• disorder
• entanglement
• localization
• many-body
• multipartite

According to quantum mechanics it is possible to prepare states that describe the global system but they do not provide information on its subparts, it is a typical feature of non-classical systems related to quantum correlations. A state that describes the global system but does not describe its subparts is called entangled. Bipartite entanglement is quantified by the von Neumann entropy and it has been used in the context of quantum information and more, in fact it plays a central role also in the field quantum phase transitions (QPTs) theory. For instance, in ground-state QPTs it has been observed that entanglement permits to differentiate critical and non-critical behaviour of the system. On the other hand, the concept of multipartite entanglement in the field of many-body physics has not been much investigated yet, anyway there exist some works in which it has been analyzed in QPTs. It has been shown that the multipartite entanglement may be a useful device to differentiate the various phases of the system. Multipartite entanglement may be studied through the quantum Fisher information (QFI) which does not permit to obtain a quantitative estimate, but it enables to differentiate between different types of it. This thesis work wants to be in continuity with the idea to study QPTs by using multipartite entanglement, we will work with a particular phase transition that is not a ground-state phase transition but it involves all the states of the energy spectrum. The phenomenon we are talking about is the many-body localization (MBL), that is a finite-temperature phase transition of a quantum system from an ergodic phase to a localized one. The models with which we work describe interacting fermions subject to a disordered external field in 1D; we generate disorder in two different ways: the first way is to build up an external random on-site potential, the second one is to use a quasi-periodic potential. It is observed that for strong disorder the eigenstate thermalization hypothesis (ETH), which is in few words the quantum generalization of classical ergodic hypothesis, fails; in a many-body localized phase the system does not thermalize and so it maintains information of its initial conditions during its evolution. Quantitatively, the quantum thermalization is studied using random matrix theory; if the system does thermalize the level spacing statistics of the Hamiltonian follows a given distribution of the Wigner-Dyson type, which is called `Gaussian orthogonal ensemble', in contrast, if the system does not thermalize, the Hamiltonian level spacing statistics follows a distribution which is Poissonian. Another important feature of MBL is the non-thermal behaviour of entanglement entropy (the von Neumann entropy of a partition of the system that quantifies bipartite entanglement). In particular, it is observed that in the ergodic phase the entropy has an extensive (thermal) behaviour by the fact that the system does thermalize; in the localized phase the system does not thermalize and so the entanglement entropy has an intensive behaviour. Phenomenologically, MBL is tested on cold-atom platforms, after a long time evolution, it is observed as for weak disorder the system erases any information of its initial conditions due to thermalization, for strong disorder the system maintains memory of initial conditions. Therefore, the original part of this thesis is a numerical study of this type of phase transition using the QFI. The numerical approach is necessary because the insertion of disorder makes the system not tractable analytically, in particular, the complexity of the problem is due to the concomitant presence of interactions and disorder. The MBL is a transition that manifests itself on Hamiltonian's excited states and for this the numerical tool used is the exact diagonalization; once the Hamiltonian is diagonalized, we can evaluate the QFI by the computation of various correlation functions and so we have information on multipartite entanglement. We study the QFI using the standard procedure described in fixing Hamiltonian's eigenstates on which to perform the computation and varying the disorder (and vice versa). By a comparison with entanglement entropy we show that this computation is incomplete, in particular it emerges that there are some states that are bipartite entangled according to the entropy computation but they are not entangled according to our QFI computation. The above computation is made complete by taking into account anisotropies that are contained in models as those considered. We show that when disorder is absent the behaviour of entanglement in the spectrum is strongly irregular, there are states that do not seem to show any form of entanglement, others that show at least bipartite entanglement and others that show entanglement more than bipartite (multipartite). What emerges is that at the edges of the spectrum the entanglement is typically bigger than at the center of the spectrum in the sense that more spins tend to correlate with each other. When the disorder is switched on, in the ergodic phase, we observe as at the center of the spectrum at least bipartite entanglement emerges, while at the edges of the spectrum states does show at least 3-partite entanglement. When the disorder exceeds a certain critical value that depends on the model with which we work, the system becomes localized; in terms of entanglement what we observe is that any form of entanglement tends to disappear. Physically, the observations in the ergodic phase are related to the fact that in this phase the system does thermalize, so its entanglement entropy is extensive and the amount of bipartite entanglement is highly non negligible anywhere in the energy spectrum excluding edges. Instead, in the localized phase, information does not spread along the chain, the system does not reach equilibrium and maintains memory of its initial conditions. In some sense, it describes a situations as if spins do not talk each other and so any form of correlation between them is absent. This thesis work represents a starting point for the study of disordered quantum systems using QFI and so multipartite entanglement. In the future, we can think to extend our analysis to other models that show MBL or, in general, to characterize new phases of matter not necessarily at zero temperature and to shed light on phases of which little is yet known.