Locality and many-body quantum spectra

This thesis is concerned with many-body quantum dynamics in lattice models. Our focus is on the spectral properties of Floquet operators that describe local interactions in one dimension. Physical properties are expressed as multiple sums over paths in the many-body Fock space, and this leads naturally to interpretations in terms of interference effects, or of paired paths. By averaging properties over ensembles of random systems we wash out non-universal interference effects, and study the structures which survive. We first address the ergodic phase of quantum dynamics. In this setting random matrix theory (RMT) and the eigenstate thermalisation hypothesis (ETH) are expected to describe spectral properties. This can be understood through the diagonal approximation to sums over paired paths. We show how the diagonal approximation must be extended in systems with local interactions, and that as a result there are deviations from RMT and the ETH which, at finite times, diverge with system size. Our focus is primarily on the spectral form factor (SFF) and related quantities, and our approach is to express averaged sums over pairs of paths in terms of a transfer matrix that acts in the space direction. We then show how, within this framework, the transition from a many-body localised (MBL) phase to an ergodic one can be viewed as symmetry breaking. This transition in dynamics has sharp signatures in spectral statistics, so it is natural to consider the SFF. The transfer matrix generating the SFF has symmetries associated with translations in time: in the ergodic phase one of these symmetries is broken, whereas in the MBL phase it is not. For the Floquet models we consider the relevant symmetry group is that of a clock model. We introduce an order parameter for the broken symmetry, and show how the applicability of the diagonal approximation can be understood as a form of long-range order.