  We describe a numerical study of the potential energy landscape for the
two-dimensional XY model (with no disorder), considering up to 100 spins and
CPU and GPU implementations of local optimization, focusing on minima and
saddles of index one (transition states). We examine both periodic and
anti-periodic boundary conditions, and show that the number of stationary
points located increases exponentially with increasing lattice size. The
corresponding disconnectivity graphs exhibit funneled landscapes; the global
minima are readily located because they exhibit relatively large basins of
attraction compared to the higher energy minima as the lattice size increases.
