This thesis centers on Bethe wave functions for one-dimensional lattice and field theories of quantum systems. The Bethe ansatz is a well established theoretical tool in the cross-cutting research field of integrable systems. However, most of the work in the literature focuses on the eigenenergies, and very limited results are known about the eigenstates. We present here two pieces of work to extend and deepen our understanding about Bethe wave functions.
In one work, we recast the Bethe states as exact matrix product states for the Heisenberg XXZ spin-½ chain and the Lieb-Liniger model with open boundary conditions, and find that the matrices do not depend on the spatial coordinate despite the open boundaries. Based on this result, we suggest generic ways of exploiting translational invariance both for finite size and in the thermodynamic limit. Our work makes the Bethe eigenstates more accessible and informs the choice of ansatz for tensor-network algorithms of both integrable and nonintegrable systems in one dimension. This achievement contributes in this way not only to the basic theory of integrable models but it will also influence the community that works using matrix product states and other tensor networks.
In another work, we use the physical interpretation of rapidities in integrable models to calculate the asymptotic expansion velocity of interacting atomic gases. Which is accessible in sudden expansion experiments as those done routinely these days using optically-trapped cold atomic gases. Through our research, the calculations of the asymptotic forms of observables of integrable models in quantum quench problems become more clear and theoretically accessible.