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arXiv:1305.5383 · Physical Review A

Harmonically trapped Fermi gas: Temperature dependence of the Tan contact

Yangqian Yan, D. Blume

Published Aug 27, 2013• 20 citations
arXivHTMLPDF

Abstract

Ultracold atomic gases with short-range interactions are characterized by a number of universal species-independent relations. Many of these relations involve the two-body Tan contact. Employing the canonical ensemble, we determine the Tan contact for small harmonically trapped two-component Fermi gases at unitarity over a wide range of temperatures, including the zero and high-temperature regimes. A cluster expansion that describes the properties of the $N$-particle system in terms of those of smaller subsystems is introduced and shown to provide an accurate description of the contact in the high-temperature regime. Finite-range corrections are quantified and the role of the Fermi statistics is elucidated by comparing results for Fermi, Bose, and Boltzmann statistics.

DOI
10.1103/physreva.88.023616

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Ultracold atomic gases with short-range interactions are characterized by a number of universal species-independent relations. Many of these relations involve the two-body Tan contact. Employing the canonical ensemble, we determine the Tan contact for small harmonically trapped two-component Fermi gases at unitarity over a wide range of temperatures, including the zero and high-temperature regimes. A cluster expansion that describes the properties of the NNN-particle system in terms of those of smaller subsystems is introduced and shown to provide an accurate description of the contact in the high-temperature regime. Finite-range corrections are quantified and the role of the Fermi statistics is elucidated by comparing results for Fermi, Bose, and Boltzmann statistics.
While the zero-temperature properties of harmonically trapped cold few-atom systems have been discussed fairly extensively over the past decade, much less is known about the finite-temperature properties. Working in the canonical ensemble, we characterize small harmonically trapped atomic systems as a function of the temperature using analytical and numerical techniques. We present results for the energetics, structural properties, condensate fraction, superfluid fraction, and superfluid density. Our calculations for the two-body system underline that the condensate and superfluid fractions are distinctly different quantities. Our work demonstrates that the path-integral Monte Carlo method yields reliable results for bosonic and fermionic systems over a wide temperature range, including the regime where the de Broglie wavelength is large, i.e., where the statistics plays an important role. The regime where the Fermi sign problem leads to reasonably large signal-to-noise ratios is mapped out for selected parameter combinations. Our calculations for bosons focus on the unitary regime, where the physics is expected to be governed by the three-body parameter. If the three-body parameter is large compared to the inverse of the harmonic oscillator length, we find that the bosons form a droplet at low temperature and behave approximately like a noninteracting Bose and eventually Boltzmann gas at high temperature. The change of the behavior occurs over a fairly narrow temperature range. A simple model that reproduces the key aspects of the phase-transition-like feature, which can potentially be observed in cold atom Bose gas experiments, is presented.
The universal Tan relations connect a variety of microscopic features of many-body quantum systems with two-body contact interactions to a single quantity, called the contact. The latter has become pivotal in the description of quantum gases. We provide a complete characterization of the Tan contact of the harmonically trapped Lieb-Liniger gas for arbitrary interactions and temperature. Combining thermal Bethe ansatz, local-density approximation, and exact quantum Monte Carlo calculations, we show that the contact is a universal function of only two scaling parameters, and determine the scaling function. We find that the temperature dependence of the contact, or equivalently the interaction dependence of the entropy, displays a maximum. The presence of this maximum provides an unequivocal signature of the crossover to the fermionized regime and it is accessible in current experiments.
The properties of two-component Fermi gases with zero-range interactions are universal. We use an explicitly correlated Gaussian basis set expansion approach to investigate small equal-mass two-component Fermi gases under spherically symmetric external harmonic confinement. At unitarity, we determine the ground-state energy for systems with up to 10 particles interacting through finite-range two-body potentials for both even and odd numbers of particles. We extrapolate the energies to the zero-range limit using a novel scheme that removes the linear and, in some cases, also the quadratic dependence of the ground-state energies on the two-body range. Our extrapolated zero-range energies are compared with results from the literature. We also calculate the two-body Tan contact and structural properties.
The Unitary Fermi Gas (UFG) is one of the most strongly interacting systems known to date, as it saturates the unitarity bound on the quantum mechanical scattering cross section.The UFG corresponds to a two-component Fermi gas in the limit of short interaction range and large scattering length, and is currently realized in ultracold-atom experiments via Feshbach resonances.While easy to define, the UFG poses a challenging quantum many-body problem, as it lacks any characteristic scale other than the density.As a consequence, accurate quantitative predictions of the thermodynamic properties of the UFG require Monte Carlo calculations.However, significant progress has also been made with purely analytical methods.Notably, in 2005 Tan derived a set of exact thermodynamic relations in which a universal quantity known as the "contact" C plays a crucial role.Recently, C has also been found to determine the prefactor of the high-frequency power-law decay of correlators as well as the right-hand-sides of shear-and bulk viscosity sum rules.The contact is therefore a central piece of information on the UFG in equilibrium as well as away from equilibrium.In this talk we describe some of the known aspects of Fermi gases at and around unitarity, show our latest Monte Carlo results for the contact at finite temperature, and summarize the open questions in the field, some of which we are starting to answer using large-scale Monte Carlo calculations by adapting methods from Lattice QCD.
The recently discovered universal thermodynamic behavior of dilute, strongly interacting Fermi gases also implies a universal structure in the many-body pair-correlation function at short distances, as quantified by the contact . Here, we theoretically calculate the temperature dependence of this universal contact for a Fermi gas in free space and in a harmonic trap. At high temperatures above the Fermi degeneracy temperature, T ≳ T F , we obtain a reliable non-perturbative quantum virial expansion up to third order. At low temperatures, we compare different approximate strong-coupling theories. These make different predictions, which need to be tested either by future experiments or by advanced quantum Monte Carlo simulations. We conjecture that in the universal unitarity limit, the contact or correlation decreases monotonically with increasing temperature, unless the temperature is significantly lower than the critical temperature, T ≪ T c ∼0.2 T F . We also discuss briefly how to measure the universal contact in either homogeneous or harmonically trapped Fermi gases.
The Unitary Fermi Gas (UFG) is one of the most strongly interacting systems known to date, as it saturates the unitarity bound on the quantum mechanical scattering cross section. The UFG corresponds to a two-component Fermi gas in the limit of short interaction range and large scattering length, and is currently realized in ultracold-atom experiments via Feshbach resonances. While easy to define, the UFG poses a challenging quantum many-body problem, as it lacks any characteristic scale other than the density. As a consequence, accurate quantitative predictions of the thermodynamic properties of the UFG require Monte Carlo calculations. However, significant progress has also been made with purely analytical methods. Notably, in 2005 Tan derived a set of exact thermodynamic relations in which a universal quantity known as the "contact" C plays a crucial role. Recently, C has also been found to determine the prefactor of the high- frequency power-law decay of correlators as well as the right-hand-sides of shear- and bulk viscosity sum rules. The contact is therefore a central piece of information on the UFG in equilibrium as well as away from equilibrium. In this talk we describe some of the known aspects of Fermi gases at and around unitarity, show our latest Monte Carlo results for the contact at finite temperature, and summarize the open questions in the field, some of which we are starting to answer using large-scale Monte Carlo calculations by adapting methods from Lattice QCD.