Scaling behaviour of quantum systems at thermal and quantum phase transitions

Experimental setups are finite in space and hardly ever in homogeneous conditions. This is very different from the ideal settings of the thermodynamic limit often adopted in condensed matter theories. Therefore, close to phase transitions, where typically long range correlations build up, it is important to correctly take into account the way in which boundaries and inhomogeneities affect the critical behaviour. This can be achieved by means of the finite-size (FSS) and trap-size (TSS) scaling theories, which generally apply to continuous phase transitions, where one can define a diverging length scale. FSS and TSS are reviewed in the first part of this work, together with some general properties of systems close to phase transitions. We then numerically study the TSS properties of the continuous finite-temperature phase transition of the Bose-Hubbard model (BH) in two and three dimension. This quantum model realistically describes experiments with ultra-cold bosonic gases trapped in optical lattices. In three dimensions, the BH exhibits a standard normal-to-superfluid transition. In two dimensions, the transition becomes of the Berezinski-Kosterlitz-Thouless type, characterised by logarithmic corrections to scaling. We perform thorough FSS analyses of quantum Monte Carlo data in homogeneous conditions to extract the value critical temperature. In two dimensions, this requires devising a matching method in which the FSS behaviour of the 2D BH is matched to the classical 2D XY model, whose transition belongs to the same universality class. We subsequently verify the validity of the TSS ansatz by simulating the trapped systems at the critical temperature. We find that the TSS theory is general and universal once one takes into account the effective way in which the trapping potential couples to the critical modes of the system. In the last part of this Thesis, we extend the FSS and TSS to discontinuous (or first order) quantum phase trnasitions. Discontinuous transitions do not develop a diverging length scale in the thermodynamic limit, but are rather characterised by the coexistence of domains of different phases at the transition. The typical size of single-phase domains induce a behaviour that closely resembles finite size scaling. We find that the scaling variable that parametrises the scaling behaviour at discontinuous transitions is the ratio of the perturbation energy driving the transition to the finite-size energy gap. We further find that inhomogeneous systems exibiting first order transitions can be treated heuristically in analogy with the TSS behaviour at continuous transitions. These findings are confirmed numerically on the quantum Ising and quantum Potts chains, which are simulated using density matrix renormalisation group techniques.