  We present a simple model of Majorana fermions on a square lattice, and study
zero-energy states due to Z$_2$ vortices. We show the relationship between the
Chern number of the ground state and the number of the zero-energy states by
numerical calculations for finite systems. We also discuss the relationship for
the bulk system by observing the change of the spectral asymmetry. We finally
present an effective continuum model with O(2) gauge potential which shows a
topological change of the index at a half-flux.
