Abstract
Recent years have seen a resurgence of interest in fundamental questions regarding the out-of-equilibrium behaviour of isolated quantum systems. Due to both new experimental techniques, such as those involving cold atomic gases, and progress in the theoretical understanding of closed quantum systems, it is now possible to explore fundamental questions about thermalisation and the applicability of statistical mechanics. The interplay between disorder, interactions and quantum interference gives rise to interesting phenomena, such as many-body localisation (MBL), which has a key role in the non-equilibrium behaviour of such systems.Realistic models of disordered, interacting quantum systems are largely intractable analytically. Their numerical study is limited by the difficulty of simulating quantum systems with classical computers, making it challenging to separate genuine \thermodynamic" results from finite size effects. In this thesis we consider some more tractable mean- field models, as a starting point to investigate specific aspects of ergodicity and localisation transitions in many-body systems.
The first part is devoted to the outstanding question of whether MBL systems undergo two separate ergodicity and localisation transitions when disorder strength is increased. The two transitions are separated by a putative \bad metal", nonergodic extended (NEE) phase, in which ergodicity is broken but the eigenstates are not exponentially localised. We show explicitly the existence of the NEE phase in a random matrix model, and characterise it using the local resolvent in an unusual scaling limit.
In the second part we consider a spin glass model, in which quantum effects are introduced by a transverse magnetic field. A refined equilibrium phase diagram, going beyond the quasi-static approximation, is obtained with a numerically exact diagrammatic Monte Carlo approach. We discuss the difference between the ergodic, eigenstate and clustering transitions.
Finally, we consider a quantum model that shares some of the peculiar low-temperature properties which have brought the Sachdev-Ye-Kitaev (SYK) model under the spotlight of the string theory community. In our model such properties are understood in terms of the glassy dynamics of a corresponding classical stochastic system.
| Date of Award | 1 Aug 2019 |
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| Original language | English |
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| Supervisor | Pierpaolo Vivo (Supervisor) & Joe Bhaseen (Supervisor) |