Cold Atoms in Topological Bands

The quantum mechanical behaviour of electrons in solids has been an important and technologically relevant subjecject over the years.
Shortly after the advent of quantum mechanics, the canonical electronic band-structure theory was introduced. This band theory has been refined over the years, and it can now describe at the quantitative a large class of materials provided the electrons in the materials are not "strongly correlated". Such a treatment typically requires large-scale computation. Band theory is relevant for numerous technological breakthroughs including the invention of the transistor.

Band theory describes electrons in crystals with infinite spatial extent. On the other hand, real physical materials, of course, have finite size. Band theory usually works because many physical observables depend on bulk rather boundary properties of the system. For instance, the heat capacity and the band gap of a material often can be described by only considering its bulk properties.

It has recently been appreciated that the canonical band theory does not give the full story for a number of materials comprised of weakly- interacting electrons. In particular, the topological properties of the bulk bands can have important implications for the behaviour of the system at the boundary. Such materials are known as topological insulators [Physics Today, Jan. 2010, Vol. 63, Pg. 33-38]. Perhaps the simplest example is the Haldane model (this work contributed to Haldane's Nobel prize in 2016). Naive band theory predicts that this system will be an insulator. On the other hand, the bands in this system possess a non-trivial topological number (known as the Chern number) which implies the existence of topologically protected conducting surface states. Such states are revealed, for instance, by performing computations on finite-sized systems.

While the first topological insulators were found in solid-state

materials, more recently they have been realised with ultra cold atoms in optical lattices [see arXiv:1803.00249 for a review]. Due to their tunability and variability, ultra cold atoms offer a unique arena to explore the physical properties of particles in topological bands. For instance, such systems allow one to explore the behaviour of higher- spin bosons in topological bands.

This PhD aims to further understand the behaviour of atoms in topological bands. In particular, using state-of-the-art techniques (including exact diagonalisation and the truncated Wigner approximation, if appropriate) we will investigate the dynamics of bosons in topologically protected edge modes. This work builds on previous work of the supervisor (see http://wwwf.imperial.ac.uk/~rlbarnet/publications_ch.html for recent publications) and is relevant for ongoing experimental efforts.

This project has major overlap with the EPSRC Strategic Themes: Mathematical Sciences and Quantum Technologies. Furthermore, it has aspects in the Research Areas: Quantum fluids and solids, Condensed matter: magnetism and magnetic materials, Mathematical physics, and Superconductivity. Partnership with industry is welcome but specific opportunities have not been identified at present.