Geometric and topological aspects of quantum defects

In this thesis we present a detailed study of the dynamics of closed, twisted quantum vortices in Bose-Einstein condensates already published in two papers (Foresti & Ricca, 2019; Foresti & Ricca 2020). The study of how geometric and topological features affect the dynamics of vortices is very important for the description of interactions between vortices and for their reconnections. We generalize twist for quantum defects defying the concept of \emph{twist phase}. Then we find a modified Gross-Pitaevskii equation for the time evolution of a twisted vortex state. We discover that such a state is unstable and its evolution is dominated by a non-Hermitian Hamiltonian underlying the non-reversibility nature of the dynamics of twisted defects. Using the hydrodynamics description of Bose-Einsteins condensates and applying Kleinert's theory (Kleinert, 2018) to manage multi-valued phase fields, we find a complete set of integro-differential equations that quantitatively describe the dynamics of twisted vortices. Depending on the nature of the twist phase injected we propose two different stabilization mechanisms: if the twist phase is global then a secondary, central vortex is produced changing the linking number of the system. This mechanism can be seen as dominated by the presence of a topological phase and it is analyzed using Kleinert's theory. We thus prove theoretically what has been discovered numerically in (Zuccher & Ricca, 2018). In case of a local twist phase, no secondary vortex will form and the system produces unstable Kelvin waves with exponentially growing amplitude in regions where $\Lapl\theta_{tw} > 0$. We demonstrate that to minimize the energy and to stabilize the system the vortex coils producing non-zero writhe and extinguishing its twist phase. This mechanism can be seen as produced by the effect of a geometric phase on the system. We also propose an experiment to inject a twist phase on a quantum vortex in order to prove or disprove such stabilization mechanisms. Bibliography. Foresti, M. and Ricca, R. L. 2019 Defect production by pure twist induction as Aharonov-Bohm effect. \textit{Phys. Rev. E} \textbf{100}, 023107. M. Foresti \& R.L. Ricca, Hydrodynamics of a quantum vortex in presence of twist. \textit{J. Fluid Mech.} \textbf{904}, A25. Kleinert, H. 2008 \textit{Multivalued Fields in Condensed Matter, Electromagnetism and Gravitation}. World Scientific, Singapore. Zuccher, S. and Ricca, R. L. 2018, Twist effects in quantum vortices and phase defects. \textit{Fluid Dyn. Res.} \textbf{50}, 1--13.