Vortices in unconventional superconductors

At extremely low temperatures, the free electrons in a superconductor bind into Cooper pairs, forming a coherent quantum fluid whose collective behaviour gives rise to remarkable properties such as zero electrical resistance. An applied magnetic field then surprisingly penetrates the superconductor at discrete points in an ordered rigid lattice. The fluid of paired electrons twists and circulates around these points, forming vortices, with no superconducting pairs at the centre of each vortex, much like water draining from a sink. In recent decades, superconductivity has also been discovered in unconventional materials, where the nature of the pairing mechanism remains an open problem. These materials can exhibit multiple pairing mechanisms and strong anisotropy, leading to exotic vortex structures with a rich variety of physical behaviour, making them both technologically promising and mathematically challenging to model.

This project aims to extend the theory of vortex solutions, lattices, and interactions beyond conventional superconductors to these unconventional materials. How does unconventional pairing reshape an individual vortex? How does anisotropy alter the interactions between vortices and the structure of the resulting vortex lattice? These questions lead naturally to the study of exotic multi-vortex configurations. Motivated by collaborations with experimental physicists across Europe, the project will focus particularly on half-quantum vortices, which exhibit fractional statistics and have been proposed as candidates for topological quantum computation.

A combination of analytical and computational approaches will be used to study the nonlinear PDEs and effective field theories describing these exotic vortices. For well-separated vortices, collective-coordinate and moduli-space methods provide a powerful way to reduce the underlying PDEs to effective dynamics on a finite-dimensional space of vortex positions and internal degrees of freedom. The project will investigate how these methods can be extended to unconventional materials, where anisotropy introduces additional internal structure that can fundamentally alter vortex interactions and dynamics.

By developing the mathematical theory of vortices in unconventional superconductors, the project aims to uncover how the microscopic physics of Cooper pairing manifests itself in the geometry, interactions, and dynamics of topological defects. The resulting mathematical models can be used to predict qualitative experimental signatures that can be investigated by our experimental collaborators, while developing simple tools for understanding how exotic vortices, particularly half-quantum vortices, might be controlled and manipulated. Such questions lie at the cutting edge of emerging quantum technologies, where the ability to create, control, and manipulate topological states of matter is central to the development of future quantum devices and topological quantum computing.

The successful candidate will join a vibrant international research community of theoretical and experimental physicists working across Europe to understand unconventional superconductivity and its exotic topological states. The project offers opportunities to work closely with experimental collaborators, participate in international workshops and conferences, and contribute to a rapidly developing field at the interface of mathematics, condensed-matter physics, and the emerging technologies of quantum computing.

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