  The fully frustrated XY model with Villain interaction on a square lattice is
studied by means of Monte Carlo simulations. On the basis of the universal jump
condition it is argued that there are two distinct transitions in the model,
corresponding to the loss of XY order and Z_2 order, respectively. The
Kosterlitz-Thouless (KT) transition is analyzed by finite size scaling of the
helicity modulus at lattices of size L = 32 through 128, giving T_KT =
0.8108(1). The vorticity-vorticity correlation function is used to determine
two different characteristic lengths, the Z_2 correlation length \xi, and the
screening length \lambda, associated with the KT transition and free vortices.
The temperature dependence of \xi is examined in order to determine T_c and the
correlation length exponent, \nu. The exponent is found to be consistent with
the 2D Ising value, \nu = 1, and the obtained critical temperature is T_c =
0.8225(5). The determinations of both \xi and \nu are done carefully, first
applying the techniques to the 2D Ising model, which serves as a convenient
testing ground.
