Symmetries and Correlations in Strongly Interacting One-dimensional Quantum Gases

TL;DR

This thesis investigates the symmetry properties and correlation behaviors of strongly interacting one-dimensional quantum gases, revealing universal features like Tan's contacts and their relation to system symmetries, with implications for ultracold atom experiments.

Contribution

It provides an exact analysis of correlation properties and symmetries in strongly interacting 1D quantum gases, introducing new scaling laws relevant for experiments.

Findings

Identification of $k^{-4}$ high-momentum tails in correlations.

Universal relations between Tan's contacts and thermodynamic properties.

Application of group theory to determine exchange symmetries.

Abstract

The main focus of this thesis is the theoretical study of strongly interacting quantum mixtures confined in one dimension and subjected to a harmonic external potential. Such strongly correlated systems can be realized and tested in ultracold atoms experiments. Their non-trivial permutational symmetry properties are investigated, as well as their interplay with correlations. Exploiting an exact solution at strong interactions, we extract general correlation properties encoded in the one-body density matrix and in the associated momentum distributions, in fermionic and Bose-Fermi mixtures. In particular, we obtain substantial results about the short-range behavior, and therefore the high-momentum tails, which display typical $k^{-4}$ laws. The weights of these tails, denoted as Tan's contacts, are related to numerous thermodynamic properties of the systems such as the two-body correlations, the derivative of the energy with respect to the one-dimensional scattering length, or the static structure factor. We show that these universal Tan's contacts also allow to characterize the spatial symmetry of the systems, and therefore is a deep connection between correlations and symmetries. Besides, the exchange symmetry is extracted using a group theory method, namely the class-sum method, which comes originally from nuclear physics. Moreover, we show that these systems follow a generalized version of the famous Lieb-Mattis theorem. Wishing to make our results as experimentally relevant as possible, we derive scaling laws for Tan's contact as a function of the interaction, temperature and transverse confinement. These laws display interesting effects related to strong correlations and dimensionality.