  The measurements of the magnetic and nematic correlation lengths in a
generalization of the two dimensional XY model on the square lattice are
presented using classical Monte Carlo simulation. The full phase diagram is
re-examined based on these correlation lengths, demonstrating their power in
studying generalized XY models. The ratio between the correlation length and
the lattice size has distinctive behaviors which can be used to distinguish
different types of phase transition. More importantly, the magnetic correlation
length give more insights into the tricritical region where the paramagnetic,
nematic and quasi-long-range phases meet. It shows signatures for the
intermediate region starting from the tricritical point, where the transition
line is neither of the same physics as the Ising transition below nor the
Berezinskii-Kosterlitz-Thouless transition far above the tricritical point.
