  We examine the melting of commensurate and incommensurate vortex lattices
interacting with square pinning arrays through the use of numerical
simulations. For weak pinning strength in the commensurate case we observe an
order-order transition from a commensurate square vortex lattice to a
triangular floating solid phase as a function of temperature. This floating
solid phase melts into a liquid at still higher temperature. For strong pinning
there is only a single transition from the square pinned lattice to the liquid
state. For strong pinning in the incommensurate case, we observe a multi-stage
melting in which the interstitial vortices become mobile first, followed by the
melting of the entire lattice, consistent with recent imaging experiments. The
initial motion of vortices in the incommensurate phase occurs by an exchange
process of interstitial vortices with vortices located at the pinning sites. We
have also examined the vortex melting behavior for higher matching fields and
find that a coexistence of a commensurate pinned vortex lattice with an
interstitial vortex liquid occurs while at higher temperatures the entire
vortex lattice melts. For triangular arrays at incommensurate fields higher
than the first matching field we observe that the initial vortex motion can
occur through a novel correlated ring excitation where a number of vortices can
rotate around a pinned vortex. We also discuss the relevance of our results to
recent experiments of colloidal particles interacting with periodic trap
arrays.
