Fock space localisation in the disordered transverse-field Ising Model

In recent years, there has been considerable interest both theoretically and experimentally in the field of many-body localisation (MBL). This thesis considers aspects of many-body localisation in Fock-space for the one-dimensional disordered transverse-field Ising model, focusing on the behaviour in the ergodic and the MBL phases.
In contrast to the ergodic phase, systems in the MBL phase fail to thermalize under their own dynamics, and memory of their initial state survives locally for arbitrarily long times. As such behaviour falls outside the paradigm of conventional statistical mechanics, the dynamics in the MBL phase is of fundamental interest. At the same time, even within the ergodic phase but at disorder strengths preceding the MBL transition, the dynamics is known to be anomalously slow. Clearly, understanding aspects of the time evolution in both phases is of interest.
One of the simplest questions one can ask is, given an initial state, how does it evolve in time on the associated Fock-space graph, in terms of the distribution of probabilities thereon? In chapters 3 and 4, we discuss our work on probability transport, where we find it exhibits a rich phenomenology, which is markedly different between the ergodic and many-body localized phases. The dynamics is, for example, found to be strongly inhomogeneous at intermediate times in both phases, but while it gives way to homogeneity at long times in the ergodic phase, the dynamics remain inhomogeneous and multifractal in nature for arbitrarily long times in the localized phase. We also show that an appropriately defined dynamical lengthscale on the Fock-space graph is directly related to the local spin autocorrelation, and as such sheds light on the (anomalous) decay of the autocorrelation in the ergodic phase, and lack of it in the localized phase.
The MBL phase transition is an eigenstate phase transition, meaning the properties of the eigenstates differ in the ergodic and the MBL phases. One of the most fundamental questions one can ask is how are the amplitudes of the eigenstates distributed across the Fock-space graph? It is well known in the MBL literature that this can be understood by examining the generalised IPR. In chapter 5, we discuss our work on IPR distributions and our understanding of the origins of multifractality in MBL. Here, we find the distributions of the IPRs are different in the ergodic and the MBL phases, with a larger inhomogeneity in the distributions in the MBL phase than in the ergodic phase. We also discuss generally the origins of multifractality in MBL, and show that the multifractality arising in the interacting problem can be regarded as having its origins in the non-interacting but many-particle limit.
It has been well known since the early days of Anderson localisation that the extent to which nearest neighbour eigenstates overlap - or equivalently are found in the same region of space - differs significantly between localised and extended states. In chapter 6, we define new order parameters for the MBL transition, which probes this difference in overlap in Fock-space. A scaling theory is then constructed, which is closely connected to that for the inverse participation ratio and Fock-space correlations. It yields a volume-scale in the ergodic phase and a length-scale in the MBL phase. The critical properties are in agreement with previous work that the MBL transition is indeed Kosterlitz-Thouless-like. We also show using an appropriately defined probability distribution that the extent to which nearest neighbour in energy eigenstates overlap has little dependence in the ergodic and strong dependence in the MBL phase on the disorder realisation or the particular eigenstate considered. By comparing the interacting and non-interacting distributions, we shed some light on the inhomogeneity in the distributions seen in the MBL phase, alongside defining a rescaling of the distributions which collapse the results for different system sizes onto a common long-tailed curve.

Advisor: Logan, D