Outline

A major discovery in contemporary condensed matter physics was the existence of phases of matter that do not fit in Landau’s theory, but instead are “topologically ordered”. There, certain properties depend on topological properties of the underlying system. This makes them attractive candidates to build a quantum computer or memory, in particular using quasi-particles called anyons. There are many interesting mathematical connections as well: besides topology, operator algebras, representation theory and modular tensor categories play an essential role.

Supported by this studentship, you will study the mathematical physics of 2D gapped topological phases, with applications to quantum information theory, to find an encompassing theory that covers all known examples. A roadblock is that there is no good way to extract the properties of the anyons from the underlying system. You will use operator algebraic techniques to develop tools to do this directly in the thermodynamic limit, using an approach pioneered by Naaijkens. Many fundamental questions are still unanswered today; some of whose you will study using this studentship. For example:

-Extend present framework to non-abelian anyons and other models. Allow for the case of approximate localisation.

-A microscopic theory of anyon condensation; development of renormalisation group (RG) flow methods.

The third issue is to develop quantum information (QI) methods for systems with infinitely many degrees of freedom:

-QI in infinite dimensions, such as channel capacities, using subfactors; QI in relativistic QFT.

Naaijkens and co-authors have shown that there is a natural connection to anyon theories. These themes each have a different mathematical focus. In conjunction with the student the supervisors will select one of them, to ensure that the studentship will be aligned with the student’s interests and strengths, and that realistic goals are set.

This studentship is embedded into the Geometry, Algebra, Mathematical Physics and Topology group (GAPT) at the School of Mathematics. This research group offers the unique possibility to engage with experts in different fields related to the topic, such as operator algebras, mathematical quantum physics, and topology. The student will furthermore benefit from the many weekly seminars offered by the School. Additional training opportunities arise from access to the MAGIC network and Cardiff’s Doctoral Academy. The student will work in an interdisciplinary international environment from the outset and will be part of an LMS Joint Research Group with Lancaster, Nottingham and York, and of the professional networks of the supervisors in the EU and the US.

What is funded

The 3.5 year studentship includes UK/EU fees, stipend (amount for 2020/21 is £15285) and a research training grant to cover costs such as research consumables, training, conferences and travel.

Eligibility

UK/EU applicants only. UK Research Council eligibility conditions apply

A 1st or upper 2nd class UK Honours degree (or equivalent) and/or a Master’s degree is required in mathematics or a related subject.

Applicants for whom English is not their first language must demonstrate their proficiency by obtaining an IELTS score of at least 6.5 overall, with a minimum of 6.0 in each skills component.

How to Apply

Applicants should apply through the Cardiff University online application portal for a Doctor of Philosophy in Mathematics with an entry point of October 2020.

In the research proposal section of your application, please specify the project title and supervisors of this project. In the funding section, please select "I will be applying for a scholarship / grant" and specify that you are applying for advertised funding from EPRSC DTP.

Shortlisted candidates will be invited to attend an interview in April.