1. Superconductivity in topological flat bands Recently, a number of experimental results in synthetic two-dimensional matter (for example stacking layers of graphene with certain pattern or interlayer twists) have shown type of superconductivity that challenges the conventional understanding of superconductivity based on weakly interactive BCS theory. These materials are not only very strongly correlated, they have exotic topological features. For example, some of them have time reversal symmetry broken topological phases interplaying with superconductivity. These scenarios are highly unusual, thus require new theoretical insights. They are also thought to be very useful for topologically protected quantum technologies. My previous work has shown that in such scenario vortex lattice structure (a crystalline order in the superconductor’s wavefunction) can be primary driver of topological phase, thus connecting these systems to the emerging field of crystalline topology. Currently, I am interested on a more detailed theory of these systems that goes beyond simple mean-field approximations but also captures fluctuation effects.

2. Polarisation, ferroelectricity, and their relations to superconductivity Polarisation is a phenomenon that is very intuitive and well understood at the high-school chemistry level. For example, water is a polar molecule, where the charge centre and the geometric centre of the molecule do not coincide. In solid-state materials, when the polarisation develops a long-range-order, it becomes a ferroelectric, which have well known technological applications, such as in memory devices. However, the conceptual understanding of polarisation, which is so simple in molecules, become very intricate for periodic solids and one needs to invoke modern quantum mechanics concepts such as Berry phase to understand them. In my recent co-authored works, I have shown a detailed picture of polarisation, where the entire real-space resolution and just the global response is a meaningful physical quantity. Moreover, they are highly relevant in recently discovered two-dimensional layered ferroelectrics and show novel topological features. I have also shown how the fluctuations in the polar order near ferroelectric domain walls can lead to a new mechanism for superconductivity.

3. Composite fermion construction without magnetic field The intuitive understanding of fractional quantum Hall effect is based on the composite Fermion theory, where one attaches external magnetic field to electrons as average flux, and then builds a field theory based on fluctuation around these mean-fields. However, recent discovery of fractional quantum Hall effect without magnetic field in synthetic two-dimensional materials raised some interesting problems on developing field theoretical approach applicable to this more generalised manifestation of fractional quantum Hall effect. I am currently interested in developing such approaches and verifying them with detailed numerical simulations.