Equal-mass two-component Fermi gases under spherically symmetric external harmonic confinement with large
s-wave scattering length are considered. Using the stochastic variational approach, we determine the lowest 286 and 164 relative eigenenergies of the
(2,2) and
(3,1) systems at unitarity as a function of the range
r0 of the underlying two-body potential and extrapolate to the ${r}_{0}\ensuremath{\rightarrow}0$ limit. Our calculations include all states with vanishing and finite angular momentum
L (and natural and unnatural parity $\ensuremath{\Pi}$) with relative energy up to 10.5 $\ensuremath{\hbar}\ensuremath{\Omega}$, where $\ensuremath{\Omega}$ denotes the angular trapping frequency of the external confinement. Our extrapolated zero-range energies are estimated to have uncertainties of 0.1
or smaller. The
(2,2) and
(3,1) energies are used to determine the fourth-order virial coefficient of the trapped unitary two-component Fermi gas in the low-temperature regime. Our results are compared with recent predictions for the fourth-order virial coefficient of the homogeneous system. We also calculate small portions of the energy spectra of the
(3,2) and
(4,1) systems at unitarity.
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.
We give exact integral expressions of the third cluster or virial coefficients of binary mixtures of ideal Bose or Fermi gases, with interspecies interactions of zero range and infinite s-wave scattering length. In general the result depends on three-body parameters Rt appearing in three-body contact conditions, because an Efimov effect is present or because the mixture is in a preefimovian regime with a mass ratio close to an Efimov effect threshold. We give a new, exact integral expression of Rt for the microscopic narrow Feshbach resonance model. A divergence of Rt in the preefimovian regime at a scaling exponent s = 1/2 is predicted and physically discussed. The analytical results are applied to typical species used in cold atom experiments.
In the free three-dimensional space, we consider a pair of identical $\ensuremath{\uparrow}$ fermions of some species or in some internal state and a pair of identical $\ensuremath{\downarrow}$ fermions of another species or in another state. There is a resonant
s-wave interaction (that is, of zero range and infinite scattering length) between fermions in different pairs and no interaction within the same pair. We study whether this
2+2 fermionic system can exhibit (as the
3+1 fermionic system) a four-body Efimov effect in the absence of three-body Efimov effect, that is, the mass ratio $\ensuremath{\alpha}$ between $\ensuremath{\uparrow}$ and $\ensuremath{\downarrow}$ fermions and its inverse are both smaller than 13.6069\dots{}. For this purpose, we investigate scale invariant zero-energy solutions of the four-body Schr\"odinger equation, that is, positively homogeneous functions of the coordinates of degree $s\ensuremath{-}7/2$, where
s is a generalized Efimov exponent that becomes purely imaginary in the presence of a four-body Efimov effect. Using rotational invariance in momentum space, it is found that the allowed values of
s are such that
M(s) has a zero eigenvalue; here the operator
M(s), that depends on the total angular momentum $\ensuremath{\ell}$, acts on functions of two real variables (the cosine of the angle between two wave vectors and the logarithm of the ratio of their moduli), and we write it explicitly in terms of an integral matrix kernel. We have performed a spectral analysis of
M(s), analytical and for an arbitrary imaginary
s for the continuous spectrum and numerical and limited to
s=0 and $\ensuremath{\ell}\ensuremath{\le}12$ for the discrete spectrum. We conclude that no eigenvalue of
M(0) crosses zero over the mass ratio interval $\ensuremath{\alpha}\ensuremath{\in}[1;13.6069\ensuremath{\cdots}]$, even if, in the parity sector ${(\ensuremath{-}1)}^{\ensuremath{\ell}}$, the continuous spectrum of
M(s) has everywhere a zero lower border. As a consequence, there is no possibility of a four-body Efimov effect for the
2+2 fermions. We also enunciated a conjecture for the fourth virial coefficient of the unitary spin-
1/2 Fermi gas, inspired from the known analytical form of the third cluster coefficient and involving the integral over the imaginary
s axis of
s times the logarithmic derivative of the determinant of
M(s) summed over all angular momenta. The conjectured value is in contradiction with the experimental results.
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 astrophysics. This work determines the fourth-order virial coefficient b_{4} 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 b_{4}, our b_{4} 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 antisymmetrization 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.