• Finite-size scaling
• Gauge theory
• Lattice simulations
• Phase transitions
• Theta dependence

Symmetries are a powerful tool for the characterization of a large amount of systems coming from all the different areas of physics, from condensed matter models to high energy systems. An interesting phenomenon in statistical mechanics and QFT is the spontaneous symmetry breaking, in which the ground state breaks the symmetry and that could be associated to a phase transition in the system. In this situation a critical point separates an "ordered" and a "disordered" phase, and the system shows non-analytical behaviors around it. In the particular case of second-order phase transitions the correlation length diverges in proximity of the critical point and this makes the microscopic details ininfluent to the determination of the global properties of the system. In this way the various systems with the same behavior around a critical point can be grouped in some "universality classes", which are associated to the fixed points of the renormalization group flow. In a phase transition associated to the breaking of a global symmetry, the addition of a gauge symmetry could lead to a more complex scenario, since it is not clear how much the pure gauge observables, which are unaffected by the global symmetry, are universal.
The aim of this thesis is to investigate this topic considering a simple system, the two-dimensional Abelian-Higgs model, that presents a global SU(N) symmetry combined with local U(1) gauge invariance and a phase transition only in the zero-temperature limit, due to the Mermin-Wagner theorem. The critical behavior of this model can be studied by using its lattice formulation by means of Monte Carlo simulations, in particular through Finite-Size Scaling techniques. Focusing on topological quantities, which are pure gauge observables that describe some non-perturbative properties of the system, we have found that they show, in some cases, a critical behavior related to what is seen for quantities associated to the global SU(N) symmetry. In the remaining cases considered the same conclusions cannot be drawn with data at our disposal, since we have found some discrepancies and the behavior is still not totally clear. This could be addressed to artefacts of the lattice discretization and further investigations in this sense are needed.