Non-perturbative and stochastic approaches to many-body localization

A central question in physics concerns the complexity of any given system. What is the nature of quantities characterising the system behaviour, and which characteristics can be safely ignored? How sensitive is the system against external perturbations and how predictable is its dynamics? How many parameters need to be controlled to manipulate the system in a specific way? These questions not only lie at the heart of fundamental fields such as statistical mechanics, but also provide the key to the characterisation of equilibrium phases and nonequilibrium dynamics and pervade practical applications in quantum information, control and sensing.

Meeting the urgent need to understand these questions in the quantum setting, the scientific community has recently identified a key paradigm that should hold many of the required answers. This concerns large but finite many-body systems with disorder, low spatial dimensionality and local interactions. These ingredients have been seen to conspire in a persistence of local memory of initial conditions, a fascinating phenomenon known as many-body localisation (MBL). The understanding of this phenomenon is still in its infancy. The theoretical arguments for MBL are thorough, but also in essence qualitative, while quantitative insight to date mainly arrives from numerical investigations of relatively small (in real-world terms) systems. This state of affairs has given rise to a proliferation of characterisations based, among others, on transport, thermalisation, entanglement, dynamics, whose detailed mutual relationships are mostly unclear. These shortcomings become even more pressing as the very first dedicated experiments target yet another set of characteristics, the directly observable consequences in real-world systems.

In this project, we take a step back and ask the question: which level of understanding could be accepted to constitute a rather complete description of many-body localisation and delocalisation? Taking a leaf out of the book from non-interacting systems, we argue that this has to involve, as a central piece, a statistical description in quasi-one dimensional settings. We approach this goal from (I.) a rich one-dimensional model system originating from lattice field theory (the Schwinger model, a version of quantum electrodynamics) which is amenable to a detailed treatment and (II.) a stochastic description based on fundamental composition rules of the density matrix (the key object to describe entanglement), allowing comprehensive access to the generic system behaviour. Amongst the observable consequences, we aim to (III.) discriminate between general and system-specific aspects in the MBL phase and of the phase transition, including experimentally testable signatures, and (IV.) extend the considerations to topological phases, where the system displays order due to intricate quantum effects.

The three main components are academic impact, training of highly qualified personnel, and long-term technological impact.

1. Academic Impact
This project is well placed to have several significant academic impacts: (a) via a detailed quantitative description which provides a game-changing level of understanding of disordered many-body systems; (b) via powerful newly developed and transferred methods involving the combination of stochastic and field-theoretical approaches; (c) via the introduction of a flexible model with scope for novel phenomena and interdisciplinary qualities arising from its origin in lattice field theory. The ensuing unified perspective on complementary MBL characteristics will also be critical to identify experimentally relevant signatures. The results will therefore be relevant for a large part of the condensed matter community, as well as notable sections of field theorists and mathematical physicist. To maximise this impact, we have selected an adventurous range of topics that connects fundamental aspects with a broad variety of specific signatures. Results will be disseminated in high-impact publications, presentations and discussions at conferences and seminars; data and where applicable code will be made publicly available via the Lancaster Research Data Management system, while preprints, postprints and publications will also be disseminated via preprint repositories and the Lancaster Research Portal.

2. Training of highly qualified personnel
The project will provide comprehensive training to a PDRA in a highly active topic of fundamental research, enabling them to acquire considerable skills covering the numerical, analytical, and conceptual aspects of the work. The PDRA will have ample opportunity to raise their standing through their involvement in high-impact publications; via conference participation with talks and discussions they will also have the opportunity to present their work to a broad international audience. The about 20 PhD and 10 MPhys project students in Lancaster's Condensed Matter Theory will benefit academically from the exposition to the proposed research in informal discussions and weekly group meetings.

3. Long-term technological impact
A principal motivation to study phenomena such as many-body localisation arises from the desire to design quantum devices with dedicated functionality. The proposed research directly addresses the critically important question of the interplay of disorder and interactions, main factors in the design and performance of any quantum device. As a main pathway to translate the insight into these questions into long-term technological applications, we will carry out dedicated work to identify the experimental signatures and will actively liaise with experimental colleagues; locally, with experimental colleagues in the Lancaster Quantum Technology Centre, Material Science Institute and Ultra-Low Temperature Group; nationally, via participation in events of the Quantum Hubs and national institutes such as Royce and NGI in Manchester as well as through the joint CDT NOWNANO; and internationally, via conferences and discussions.