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ResearchMy research concerns exploring topological aspects of condensed matter systems, both in the context of cold atoms and electronic systems, often using a blend of analytical and numerical techniques. The projects I have worked on or are currently working on fall under the following broad research areas:
- Ultracold atoms: Developed a theory of the dynamics of atoms in two-dimensional shallow quasicrystalline optical lattices. Studied Bloch oscillations, Berry curvature and Chern number defined over a pseudo-Brillouin-zone. Key finding of an unusual 'spiral holonomy' in the effective band structure.
- Quantum oscillations: Provided the first account of quantum oscillations in two or three dimensional 'weakly coupled' quasiperiodic materials---a key example being 30° twisted bilayer graphene. Found that these reveal a 'spiral Fermi surface' characterised by a nontrivial topological invariant.
- Artificial gauge fields: Explored ways of engineering topologically nontrivial phases in quasicrystalline optical lattices via the breaking of time reversal symmetry, approaches included Floquet engineering and coupling of internal states.
- Topological insulators: Investigating possible higher order topological phases in materials with non-Bravais lattice rotational symmetries.
In Plain English
After the discovery of X-ray diffraction it was generally considered that ordered materials (those which displayed sharp Bragg peaks in their diffraction patterns) were periodic and a simple theorem known as the `crystallographic restriction theorem' ensured that the only allowed rotational symmetries of a periodic structure are 1-,2-,3-,4- and 6-fold. Despite this, in 1983, Dan Schectman discovered that the X-ray diffraction pattern of an aluminium alloy remarkably displayed both sharp Bragg peaks but with a crystallographically disallowed 5-fold rotational symmetry. These and similar materials were soon termed `quasicrystals', and their structure was found to be related to aperiodic tilings such as the well-known Penrose tiling.
In recent years there has been a growing interest in the `simulation' of models from condensed matter by using systems of so-called `ultracold atomic gases'. Here the atoms play the role of electrons and an `optical lattice' potential created from the interference of overlapping laser beams, models the background ionic lattice. The key advantage of these systems is that the optical lattice is highly controllable and free from defects that can obscure various quantum phenomena. By simply arranging the lasers to have a rotational symmetry that is disallowed for a periodic structure one can then extend the control available in cold-atom systems to simulate quasicrystals. In my research I have been exploring novel theoretical questions that were previously less relevant in condensed matter quasicrystals (and therefore remained uninvestigated), but which are now highly relevant for their optical lattice analogues.
- Spiral Fermi Surfaces in Quasicrystals and Twisted Bilayer Graphene: Signatures in Quantum Oscillations arXiv:1811.03652
- Semiclassical dynamics, Berry curvature, and spiral holonomy in optical quasicrystals Phys. Rev. A 97 043603 (2018) , Editors' Suggestion , arXiv:1710.10659
- Quantum Oscillations in Quasicrystals and Twisted Bilayer Graphene Stephen Spurrier and Nigel R. Cooper