A tour through Hilbert space Exploring the Lieb-Liniger model and the interaction quench

In this thesis we consider three different problems, all revolving around the Lieb-Liniger model. What makes the Lieb-Liniger model interesting to study from a physical perspective, is the lack of thermalization it exhibits when prepared in an out-of-equilibrium initial state, even in experimental setups where integrability-breaking perturbations inevitably exist. From a technical standpoint its study is enabled by the Bethe Ansatz, which leverages the integrable nature of the model in order to compute all of its eigenstates as well as the matrix elements of physically interesting operators. The first problem we study is how we can best use our knowledge of the eigenstates and matrix elements to compute correlation functions. This generally requires the numerical evaluation of summations over all eigenstates in Hilbert space. In an effort to obtain optimal results, we study Hilbert space exploration algorithms aimed at finding the eigenstates most important to a given calculation, and develop several algorithms to achieve this. Having studied Hilbert space exploration algorithms, we turn to the problem of computing the time evolution following a quench in the interaction strength. By choosing an appropriate weighing metric for eigenstates we show how the Hilbert space exploration algorithms we studied can be used to accurately capture the time evolution following the quench. Finally, we switch gears and study the more fundamental properties of matrix elements in the Lieb-Liniger model. In doing so we study Hilbert space sampling algorithms and discover that the distribution of certain off-diagonal matrix elements is described by a Fréchet distribution.