We study the strange nature of low-dimensional quantum systems in the presence of disorder, with a particular focus on a broad class of closed quantum system that fails to equilibrate under its own dynamics; so-called many-body localised systems. These systems comprise particles that are subject to disorder---usually introduced via an inhomogeneous magnetic field---which localise in real space as disorder passes a critical threshold. The transition into this localised regime is characterised by the spontaneous emergence of an extensive set of local conserved quantities, leading to the notion of emergent integrability.
It is precisely the nature of these conserved quantities that we tackle first. The emergence of these conserved quantities is mathematically undeniable; however, there still lies the question of what form these conserved quantities take, and moreover, how should they be constructed? In the literature, attempts to construct conserved quantities via perturbative methods are common; and rightfully so, as it is natural to attempt to extend the notion of single-particle Anderson localisation---which has been analytically solved---to the many-body regime. This involves ``dressing'' the single-particle operators of the noninteracting case with extra terms, thus creating a complete single-quasiparticle basis of the interacting system. Despite the convenience, however, such perturbative constructions depend upon assumptions that are not necessarily guaranteed. We attack these assumptions directly, and show---both analytically and numerically---that conserved quantities are distinctly nonperturbative in a paradigmatic model of many-body localisation.
Next, we consider the effect of global symmetries on the nature of many-body localisation; in particular, chiral symmetry. By definition, chiral symmetry produces a eigenspectrum that is symmetrical about zero, thus ``pairing'' each eigenstate with another of mirrored eigenenergy; however, that is only true if there exists an even number of eigenstates---what of an odd number? It is this question that motivates our use of spin-1 particles, as their odd number of spin-degrees of freedom produce a many-body Hilbert space that is, by necessity, odd-dimensioned. The joint constraints of chiral symmetry and an odd number of eigenstates produce at least one state that is pinned to zero eigenenergy---a zero mode robust to all parameter variation. We explore the phenomenology of this zero mode in the context of many-body localisation, and find that it possesses fragmented correlations that clearly distinguish it from nonzero modes that localise in a more typical fashion.
Finally, we conclude this work with an initial study into the nature of entanglement transitions in general, via the consideration of new, more-recent models. A well-known consequence of the emergent integrability central to many-body localised systems is a stark shift in the entanglement of eigenstates. Whereas, in the ergodic regime, entanglement spreads ballistically and scales extensively with the volume of the system---a so-called volume law---the localised regime is characterised by entanglement that spreads logarithmically, with a subextensive area-law scaling. This area-law scaling is not unique to many-body localised systems, as we also see it emerge in quantum circuit models with ``brick-layer'' structure. We compare these two different types of entanglement transition directly using entropy-like quantities, and attempt to map their behaviour onto random-matrix models (that are more easily understood from an analytical standpoint).
