Bound states in integrable models Formation and structure

In this thesis, we examine various aspects of bound states that appear as fundamental excitations in certain paradigmatic integrable models. The number and structure of these bound states depend on the parameter regime of the model. The overarching theme of this thesis is to tweak parameters to deform or create bound states and better understand their mathematical and physical properties.
Concretely, two lines of research are pursued. Firstly, we consider the production of bound states in the Lieb-Liniger model by slowly driving the interaction strength from the repulsive to the attractive regime using generalized hydrodynamics. In this out-of-equilibrium protocol, bound states are created at the crossover between repulsive and attractive interactions, for which we derive an exact analytical expression using an entropy maximization ansatz. To benchmark our results, we take the semi-classical limit of the Lieb-Liniger model and the entire protocol, arriving at the non-linear Schrödinger Equation (NLS). Surprisingly, the excitation spectrum of the attractive NLS consists solely of solitons, with radiative modes absent. To further explore the interplay between radiative and solitonic modes, we also study the semi-classical limit of the sine-Gordon model.
Secondly, we examine bound state eigenstates of the XXZ-Heisenberg spin-1/2 chain, where a rigorous classification for finite system sizes and arbitrary anisotropy is absent. We address this by classifying the eigenstates in the Ising limit and mapping them back through the gapped phase all the way to the isotropic point. We find a system size and anisotropy dependent critical equation for the unbinding of bound states.