Finite-size effects on the statistical angle of an electron induced dyon in proximity to a topological insulator

A pointlike electric charge close to the surface of a three-dimensional topological insulator induces a magnetic monopole mirror charge. We study the distance dependence of the statistical angle describing this induced dyon system. We find that the total angular momentum, which sometimes is used as the signature of the statistical angle, for an electron outside a finite-size spherical or tube-shaped topological insulator is zero for any finite distance between the electron and the surface. However, we show that in the two-electron system one can indeed isolate a nontrivial statistical angle for intermediate size loops, that is, loops much larger than the distance of the charge to the sample but much smaller than the size of the sample. The necessity for this limit confirms the (2+1)-dimensional nature of the nontrivial statistical angle found in previous work. Our results clarify the conditions under which the statistical angle of this system could be measured in real experiments.