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  EPSRC DTP PhD Studentship in The Mathematics of Exotic Phases: Fractons

   Cardiff School of Mathematics

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  Dr P Naaijkens, Dr F Flicker, Dr S Wood  No more applications being accepted  Funded PhD Project (Students Worldwide)

About the Project

The School of Mathematics at Cardiff University are delighted to offer a fully funded EPSRC studentship as part of the Quantum Materials (QuMat) Interdisciplinary Doctoral Training Hub (IDTH), starting in July or October 2023.

Project Summary

The goal of this project is to develop a rigorous and general mathematical framework for fractonic phases of matter. Fractons are exotic quasiparticles which have restricted mobility, and have potential applications in e.g., quantum error correcting. There are several proposals on the realisation of such exotic phases in actual quantum materials.

The starting point is 2D gapped ground states with topological order. Such states support anyonic quasiparticles with exotic exchange statistics. Mathematically, anyons are described by what is a called a fusion category. This rich algebraic structure appears in many different contexts in mathematics. Here, it encodes all interesting physical properties of the anyons and can be found by studying (superselection) sectors of the theory. These correspond to different irreducible representations of the observable algebra. This structure is invariant under enlarging the system (if we only add product states), or gently changing the Hamiltonian whilst preserving the spectral gap.

Foliated fracton order (FFO) in 3D is a generalisation of this: instead of adding product states, we are allowed to add 2D topologically ordered systems. Hence a stack of uncoupled 2D topologically ordered states is trivial from this point of view. Many non-trivial examples with fracton quasiparticles have been found. These examples have a “foliation” structure, where the models are built up from 1D or 2D “leaves” coupled in non-trivial ways.

The goal of this project is to develop the mathematical tools to look at classes (or phases) of FFOs at the same time, based on the sector theory for 2D topological order. Our approach, rooted in operator algebra theory, allows us to work directly in the thermodynamic limit. Guided by various examples, we will develop a rigorous version of the notion of FFO. Fracton models have infinitely many superselection sectors, but many can be considered equivalent. This leads to the notion of a quotient superselection sector. One of the main tasks will be to translate this notion into the operator algebraic setting. This will pave the way for a theory of fusion for fractons.

A second goal of the project is to systematically develop the “gauging” of a global symmetry in this framework. This is not only useful to describe various FFOs, but is also relevant for the study of symmetry enriched topological (SET) phases in 2D. Hence we can describe many examples in a unified framework.

Research Environment

The successful candidate will be joining the Geometry, Algebra, Mathematical Physics & Topology (GAPT) group. The group organises a weekly seminar and regular reading groups focussing on a particular topic. You will also be joining 5 other students as part of the QuMat cohort. The School of Mathematics is housed in a new, purpose-built building, which it shares with the School of Computer Science

Training and Development Opportunities

In this project you will become an expert in an exciting area of research, which combines diverse fields such as operator algebra, quantum information theory, tensor categories, and condensed matter physics. This learning will be supported by weekly meetings with the primary supervisor, Pieter Naaijkens (Mathematics), and regular meetings (at least once a month) with the second supervisor, Felix Flicker (Physics), to allow to consider the project both from a mathematics as well as a physics point of view.

You will have access to the UK-wide MAGIC courses, which provide a wide range of courses in advanced mathematics. As part of the QuMat IDTH Cohort you will in addition benefit from bespoke training in quantum materials, the opportunity to regularly interact with the PhD students from physics, and the chance to get involved in the organisation of student-led events.

How to apply

The deadline for applications is Friday 27th January 2023.

Applicants should apply through the Cardiff University online application portal. Applicants should select Doctor of Philosophy in Mathematics, with a start date of July or October 2023. In the research proposal section of your application, please specify the project title “The Mathematics of Exotic Phases: Fractons” and supervisor “Dr Pieter Naaijkens”. In the funding section, please specify that you are applying for advertised funding from the EPSRC-funded QuMat IDTH.

Please include:

  • your academic CV
  • a personal statement
  • two completed references
  • current academic transcripts

The typical academic requirement is a bachelor’s or master’s degree with a minimum classification of 2:1 in a relevant discipline (mathematics, theoretical physics, or closely related). We also welcome applications from non-traditional candidates. 


EPSRC DTP studentships are available to home and international students. Up to 30% of our cohort can comprise international students, once the limit has been reached, we are unable to make offers to international students.  

We are still accepting applications from international applicants. International students will not be charged the fee difference between the UK and international rate. Applicants should satisfy the UKRI eligibility requirements. 

Cardiff University is committed to support students from a range of backgrounds and circumstances. Where needed, we will work with you to take into consideration reasonable project adaptations (for example to support caring responsibilities, disabilities, other significant personal circumstances) as well as flexible working and part‐time study requests, to enable greater access to a PhD. All our supervisors support us with this aim, so please feel comfortable in discussing further with the listed PhD project supervisor to see what is feasible. 

Cardiff University is committed to supporting and promoting equality and diversity and to creating an inclusive environment for all. We welcome applications from all members of the global community irrespective of age, disability, sex, gender identity, gender reassignment, marital or civil partnership status, pregnancy or maternity, race, religion or belief and sexual orientation.

English language requirements

Applicants for whom English is not their first language must demonstrate proficiency by obtaining an IELTS score of at least 6.5 overall, with a minimum of 5.5 in each skills component or (or equivalent qualification).

Computer Science (8) Mathematics (25) Physics (29)

Funding Notes

The studentship funded by the EPSRC DTP is for 3.5 years and covers tuition fees, an annual tax-free living stipend in line with UKRI minimum rates (currently £17,668) and includes access to a Research Training Support Grant (currently £4000).
If you have a disability you may be entitled to a Disabled Students’ Allowance (DSA) on top of your studentship.

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