Periodic Reporting for period 1 - quasiTENS (Quantum Systems Investigated through Tensor Network States) Reporting period: 2017-10-01 to 2019-09-30 Summary of the context and overall objectives of the project The results of this project all lie in the area of many-body localisation (MBL). Systems which display MBL are perfect heat insulators. That is, if part of an MBL system is heated up, the heat does not propagate but instead gets locally stuck in the system. MBL systems have been proposed as constituents of quantum computers due to their ability to store quantum information at non-zero temperature. So far, MBL has been realised experimentally in optical lattices filled with ultra-cold atomic gases, chains of trapped ions and nitrogen-vacancy centres. There is also ongoing work on realising MBL in genuine solid state systems.This project gives important new insights into MBL systems and their ability to store quantum information: In the first part of the project, the length scales over which heat locally propagates in effectively one-dimensional MBL systems were calculated. In the second part, it was shown mathematically rigorously that MBL systems with certain symmetries are able to protect quantum information at non-zero temperature. Finally, the third part sheds light on the hotly debated question on whether MBL exists in effectively two dimensions. These insights will help to quantify the practical importance of MBL systems for technological applications. Moreover, novel analytical and numerical tensor network methods are developed for the description of MBL systems, further establishing the high relevance of tensor network techniques for the description of quantum matter. Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far The research carried out during the course of this fellowship shed light on the following three areas of MBL:1. We obtained an algorithm to exactly calculate quantities which describe how far heat can locally propagate in small MBL systems. We also found an approach to approximately determine those quantities for larger MBL systems, partially improving over previous methods.2. Dr Wahl showed mathematically rigorously that effectively one-dimensional MBL systems with certain symmetries can protect quantum information at arbitrary temperature. 3. A code was written and used to simulate two-dimensional MBL systems. The strict existence of MBL in effectively two dimensions is debated, with theoretical work refuting it, whereas optical lattice experiments are affirmative. This apparent contradiction is resolved by the insight that strongly disordered two-dimensional systems might be perfect heat insulators (i.e. have MBL) on all experimentally relevant time scales, whereas in the theoretical limit of infinite observation times, they do conduct heat over arbitrary distances. Dr Wahl's code describes the behaviour of such systems on experimental time scales, which is the practically most relevant issue. On the available time scales, we obtained a transition from a heat-conducting regime to a heat-insulating regime as the disorder strength is increased, just as experimentally observed. The extracted transition point is consistent with the latest experiments.All results were published and presented at various conferences and seminars. Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far) 1. We provided the first method to exactly construct local integrals of motion for one-dimensional MBL systems of system sizes that can be captured by exact diagonalisation. 2. Dr Wahl provided a mathematical proof for the stability of symmetry-protected topological MBL systems. This is a major advancement, as rigorous proofs are extremely rare in the field of MBL, and it further emphasises the versatility of tensor network methods.3. A specific tensor network approach - two dimensional quantum circuits - provided the first simulations of optical lattice experiments which display two-dimensional MBL. Our results constitute the first theoretical piece of evidence for the existence of MBL in two dimensions on experimental time scales.