Topology is like geometry where I forget my glasses. I can see rough shapes, but no details, and can only distinguish shapes by their global properties – like whether they have a hole or not. These global properties are called topological invariants. How does topology enter quantum physics? Through the topological properties of the wavefunction – the wavefunction itself can have a hole in it over some parameter space, and can then be classified by topological invariants. These are helpful because the wavefunction is weak (in the sense that it is easily perturbed) but the topology is strong (except when it isn’t but that’s a different story).
The topological properties of the energy eigenstates in periodic systems have been extensively classified. But a more recent idea is to instead study the spin eigenstates in periodic systems (defined as eigenstates of the operator PSP , where P is a ground-state projector).
These spin eigenstates reveal several interesting features, including the possibility to define new integer-valued topological invariants (spin Chern numbers), which also turn out to give new interesting bounds for optical properties realizable in materials!
The full physical consequences of these phases, however, remain largely unexplored. As such, there are multiple interesting directions in which to take this project, and it could potentially be interesting for more than one student. Concrete possibilities include:
- Spin-topology without energy gap closings:
In all spin topological toy models which have been studied, the energy gap (defined as the point where occupied and unoccupied energy levels become degenerate) closes at the same point as the spin gap (defined analogously for the eigenstates of PSP). But this does not have to be the case! When this does not happen, we could have a large response in optical response functions, which would be very unexpected from an energetics viewpoint. This has interesting connections also to critical quantum sensing.
- Spin-topology in altermagnets:
Altermagnets are fascinating new phases of magnetism which were discovered only recently and break with our conventional paradigm of what a magnet is, as they display surprising spin ordering in their crystal lattice. They have not been studied using a spin topology framework. Do they display equally surprising features in their spin bands? Are there bounds on optical properties in analogy to those for other spin topological systems?
Method
This project relies first and foremost on knowing quantum mechanics and solid-state physics, particularly band theory. Depending on your interest, it could also involve any amount of pure mathematics that you would like. Alternatively, the project can be tailored more towards a numerical project, with materials simulations.
The methods of topological band theory are not currently taught in any course at UiO, which is why I will be offering it as a special curriculum in the spring for those writing a thesis with me.
Milestones
- Understand band theory and the role of topology in band theory
- Reproduce selected model results from the quoted papers
Further progress depends on chosen path:
- If gap closing: Study model without gap closing in the energy bands, look at Quantum Fisher information and
sensing possibilities.
- If material: Study relevant materials and write code for spin topology
- If altermagnet: Investigate standard altermagnet models and investigate interplay of symmetry and topology.
Courses relevant from the bachelor's degree:
Recommended courses for theoretical curriculum in the master's programme: