Aspects of localization in disordered many-body quantum systems

For a quantum system to be permanently out-of-equilibrium, some non-trivial mechanism must be at play, to counteract the general tendency of entropy increase and flow toward equilibration. Among the possible ways to protect a system against local thermalization, the phenomenon of localization induced by quenched disorder appears to be one of the most promising. Although the problem of localization was introduced almost sixty years ago, its many-body version is still partly unresolved, despite the recent theoretical effort to tackle it. In this thesis we address a few aspects of the localized phase, mainly focusing on the interacting case. A large part of the thesis is devoted to investigating the underlying “integrable” structure of many-body localized systems, i.e., the existence of non-trivial conservation laws that prevent ergodicity and thermalization. In particular, we show that such conserved operators can be explicitly constructed by dressing perturbatively the non-interacting conserved quantities, in a procedure that converges when scattering processes are weak enough. This is reminiscent of the quasiparticle theory in Fermi liquids, although in the disordered case the construction extends to the full many-body energy spectrum, and it results in operators that are exactly conserved. As an example of how to use the constructive recipe for the conserved quantities, we compute the long-time limit of an order parameter for the MBL phase in antiferromagnetic spin systems. Similar analytical tools as the ones exploited for the construction of the conserved operators are then applied to the problem of the stability of single-particle localization with respect to the coupling to a finite bath. In this context, we identify a quantum-Zeno-type effect, whereby the bath unexpectedly enhances the particle’s localization. In the final part of the thesis, we discuss several mechanisms by which thermal fluctuations may influence the spatial localization of excitations in interacting many-body states.