  The Witten effect predicts that a magnetic monopole acquires a fractional
electric charge inside topological insulators.
  In this work, we give a microscopic description of this phenomenon, as well
as an analogous two-dimensional system with a vortex. We solve the Dirac
equation of electron field both analytically in continuum and numerically on a
lattice, by adding the Wilson term and smearing the gauge field within a finite
range to regularize the short-distance behavior of the system. Our results
reveal that the Wilson term induces a strong positive mass shift, creating a
domain-wall around the monopole/vortex. This small, yet finite-sized
domain-wall localizes the chiral zero modes and ensures their stability through
the Atiyah-Singer index theorem, whose cobordism invariance is crucial in
explaining why the electric charge is fractional.
