  The 2d XY model exhibits an essential phase transition, which was predicted
long ago --- by Berezinskii, Kosterlitz and Thouless (BKT) --- to be driven by
the (un)binding of vortex--anti-vortex pairs. This transition has been
confirmed for the standard lattice action, and for actions with distinct
couplings, in agreement with universality. Here we study a highly
unconventional formulation of this model, which belongs to the class of
topological lattice actions: it does not have any couplings at all, but just a
constraint for the relative angles between nearest neighbour spins. By means of
dynamical boundary conditions we measure the helicity modulus Upsilon, which
shows that this formulation performs a BKT phase transition as well. Its finite
size effects are amazingly mild, in contrast to other lattice actions. This
provides one of the most precise numerical confirmations ever of a BKT
transition in this model. On the other hand, up to the lattice sizes that we
explored, there are deviations from the spin wave approximation, for instance
for the Binder cumulant U_4 and for the leading finite size correction to
Upsilon. Finally we observe that the (un)binding mechanism follows the usual
pattern, although free vortices do not require any energy in this formulation.
Due to that observation, one should reconsider an aspect of the established
picture, which estimates the critical temperature based on this energy
requirement.
