  We study vortex unbinding for the classical two-dimensional XY model in a
magnetic field on square and triangular lattices. A renormalization group
analysis combined with duality in the model shows that at high temperature and
high field, the vortices unbind as the magnetic field is lowered in a two-step
process: strings of overturned spins first proliferate and then vortices
unbind. The transitions are highly continuous but are not of the
Kosterlitz-Thouless type. The unbound vortex fixed point is shown to inherit
properties of the underlying lattice, in particular containing a set of nodal
lines that reflect the lattice symmetry.
