Fermionic edge states and new physics

TL;DR

This paper studies fermionic edge states on manifolds with boundaries, providing exact counts, mapping boundary conditions to delta potentials, and addressing self-adjointness issues with moving boundaries.

Contribution

It introduces a method to exactly count edge states, maps boundary conditions to delta potentials, and simplifies the analysis of moving boundary problems in fermionic systems.

Findings

Exact counting of edge states for spherical boundaries.

Mapping boundary conditions to delta function potentials.

Simplification of moving boundary problems via transformations.

Abstract

We investigate the properties of the Dirac operator on manifolds with boundaries in presence of the Atiyah-Patodi-Singer boundary condition. An exact counting of the number of edge states for boundaries with isometry of a sphere is given. We show that the problem with the above boundary condition can be mapped to one where the manifold is extended beyond the boundary and the boundary condition is replaced by a delta function potential of suitable strength. We also briefly highlight how the problem of the self-adjointness of the operators in the presence of moving boundaries can be simplified by suitable transformations which render the boundary fixed and modify the Hamiltonian and the boundary condition to reflect the effect of moving boundary.