PDEA Seminar: Non-Linear Landau Levels: Density Parametrization, Energy Optimization and Berry Phases

Abstract: Anyons - quasiparticles obeying fractional exchange statistics - can be viewed as excitations of fractional quantum Hall liquids and form the basis of fault-tolerant, topological quantum computing (TQC). In this project we explore a mathematical toy model for anyon physics based on the Chern-Simons-Schrödinger (CSS) functional. When the coupling constant is critical and in the absence of an external potential, the zero energy ground states of the CSS functional are described by the finite dimensional manifold of nonlinear Landau levels (NLL). The ground states are known and can be calculated explicitly from the position of their zeros (vortices). We test how well the wave functions in the NLLs approximate the ground state when the coupling is not at its critical value. To do this, we parametrize wave functions and minimize their energy in a harmonic trap while simplifying the problem by using symmetrically placed vortices. Further, we can treat the vortex positions of the wave functions in the NLL as slowly moving coordinates and adiabatically move them around closed loops to compute the emerging Berry phases. Whilst the connection to TQC remains tentative, the results suggest that an emergent abelian anyon gas with few vortices could be captured by the CSS ground states and therefore the NLLs.

Participate on site or via Zoom link, meeting ID: 67226102216. Ensure Zoom access by contacting the organisers in advance.

This is a seminar in the PDEs and Applications seminar series