Strongly-entangled topological matter

This project will advance the theoretical understanding of the new type of matter called topological matter, which emerges in strongly-interacting quantum systems. By performing numerical simulations, the project will investigate fundamental properties of topological matter, such as its geometry and quantum entanglement. This will provide feedback to experiments on how to realise new topological matter in materials like bilayer graphene.

Topology is a branch of mathematics that describes properties of objects which do not change under local perturbations. For example, a soccer ball is the same as a rugby ball because we can slowly stretch one into the other. Curiously, in certain semiconductor materials (like the ones used to build transistors and solar cells) there are phases of matter which are also insensitive to local perturbations. This topological matter is very different from ordinary matter (like water or ice) because it represents a collective state that emerges when many quantum particles interact, similar to superfluids and superconductors.

Topological matter forms a very active field of modern condensed matter physics, for at least three reasons. First, topological matter has been seen in many beautiful experiments, starting with the original discovery of the fractional quantum Hall effect in the 1980s. Second, topological matter represents a major challenge for theoretical physics, because it cannot be explained by traditional solid state theories based on "symmetry breaking". Third, topological phases have very rich and unexpected properties, for example their low-energy excitations behave as "quasiparticles" which are more general than the Standard Model of particle physics (i.e., they are neither bosons nor fermions). Recent discovery of one such quasiparticle - the "Majorana fermion" - has attracted much public attention, and current research focuses on harnessing the power of the Majoranas to perform quantum computing. Thus, topological matter may have an important role to play in future quantum technologies.

This project will advance the understanding of topological matter in the systems of strongly interacting particles, where many fundamental problems remain open. The project will investigate the role of geometry in topological matter, which determines their elastic and thermal properties. Furthermore, the project will investigate quantum correlations ("entanglement") in topological matter, with the goal of understanding how topological order could be enabled to survive at high temperatures. This would represent an important practical advance as most of topological matter is currently realised only at cryogenic conditions. Finally, the project will establish close connection to experiments that seek to realise topological matter in new materials. By developing and applying new numerical algorithms, the project will identify interaction-driven topological phenomena that can be experimentally accessed in bilayer graphene, in particular the phases that host the Majorana fermions or even more exotic "parafermion" quasiparticles.

Physics has traditionally been reductionist: to understand something, we divide it into smaller parts. However, some of the most interesting phenomena in modern condensed matter physics cannot be understood in this way because of interactions between particles. Topological phases of matter are such emergent phenomena. The research into topological matter currently lies at the frontier of modern physics; for example, this has been acknowledged by the EPSRC as one of the "Grand Challenges", and also by the international collaboration on the "Many-Electron Problem" supported by the Simons Foundation.

Apart from contributing to a fundamental challenge of theoretical physics, this project will also impact the experiments in two different ways, which may lead to technological applications in the long term. First, the project will benefit from being at the interface between condensed matter physics and quantum information. A synergy between these two areas is expected to be the key for designing robust quantum devices for real world applications. The proposal focuses on bilayer graphene, which connects with the UK's focus on graphene technology. Second, although the research in this proposal is theoretical, one of its goals is to perform extensive numerical modeling of real systems and thereby guide the experiments towards realising topological matter. In particular, the project aims to provide feedback to experiments on how to realise complex topological states whose quasiparticles (the parafermions) may be vital for designing fault-tolerant quantum computers which may revolutionise future technology.

The impact of this project will foremost be generated by establishing my group in Leeds as a hub for research on topological phenomena in strongly-interacting systems. In particular, this will be achieved by developing computational methods based on exact diagonalisation and tensor networks. This will strengthen the UK position in state-of-the-art numerical simulations of topological systems, which is currently not on the level of the US, Canada and Europe (particularly Germany).

The impact and visibility of this research will be achieved via the following routes.

1. The dissemination plan consists of several high-impact publications (of the level of Phys. Rev. Lett. or Phys. Rev. X). Additionally, a review article on topological phenomena in bilayer graphene, co-authored with at least one experimental colleague, is planned to maximise the impact.

2. Invited talks at workshops, conferences and academic institutions, in which I have extensive track-record. In February 2017, I am also co-organising a large international meeting funded by the Royal Society, together with colleagues at Oxford (Arijeet Pal, Steve Simon), Cambridge (Ulrich Schneider) and London (Sir Michael Pepper). This meeting will be an additional opportunity to generate international impact.

3. A project-dedicated webpage which will contain, on the one hand, a popular introduction and summaries to the publications. On the other hand, the page will be an interface to an open-source software repository. My existing software is GPL-licensed and available on my homepage.

4. Train a PDRA in both scientific and transferrable skills, and contribute to the education of students (especially via summer research projects).

5. Engage young audiences through UCAS School Open Days, by participating at Leeds Science Festival, and on a regular basis via social media (Twitter, Facebook pages).

6. Develop personal contacts and collaborations within the UK, and bring the international experts for shorter and longer term stays in Leeds. In the second year of the grant, I will organise a workshop on many-body topological phenomena. The workshop will fit naturally into the existing framework of the Leeds Symposia on Topological Quantum Computation (whose 19th session I have co-organised this year), which will also minimise its cost on this project.