Exploring the fuzzy nature of quantum criticality

From the boiling of water to the birth of the universe, continuous phase transitions reveal nature’s ability to organise itself in strikingly universal ways. At these critical points, microscopic details are washed away and systems display scale invariance, emergent collective excitations, and mathematical elegance captured by conformal field theories (CFTs). CFTs have transformed our understanding of two-dimensional criticality, enabling exact solutions and deep links to quantum gravity and string theory. Yet in three dimensions where many of the most important transitions occur, from magnets and superconductors to strongly interacting quantum matter, the full structure of CFTs remains largely uncharted, limited by the analytical difficulty of solving such interacting theories or even numerically extracting conformal data. Recently, an exciting way forward opened up with the discovery of the fuzzy sphere, a non-commutative geometry born from the physics of the quantum Hall effect. When electrons (or composite fermions) are confined to the lowest Landau level on a sphere threaded by a magnetic monopole, their spatial coordinates no longer commute, effectively “fuzzing” the sphere and allowing a seamless implementation of the state–operator correspondence of CFT: the first direct numerical evidence for emergent conformal symmetry in the 3D Ising transition, revealing the full operator spectrum predicted by CFT, see Zhu et al., Phys. Rev. X 13, 021009 (2023). Following up on this work, our group has shown the ground-breaking potential of this approach: we provided, and we recently extended the method to fractional quantum Hall backgrounds, demonstrating that the critical spectrum remains sharply defined even when embedded in a topologically ordered charge sector [Phys. Rev. X 15, 031007 (2025)]. These results open an entirely new arena where quantum criticality and topological order intertwine.

<div data-olk-copy-source="MailCompose" style="text-align: left; text-indent: 0px; margin-top: 1em; margin-bottom: 1em; font-family: Helvetica; font-size: 11pt; color: rgb(0, 0, 0);"> <p style="margin-bottom:11px">In this PhD project you will build on these breakthroughs to explore new universality classes and topologically enriched critical points using fuzzy-sphere regularisation. You will develop and apply advanced numerical techniques, exact diagonalisation, density-matrix renormalisation group, and conformal perturbation theory, to extract operator spectra, entanglement signatures, and universal quantities such as F-functions and operator-product coefficients. Crucially, the models studied here are directly motivated by, and can inform quantum Hall bilayer experiments in semiconductor heterostructures and graphene, where similar Landau-level physics and interlayer couplings can be realised. Our group has a strong track record of collaborating with leading experimental teams in these platforms, ensuring that the theoretical predictions you develop will be closely connected to ongoing and future measurements of exotic critical phenomena in real materials.</p> </div>

<p>Formal applications for research degree study should be made online through the <a href="https://www.leeds.ac.uk/research-applying/doc/applying-research-degrees">University's website</a>. Please state clearly the Planned Course of Study that you are applying for <em><strong>PHD Physics & Astronomy FT,</strong></em> in the research information section that the research degree you wish to be considered for is <em><strong>Exploring the fuzzy nature of quantum criticality </strong></em>as well as <a href="https://eps.leeds.ac.uk/physics/staff/4124/dr-zlatko-papic">Prof Zlatko Papic</a> as your proposed supervisor and in the finance section, please state clearly <em><strong>the funding that you are applying for are EPSRC Doctoral Landscape Award 2026/27 and School of Physics & Astronomy Studentship 2026/27.</strong></em></p> <p>Applications will be considered after the closing date of Friday 30 January 2026.  Potential applicants are strongly encouraged to contact the supervisors for an informal discussion before making a formal application.  We also advise that you apply at the earliest opportunity as the application and selection process may close early, should we receive a sufficient number of applications or that a suitable candidate is appointed.</p> <p>Please note that you must provide the following documents in support of your application by the closing date of 30 January 2026:</p> <ul> <li>Full Transcripts of all degree study or if in final year of study, full transcripts to date including grading scheme</li> <li>Personal Statement outlining your interest in the project</li> <li>CV</li> </ul> <p>If English is not your first language, you must provide evidence that you meet the University's minimum English language requirements (below).</p> <p><em>As an international research-intensive university, we welcome students from all walks of life and from across the world. We foster an inclusive environment where all can flourish and prosper, and we are proud of our strong commitment to student education. Across all Faculties we are dedicated to diversifying our community and we welcome the unique contributions that individuals can bring, and particularly encourage applications from, but not limited to Black, Asian, people who belong to a minority ethnic community, people who identify as LGBT+ and people with disabilities. Applicants will always be selected based on merit and ability.</em></p>

Applicants to research degree programmes should normally have at least a first class or an upper second class British Bachelors Honours degree (or equivalent) in an appropriate discipline. The criteria for entry for some research degrees may be higher, for example, several faculties, also require a Masters degree. Applicants are advised to check with the relevant School prior to making an application. Applicants who are uncertain about the requirements for a particular research degree are advised to contact the School or Graduate School prior to making an application.

The minimum English language entry requirement for research postgraduate research study is an IELTS of 6.0 overall with at least 5.5 in each component (reading, writing, listening and speaking) or equivalent. The test must be dated within two years of the start date of the course in order to be valid.

<p class="MsoNoSpacing" style="text-align:start; margin-bottom:24px"><strong>EPSRC Doctoral Landscape Award 2026/27</strong></p> <p class="MsoNoSpacing" style="text-align:start; margin-bottom:24px">A highly competitive EPSRC Doctoral Landscape Award, providing full academic fees, together with a tax-free maintenance grant at the standard UKRI rate (£20,780 in academic session 2025/26) for 3.5 years.  Training and support will also be provided.<br /> <br /> This opportunity is open to UK applicants only.  All candidates will be placed into the EPSRC Doctoral Landscape Award Competition and selection is based on academic merit.</p> <p>Please note that there is only 2 funded places available to UK applicants only and this project is in competition with 4 other projects to secure this funding.  If you are successful in securing an academic offer for PhD study, this does not mean that you have been successful in securing an offer of funding.</p> <p>Please refer to the <a href="https://www.ukcisa.org.uk/">UKCISA</a> website for information regarding Fee Status for Non-UK Nationals.</p> <p class="MsoNoSpacing"><strong>School of Physics & Astronomy Studentship 2026/27</strong></p> <p class="MsoNoSpacing">A highly competitive School of Physics & Astronomy Studentship providing the award of full academic fees, together with a tax-free maintenance grant at the standard UKRI rate (£20,780 in academic session 2025/26) for 3.5 years. There are no additional allowances for travel, research expenses, conference attendance or any other costs.</p> <p>You will be responsible for paying the overtime fee in full in your writing up/overtime year (£340 in Session 2025/26), but the scholarship maintenance allowance will continue to be paid for up to 6 months in the final year of award.</p> <p>Please note that there is only 1 funded place available to UK applicants only and this project is in competition with 4 other projects to secure this funding.  If you are successful in securing an academic offer for PhD study, this does not mean that you have been successful in securing an offer of funding.</p> <p>Please refer to the <a href="https://www.ukcisa.org.uk/">UKCISA</a> website for information regarding Fee Status for Non-UK Nationals.</p>

<p>For further information about your application, please contact PGR Admissions by email to <a href="mailto:phd@engineering.leeds.ac.uk">phd@engineering.leeds.ac.uk</a></p> <p>For further information about this project, please contact Prof Zlatko Papic by email to <a href="mailto:Z.Papic@leeds.ac.uk">Z.Papic@leeds.ac.uk</a></p>