arXiv:1908.00070 · Physical Review A

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

K. J. Morrell, Casey Berger, Joaquín E. Drut

Published Dec 12, 2019• 7 citations

Abstract

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.

DOI: 10.1103/physreva.100.063626

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arXiv:1908.00070 · Physical Review A

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

K. J. Morrell, Casey Berger, Joaquín E. Drut

Published Dec 12, 2019• 7 citations

arXiv HTML PDF

Abstract

\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.

DOI: 10.1103/physreva.100.063626

Related papers

Fourth- and fifth-order virial expansion of harmonically trapped fermions at unitarity

Y. Hou, K. J. Morrell, A. J. Czejdo +1 more

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- $\frac{1}{2}$ 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). In addition to the conventional coefficients, we provide estimates for the contributions from subspaces with varying polarization. We use our results for the virial coefficients, together with resummation techniques, to calculate the compressibility and spin susceptibility.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-$\frac{1}{2}$ 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). In addition to the conventional coefficients, we provide estimates for the contributions from subspaces with varying polarization. We use our results for the virial coefficients, together with resummation techniques, to calculate the compressibility and spin susceptibility.

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arXiv:2004.02344Physical Review AAug 6, 20205 citations

\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.

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-

\frac{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). In addition to the conventional coefficients, we provide estimates for the contributions from subspaces with varying polarization. We use our results for the virial coefficients, together with resummation techniques, to calculate the compressibility and spin susceptibility.

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 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.

We present highly accurate solutions of the Schr\"odinger equation for three fermions in two different spin states with zero-range

s

-wave interactions under harmonic confinement. Our approach covers spherically symmetric, strictly two-dimensional, strictly one-dimensional, cigar-shaped, and pancake-shaped traps. In particular, we discuss the transition from quasi-one-dimensional to strictly one-dimensional and from quasi-two-dimensional to strictly two-dimensional geometries. We determine and interpret the eigenenergies of the system as a function of the trap geometry and the strength of the zero-range interactions. The eigenenergies are used to investigate the dependence of the second- and third-order virial coefficients, which play an important role in the virial expansion of the thermodynamic potential, on the geometry of the trap. We show that the second- and third-order virial coefficients for anisotropic confinement geometries are, for experimentally relevant temperatures, very well approximated by those for the spherically symmetric confinement for all

s

-wave scattering lengths.

{b}_{n}

for all

n

in the noninteracting case, as functions of the trapping and rotational frequencies. We also report on the virial coefficients for the angular momentum and associated moment of inertia. Using the

{b}_{n}

coefficients, we analyze the deconfined limit (in which the angular frequency matches the trapping frequency) and derive explicitly the limiting form of the partition function, showing from the thermodynamic standpoint how both the 2D and 3D cases become effectively homogeneous 2D systems. To tackle the virial coefficients in the presence of weak interactions, we implement a coarse temporal lattice approximation and obtain virial coefficients up to third order.