Comparative Analysis of Topological Effects in Fermionic and Bosonic Systems

The thesis aims to describe a paradigmatic model of condensed matter systems: the XY model. A comparative analysis is performed by analogy with superfluid and superconducting systems, focusing on the variation of characteristic observables, such as the superfluid density, within the path-integral formalism. The work extends to the natural involvement of topological defects, such as dark solitons in one dimension and vortices in two dimensions, in modifying the dynamics of the system and in driving the peculiar topological Berezinskii–Kosterlitz–Thouless (BKT) transition. The last chapter analyses the effects of adding topological twists and supercurrents in one and two dimensions, focusing on their impact and on the relation between two different definitions of the superfluid density.