Register a new userThe fourth- and fifth-order virial coefficients from weak-coupling to unitarity
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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 temperature, 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. Implementing these factorizations and automated algebra codes, we obtain the interaction-induced change $Delta b_n$ from weak coupling to unitarity. At unitarity, we find: $Delta b_3 = -0.356(4)$, in agreement with previous results; $Delta b_4 = 0.062(2)$, in agreement with all previous theoretical estimates but at odds with experimental determinations; and $Delta b_5 = 0.078(6)$, which is a prediction. We show the impact of those answers on the density equation of state and Tan contact, and track their origin back to their polarized and unpolarized components.
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By generalizing our automated algebra approach from homogeneous space to harmonically trapped systems, we have calculated the fourth- and fifth-order virial coefficients of universal spin-1/2 fermions in the unitary limit, confined in an isotropic harmonic potential. We present results for said coefficients as a function of trapping frequency (or, equivalently, temperature), which compare favorably with previous Monte Carlo calculations (available only at fourth order) as well as with our previous estimates in the untrapped limit (high temperature, low frequency). We use our estimates of the virial expansion, together with resummation techniques, to calculate the compressibility and spin susceptibility.
The virial expansion characterizes the high-temperature approach to the quantum-classical crossover in any quantum many-body system. Here, we calculate the virial coefficients up to the fifth-order of Fermi gases in 1D, 2D, and 3D, with attractive contact interactions, as relevant for a variety of applications in atomic and nuclear physics. To that end, we discretize the imaginary-time direction and calculate the relevant canonical partition functions. In coarse discretizations, we obtain analytic results featuring relationships between the interaction-induced changes $Delta b_3$, $Delta b_4$, and $Delta b_5$ as functions of $Delta b_2$, the latter being exactly known in many cases by virtue of the Beth-Uhlenbeck formula. Using automated-algebra methods, we push our calculations to progressively finer discretizations and extrapolate to the continuous-time limit. We find excellent agreement for $Delta b_3$ with previous calculations in all dimensions and we formulate predictions for $Delta b_4$ and $Delta b_5$ in 1D and 2D. We also provide, for a range of couplings,the subspace contributions $Delta b_{31}$, $Delta b_{22}$, $Delta b_{41}$, and $Delta b_{32}$, which determine the equation of state and static response of polarized systems at high temperature. As a performance check, we compare the density equation of state and Tan contact with quantum Monte Carlo calculations, diagrammatic approaches, and experimental data where available. Finally, we apply Pade and Pade-Borel resummation methods to extend the usefulness of the virial coefficients to approach and in some cases go beyond the unit-fugacity point.
Using a leading-order semiclassical approximation, we calculate the third- and fourth-order virial coefficients of nonrelativistic spin-1/2 fermions in a harmonic trapping potential in arbitrary spatial dimensions, and as functions of temperature, trapping frequency and coupling strength. Our simple, analytic results for the interaction-induced changes $Delta b_3$ and $Delta b_4$ agree qualitatively, and in some regimes quantitatively, with previous numerical calculations for the unitary limit of three-dimensional Fermi gases.
Using a coarse temporal lattice approximation, we calculate the first few terms of the virial expansion of a three-species fermion system with a three-body contact interaction in $d$ spatial dimensions, both in homogeneous space as well as in a harmonic trapping potential of frequency $omega$. Using the three-body problem to renormalize, we report analytic results for the change in the fourth- and fifth-order virial coefficients $Delta b_4$ and $Delta b_5$ as functions of $Delta b_3$. Additionally, we argue that in the $omega to 0$ limit the relationship $b_n^text{T} = n^{-d/2} b_n$ holds between the trapped (T) and homogeneous coefficients for arbitrary temperature and coupling strength (not merely in scale-invariant regimes). Finally, we point out an exact, universal (coupling- and frequency-independent) relationship between $Delta b_3^text{T}$ in 1D with three-body forces and $Delta b_2^text{T}$ in 2D with two-body forces.
Following up on recent calculations, we investigate the leading- and next-to-leading order semiclassical approximation to the virial coefficients of a two-species fermion system with a contact interaction. Using the analytic result for the second-order virial coefficient as a renormalization condition, we derive expressions for up to the seventh-order virial coefficient $Delta b_7$. Our results at leading order, though approximate, furnish simple analytic formulas that relate $Delta b_n$ to $Delta b_2$ for arbitrary dimension, providing a glimpse into the behavior of the virial expansion across dimensions and coupling strengths. As an application, we calculate the pressure and Tans contact of the 2D attractive Fermi gas and examine the radius of convergence of the virial expansion as a function of the coupling strength.
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