• cold atoms
• inhomogeneous quantum systems
• critical phenomena

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.