Quasi-random systems: duality transformations and numerical simulations

For classical systems, the concept of thermalization, introduced by Boltzmann in 19th century, is the main instrument for explaining the equilibrium thermodynamics. For quantum systems, although the Schrodinger unitary evolution is deterministic and invertible, it is possible to extend the thermalization concept by considering a subsystem that thermalizes with the rest of the system. However, as in the classical case, there is a class of systems that violate the ergodic principle and do not thermalize. This phenomenon of egodicity breaking is present in models with disorder introduced by P.W. Anderson in 1958. In chapter 1 of our work, we present the class of quasi-random systems, showing their most important features like the ergodicity breaking, the self-duality and the presence of mobility edges. In chapter 2, as original work, we found the phase transition lines of the superconducting Aubry-André model using the analytical tool of the duality transformations. Moreover, we investigate directly the many body localization performing simulations of several quantities like the IPR, the imbalance and the fidelity to detect the many body localization. In the last chapter, we analyse the Jordan-Wigner and Bravy-Kitaev transformations for the quantum simulation of our fermionic systems.