Quantum Order, Entanglement and Localization in Many-Body Systems

Abstract

The interplay of disorder and interactions can have remarkable effects on the physics of quantum systems. A striking example is provided by the long conjectured---and recently confirmed---phenomenon of many-body localization. Many-body localized (MBL) phases violate foundational assumptions about ergodicity and thermalization in interacting systems, and represent a new frontier for non-equilibrium quantum statistical mechanics.

We start with a study of the dynamical response of MBL phases to time-dependent perturbations. We find that that an asymptotically slow, local perturbation induces a highly non-local response, a surprising result for a localized insulator. A complementary calculation in the linear-response regime elucidates the structure of many-body resonances contributing to the dynamics of this phase

We then turn to a study of quantum order in MBL systems. It was shown that localization can allow novel high-temperature phases and phase transitions that are disallowed in equilibrium. We extend this idea of ``localization protected order'' to the case of symmetry-protected topological phases and to the elucidation of phase structure in periodically driven Floquet systems. We show that Floquet systems can display nontrivial phases, some of which show a novel form of correlated spatiotemporal order and are absolutely stable to all generic perturbations.

The next part of the thesis addresses the role of quantum entanglement, broadly speaking. Remarkably, it was shown that even highly-excited MBL eigenstates have low area-law entanglement. We exploit this feature to develop tensor-network based algorithms for efficiently computing and representing highly-excited MBL eigenstates. We then switch gears from disordered, localized systems and examine the entanglement Hamiltonian and its low energy spectrum from a statistical mechanical lens, particularly focusing on issues of universality and thermalization.

We close with two miscellaneous results on topologically ordered phases. The first studies the nonequilibrium ``Kibble-Zurek'' dynamics resulting from driving a system through a phase transition from a topologically ordered phase to a trivial one at a finite rate. The second shows that the four-state Potts model on the pyrochlore lattice exhibits a ``Coulomb Phase'' characterized by three emergent gauge fields.