The applicant will read the paper "Anyons in an exactly solvable model and beyond" by Kitaev and reproduced the solution of Kitaev honeycomb model introduced therein.
Then the applicant will solve the model and find the phase diagram. It needs to be shown that the gapped phase is described by a Z2 gauge theory, i.e. toric code.
Last, the applicant will derive the phenomena when one adiabatically goes around an incontractible circle that encloses the gapless region of the model, in the parameter space. This will presumably be mapped into a path of the mass that gap out the Majorana cone of the matter fields and derive a topological term associated with the evolution of the mass under a closed circle.