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.<br /> 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.

<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 class="MsoNoSpacing">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.</p> <p>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, if you are self-funding or externally sponsored</strong></em>.</p> <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>Applications will be considered on an ongoing basis.  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><strong>If you are applying for University or School Scholarships for 2026/27 entry, with external sponsorship or you are funding your own study, please ensure you provide your supporting documents at the point you submit your application:</strong></p> <ul> <li>Full Transcripts of all degree study or if in final year of study, full transcripts to date including the grading scheme</li> <li>Personal Statement outlining your interest in the project</li> <li>CV</li> </ul> <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><strong>Self-Funded or externally sponsored students are welcome to apply.</strong></p> <p><strong>Scholarship opportunities open from October 2025</strong></p> <p><strong>UK</strong> – The <a href="https://phd.leeds.ac.uk/funding/102-bell-burnell-scholarship-award-awarded-by-the-iop">Bell Burnell Scholarship Award</a> is available to support applicants who are from groups that are currently under-represented in physics (awarded by the IOP) (closing date: December 2025). The <a href="https://phd.leeds.ac.uk/funding/138-leeds-doctoral-scholarship-2025-faculty-of-engineering-and-physical-sciences#:~:text=Key%20facts&text=One%20Leeds%20Doctoral%20Scholarship%20is,rata%20for%20part%2Dtime%20study.">Leeds Doctoral Scholarship</a> (closing date: February 2026) and <a href="https://phd.leeds.ac.uk/funding/234-leeds-opportunity-research-scholarship-2022">Leeds Opportunity Research Scholarship</a> (closing date: April 2026) are available to UK applicants.  <a href="https://phd.leeds.ac.uk/funding/60-alumni-bursary">Alumni Bursary</a> is available to graduates of the University of Leeds.</p> <p><strong>Non-UK</strong> – The <a href="https://phd.leeds.ac.uk/funding/48-china-scholarship-council-university-of-leeds-scholarships-2021">China Scholarship Council - University of Leeds Scholarship</a> is available to nationals of China (closing date: January 2026). The <a href="https://phd.leeds.ac.uk/funding/73-leeds-marshall-scholarship">Leeds Marshall Scholarship</a> is available to support US citizens. <a href="https://phd.leeds.ac.uk/funding/60-alumni-bursary">Alumni Bursary</a> is available to graduates of the University of Leeds.</p> <p>Please note that 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>If you are applying for the Leeds Doctoral Scholarship, Leeds Opportunity Research Scholarship, China Scholarship Council-University of Leeds Scholarship or Leeds Marshall Scholarship, you will need to complete a separate application, specific to these scholarships, to be considered for funding.</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><strong>Important</strong>: Please note that that the award does <em><strong>not </strong></em>cover the costs associated with moving to the UK.  All such costs (<a href="https://www.leeds.ac.uk/international-visas-immigration/doc/applying-student-visa">visa, Immigration Health Surcharge</a>, flights etc) would have to be met by yourself, or you will need to find an alternative funding source. </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>