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arXiv:1907.10120 · Physical Review A

Leading- and next-to-leading-order semiclassical approximation to the first seven virial coefficients of spin-1/2 fermions across spatial dimensions

Y. Hou, A. J. Czejdo, Jared De Chant, C. R. Shill, Joaquín E. Drut

Published Dec 12, 2019• 9 citations
arXivHTMLPDF

Abstract

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 $\mathrm{\ensuremath{\Delta}}{b}_{7}$. Our results at leading order, though approximate, furnish simple analytic formulas that relate $\mathrm{\ensuremath{\Delta}}{b}_{n}$ to $\mathrm{\ensuremath{\Delta}}{b}_{2}$ 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 Tan's contact of the two-dimensional attractive Fermi gas and examine the radius of convergence of the virial expansion as a function of the coupling strength.

DOI
10.1103/physreva.100.063627

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•Jan 9, 2020•0 citations
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 \ensuremathΔb7\mathrm{\ensuremath{\Delta}}{b}_{7}\ensuremathΔb7​. Our results at leading order, though approximate, furnish simple analytic formulas that relate \ensuremathΔbn\mathrm{\ensuremath{\Delta}}{b}_{n}\ensuremathΔbn​ to \ensuremathΔb2\mathrm{\ensuremath{\Delta}}{b}_{2}\ensuremathΔb2​ 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 Tan's contact of the two-dimensional attractive Fermi gas and examine the radius of convergence of the virial expansion as a function of the coupling strength.
We map out the interaction effects on the first six virial coefficients of one-dimensional Fermi gases with zero-range attractive and repulsive interactions and the first four virial coefficients of the two-dimensional analog with attractive interactions. To that end, we use two nonperturbative stochastic methods: projection by complex stochastic quantization, which allows us to determine high-order coefficients at weak coupling and estimate the radius of convergence of the virial expansion, and a path-integral representation of the virial coefficients. To complement our numerical calculations, we present leading-order results in a semiclassical lattice approximation, which we find to be surprisingly close to the expected answers.
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 $\mathrm{\ensuremath{\Delta}}{b}_{3}$ and $\mathrm{\ensuremath{\Delta}}{b}_{4}$ agree qualitatively, and in some regimes quantitatively, with previous numerical calculations for the unitary limit of three-dimensional Fermi gases.
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 one, two, and three dimensions, 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 $\mathrm{\ensuremath{\Delta}}{b}_{3}, \mathrm{\ensuremath{\Delta}}{b}_{4}$, and $\mathrm{\ensuremath{\Delta}}{b}_{5}$ as functions of $\mathrm{\ensuremath{\Delta}}{b}_{2}$, 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 $\mathrm{\ensuremath{\Delta}}{b}_{3}$ with previous calculations in all dimensions and we formulate predictions for $\mathrm{\ensuremath{\Delta}}{b}_{4}$ and $\mathrm{\ensuremath{\Delta}}{b}_{5}$ in one and two dimensions. We also provide, for a range of couplings, the subspace contributions $\mathrm{\ensuremath{\Delta}}{b}_{31}, \mathrm{\ensuremath{\Delta}}{b}_{22}, \mathrm{\ensuremath{\Delta}}{b}_{41}$, and $\mathrm{\ensuremath{\Delta}}{b}_{32}$, 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 Pad\'e and Pad\'e-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 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 ddd spatial dimensions, both in homogeneous space and in a harmonic trapping potential of frequency $\ensuremath{\omega}$. Using the three-body problem to renormalize, we report analytic results for the change in the fourth- and fifth-order virial coefficients $\mathrm{\ensuremath{\Delta}}{b}_{4}$ and $\mathrm{\ensuremath{\Delta}}{b}_{5}$ as functions of $\mathrm{\ensuremath{\Delta}}{b}_{3}$. Additionally, we argue that in the $\ensuremath{\omega}\ensuremath{\rightarrow}0$ limit the relationship ${b}_{n}^{\text{T}}={n}^{\ensuremath{-}d/2}{b}_{n}$ holds between the trapped (T) and the 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 $\mathrm{\ensuremath{\Delta}}{b}_{3}^{\text{T}}$ in one dimension with three-body forces and $\mathrm{\ensuremath{\Delta}}{b}_{2}^{\text{T}}$ in two dimensions with two-body forces.
Using ab initio lattice methods, we calculate the finite temperature thermodynamics of homogeneous two-dimensional spin-1/2 fermions with attractive short-range interactions. We present results for the density, pressure, compressibility, and quantum anomaly (i.e., Tan's contact) for a wide range of temperatures (mostly above the superfluid phase, including the pseudogap regime) and coupling strengths, focusing on the unpolarized case. Within our statistical and systematic uncertainties, our prediction for the density equation of state differs quantitatively from the prediction by Luttinger-Ward theory in the strongly coupled region of parameter space, but otherwise agrees well with it. We also compare our calculations with the second- and third-order virial expansion, with which they are in excellent agreement in the low-fugacity regime.
Many-body quantum systems provide an interesting playground for experimentalists and theorists alike. In particular, ultracold atomic gases provide not only a realm of study for which experiments have a high degree of control but also provides theorists with intriguing, yet challenging systems that may be accessed computationally. It is for these reasons that it is here where the majority of our work will focus. We discuss the necessary physics, thermodynamics, and computational methods for carrying out this work, and present several methods for computing coefficients of the quantum virial expansion across dimensions for Fermi gases with zero-range interactions. The techniques discussed are non-perturbative and involve stochastic methods as well as a semi-classical lattice approximation. Through these algorithms, we have computed the interaction dependence (both attractive and repulsive where applicable) of the virial coefficients to a relatively high order as compared to current literature in one, two, and three spatial dimensions. Through one of said methods, we are able to provide a prediction for the radius of convergence of the virial expansion in one spatial dimension.