We study Tan's contact of a one-dimensional quantum gas of
N repulsive identical bosons confined in a harmonic trap at finite temperature. This canonical ensemble framework corresponds to the experimental conditions, with the number of particles being fixed for each experimental sequence. We show that in the strongly interacting regime, the contact rescaled by the contact at the Tonks-Girardeau limit is a universal function of two parameters: the rescaled interaction strength and temperature. This means that all pair and correlation effects in Tan's contact are embedded in Tan's contact in the Tonks-Girardeau limit.
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.
We determine the finite-temperature momentum distribution of a strongly interacting 1D Bose gas in the Tonks-Girardeau (impenetrable-boson) limit under harmonic confinement and explore its universal properties associated to the scale invariance of the model. We show that, at difference from the unitary Fermi gas in three dimensions, the weight of its large-momentum tails--given by Tan's contact--increases with temperature and calculate the high-temperature universal second contact coefficient using a virial expansion.
Following the idea of the density-functional approach, we develop a generalized Bogoliubov theory of an interacting Bose gas confined in a one-dimensional harmonic trap by using a local chemical potential, calculated with the Lieb-Liniger exact solution, as the exchange energy. At zero temperature, we use the theory to describe collective modes of a finite-particle system in all interaction regimes from the ideal gas limit to the mean-field Thomas-Fermi regime and to the strongly interacting Tonks-Girardeau regime. At finite temperature, we investigate the temperature dependence of collective modes in the weak-coupling regime by means of a Hartree-Fock-Bogoliubov theory with Popov approximation. By emphasizing the effects of finite particle number and nonzero temperature on collective-mode frequencies, we compare our results with a recent experimental measurement [E. Haller et al., Science 325, 1224 (2009)] and some previous theoretical predictions. We show that the experimental data are still not fully explained within current theoretical framework.
We present an exact numerical study of the scaling of density and momentum distribution functions of harmonically trapped one-dimensional bosons with repulsive contact interactions at zero and finite temperatures. We use path integral quantum Monte Carlo with worm updates in our calculations at finite interaction strengths, and the Bose-Fermi mapping in the Tonks-Girardeau regime. We discuss the homogeneous case and, within the local density approximation, use it to motivate the scaling in the presence of a harmonic trap. For the momentum distribution function, we pay special attention to the high momentum tails and their ${k}^{\ensuremath{-}4}$ asymptotic behavior.
The scaling of the largest eigenvalue ${\ensuremath{\lambda}}_{0}$ of the one-body density matrix of a system with respect to its particle number
N defines an exponent
C and a coefficient
B via the asymptotic relation ${\ensuremath{\lambda}}_{0}\ensuremath{\sim}\mathcal{B}\phantom{\rule{0.16em}{0ex}}{N}^{\mathcal{C}}$. The case
C=1 corresponds to off-diagonal long-range order. For a one-dimensional homogeneous Tonks-Girardeau gas, a well-known result also confirmed by bosonization gives instead
C=1/2. Here we investigate the inhomogeneous case, initially addressing the behavior of
C in the presence of a general external trapping potential
V. We argue that the value
C=1/2 characterizes the hard-core system independently of the nature of the potential
V. We then define the exponents $\ensuremath{\gamma}$ and $\ensuremath{\beta}$, which describe the scaling of the peak of the momentum distribution with
N and the natural orbital corresponding to ${\ensuremath{\lambda}}_{0}$, respectively, and we derive the scaling relation $\ensuremath{\gamma}+2\ensuremath{\beta}=\mathcal{C}$. Taking as a specific case the power-law potential $V(x)\ensuremath{\propto}{x}^{2n}$, we give analytical formulas for $\ensuremath{\gamma}$ and $\ensuremath{\beta}$ as functions of
n. Analytical predictions for the coefficient
B are also obtained. These formulas are derived by exploiting a recent field theoretical formulation and checked against numerical results. The agreement is excellent.