Two of the exciting forefronts in condensed matter theory revolve around the ideas of topology and non-equilibrium dynamics in quantum systems of many particles. For example, in low-dimensional electronic systems (such as graphene) interactions can conspire with topology to give rise to new phases of matter, whose quasiparticles are radically different from ordinary electrons in our 3D world – they have a fraction of an electron charge or even fractional statistics . Another exciting direction stems from asking a seemingly basic question: what kind of systems can be described by statistical mechanics? We know that is true for most systems in our classical everyday world; in the quantum world things are more subtle and recently a class of systems (called “many-body localized” systems) have attracted a lot of attention because they fail to thermalize, even at infinite times . How do these two sets of ideas connect together? Through an unexpected possibility of using them to build a quantum computer! For example, states with quasiparticles that have fractional statistics may be used as "0" and "1" qubit states. Such qubits are “protected” by topology, i.e. they are more resilient to decoherence effects. On the other hand, many-body localization also protects the system from decoherence and may be used to extend the lifetimes of qubit devices. In this project we will rely on the connections between condensed matter and quantum information to learn more about the systems above. We will explore the possibility of creating new types of phases with exotic quasiparticles in systems made of graphene and semiconductor materials. In particular, we will search for the so-called non-Abelian quasiparticles, such as “Majorana fermions” . We will also study their dynamical properties, which will reveal connections between topology, geometry and quantum mechanics that are surprisingly reminiscent of the theory of quantum gravity!  Finally, we will investigate thermalization and many-body localization in different models, trying to gain a better understanding of what systems can in principle reach thermal equilibrium and which ones can not, in anticipation of the first experiments on this type of phenomena in systems of ultracold atoms . This project will allow you to expand your analytical skills (topological field theory, advanced quantum mechanics), as well as numerical skills through the algorithms based on entanglement (matrix-product states and density-matrix renormalization group). It will provide an opportunity to interact with other groups in Leeds and worldwide. Although we will study fundamental questions, most of what we will do will have strong experimental relevance, therefore we will also be talking to the experimentalists.