The interest in vortices and vortex lattices was sparked by the prediction of quantisation of circulation by Onsager in the 1940s. The field has since developed dramatically and attracted a lot of interest across the physics community. In this dissertation we study vortices in two different systems: a rotating, Rabi-coupled, two-component Bose--Einstein condensate (BEC) and a rotating spinor-BEC, in two spatial dimensions.

Vortex molecules can form in a two-component superfluid when a Rabi field drives transitions between the two components. We study the ground state of an infinite system of vortex molecules in two dimensions, using a numerical scheme which makes no use of the lowest Landau level approximation. We find the ground state lattice geometry for different values of intercomponent interactions and strength of the Rabi field. In the limit of large field, when molecules are tightly bound, we develop a complementary analytical description. The energy governing the alignment of molecules on a triangular lattice is found to correspond to that of an infinite system of two-dimensional quadrupoles, which may be written in terms of an elliptic function $Q(zij;\omega 1,\omega 2)$. This allows for a numerical evaluation of the energy enabling us to find the ground state configuration of the molecules.

In the $polar$ phase of a two-component BEC, in which the spin density is zero, the emergent $spin-gauge\; rotation\; symmetry$ of the order parameter allows for the presence of half-quantum vortices (HQVs). We numerically search for this object in the variational ground state of a spinor-BEC and find it in certain region of the phase diagram. We provide analytical arguments that suggest that this object is energetically favorable in the ground state.

Matrix product state (MPS) based methods are currently regarded as one of the most powerful tools to study the low-energy physics of one-dimensional many-body quantum systems. In this work we find a connection between MPS in the left canonical form and the Stiefel manifold. This relation allows us to constrain the optimisation to this subspace of the otherwise larger MPS manifold. We find that our method suffers from two undesirable features. First, the need of a large unit cell to achieve machine precision. Second, because of the presence of the power method in the variational energy expression, it is possible for the convergence process to get stuck in regions of the Stiefel manifold where the modulus of the second largest eigenvalue of the transfer matrix is very close to one.

Since the foundation of the field of $Artificial\; Intelligence$ (AI) in 1956, at a workshop held in Dartmouth College (New Hampshire, US), the excitement and optimism towards it has oscillated throughout the years. The last AI boom started in 2012 and we live in a time where people from $all$ disciplines, both in industry and academia, are getting involved in machine learning. We contribute to the field with a $quantum-inspired$ generative model for raw audio. Our model is based on continuous matrix product states and it takes the form of a $stochastic\; Schrödinger\; equation$, describing the continuous time measurement of a quantum system. We test our model on three different synthetic datasets and we find its performance promising.