Quantum correlations in systems of indistinguishable particles/ topological states of matter in a number conserving setting

Abstract

This thesis is a contribution to the research field of three distinct subjects: quantum correlations in systems of indistinguishable particles, topological states of matter, and non-Markovian dynamics. The first two subjects are the main subjects of the thesis, in which was kept an exclusive dedication, while the last one fits as a satellite part in the thesis. The first part of this thesis concerns to a proper understanding of quantum correlations in systems of indistinguishable particles. In this case, the space of quantum states is restricted to symmetric or antisymmetric subspaces, depending on the bosonic or fermionic nature of the system, and the particles are no longer accessible individually, thus eliminating the usual notions of separability and local measurements, and making the analysis of correlations much subtler. We completely review the distinct approaches for the entanglement in these systems, and based on its definitions we elaborate distinct methods in order to quantify the entanglement between the indistinguishable particles. Such methods have proven to be very useful and easy to handle, since they adapt common tools in the usual entanglement theory of distinguishable systems for the present indistinguishable case. We further propose a general notion of quantum correlation beyond entanglement (the quantumness of correlations) in these systems, by means of an ¿activation protocol¿. Such general notion is very helpful at the ongoing debate in the literature regarding the correct definition of particle entanglement, since it allows us to analyze the correlations between indistinguishable particles in a different, and more general, framework, settling some of its controversies. We then use our quantifiers to study the entanglement of indistinguishable particles on its particle partition in a specific model, namely extended Hubbard model, with focus on its behavior when crossing its quantum phase transitions. The second part of this thesis concerns to the study of topological states of matter. Our analysis and results have as a basis the paradigmatic (non-number conserving) Kitaev model, which provides a minimal setting showcasing all the key aspects of topological states of matter in fermionic systems. Our analysis focus, however, in a number conserving setting. In such setting, we present two distinct ways to generate topological states completely similar to the Kitaev model: (i) in a Hamiltonian setting, we present an exactly solvable two-wire fermionic model which conserves the number of particles and features Majorana-like exotic quasiparticles at the edges; (ii) by means of a suitably engineered dissipative dynamics, we present how to generate such topological states as the dark states (steady-states) of the evolution.