## Periodic Reporting for period 1 - TopOutEq (Topological and Correlated Quantum Systems out of Equilibrium)

Reporting period: 2017-01-01 to 2018-12-31

Understanding systems out of equilibrium is one of the great challenges of modern science. In nature non-equilibrium behaviour is very common. In contrast, true equilibrium situations in which a system is characterised by a small number of time independent thermodynamic quantities such as temperature appear only as idealised cases. Moreover, it is also of technological relevance, for example for the fast operation of electronic devices. In condensed matter physics complex behaviour arises from the interaction of a large number of basic degrees of freedom. It is fascinating to uncover the richness of this behaviour, and to understand the universal principles that organise the physical world. One of the established principles is the Landau theory whereby states of matter are classified according to their broken symmetries. In recent years distinct ways were uncovered showing how this description can break down. This project has focused on condensed matter systems out of equilibrium beyond the Landau paradigm in regimes in which the laws of quantum mechanics are important. The objective was to investigate the non-equilibrium behaviour of interacting and topologically ordered quantum systems. I have focused on analytically tractable model systems and on calculating various observables in order to make direct contact with recent experiments.

One of the paradigmatic settings of non-equillibrium physics is a quantum quench, e.g. a sudden change of a parameter in the Hamiltonian. From the response of the system one can directly learn about the excitations and relaxation processes. At the beginning of this project and following Project B (PB3+PB4 of my proposal) I have work on these quantum quenches in disordered systems. The study of localization phenomena – pioneered in Anderson’s seminal work on the absence of diffusion in certain random lattices – is receiving redoubled attention in the context of the physics of interacting systems showing many- body localization. While in these systems the presence of quenched disorder plays a central role, suggestions for interaction-induced localization in disorder-free systems appeared early in the context of solid Helium. However, all of these are limited to settings having inhomogeneous initial states. Whether quenched disorder is a general precondition for localization has remained an open question. I have provided an explicit example to demonstrate that a disorder-free system can generate its own randomness dynamically, which leads to localization in one of its subsystems. The model is exactly soluble, thanks to an extensive number of conserved quantities, which can be identified, allowing access to the physics of the long-time limit. The model can be extended while preserving its solubility, in particular towards investigations of disorder-free localization in higher dimensions. Localization phenomena are often diagnosed, in experiment and simulation, via the dynamical response to a global quantum quench. The underlying idea is to examine if a system thermalizes, thereby losing memory of the initial state, or whether this memory persists in the long-time limit. We have used this as a tool for diagnosing non-ergodicity and for making experimental predictions.

I have also worked on dynamical excitations in topological magnets, so called Quantum Spin liquids (QSL). In a collaboration with inelastic neutron experimentalists I have calculated the dynamical response for the seminal Kitaev QSL in a a magnetic field, which has allowed us to observe signatures of spin fractionalisation into Majorana fermions. I have developed a new real space quench formalism for this calculation as proposed in subproject PB4.

Finally, I have worked on interacting superconducting wires with Majorana zero modes which are discussed in the context of topological quantum computations.

I have also worked on dynamical excitations in topological magnets, so called Quantum Spin liquids (QSL). In a collaboration with inelastic neutron experimentalists I have calculated the dynamical response for the seminal Kitaev QSL in a a magnetic field, which has allowed us to observe signatures of spin fractionalisation into Majorana fermions. I have developed a new real space quench formalism for this calculation as proposed in subproject PB4.

Finally, I have worked on interacting superconducting wires with Majorana zero modes which are discussed in the context of topological quantum computations.

First, I have invented a new model of disorder-free localisation which has potential applications in quantum information technology. For example, it could possibly extend the regime for topological quantum computations to much higher temperatures than currently possible. Second, my collaboration with neutron scattering experimentalists has found signatures of spin fractionalization in a QSL candidate materials. Hence it has brought us an important step closer towards the discovery of this long-sought after phase of matter. Third, the methodological advances of these works are of much wider use than these concrete projects.