  We study the melting transition of the low-temperature vortex solid in
strongly anisotropic layered superconductors with a concentration of random
columnar pinning centers small enough so that the areal density of the pins is
much less than that of the vortex lines. Both the external magnetic field and
the columnar pins are assumed to be oriented perpendicular to the layers Our
method, involving numerical minimization of a model free energy functional,
yields not only the free energy values at the local minima of the functional
but also the detailed density distribution of the system at each minimum: this
allows us to study in detail the structure of the different phases. We find
that at these pin concentrations and low temperatures, the thermodynamically
stable state is a topologically ordered Bragg glass. This nearly crystalline
state melts into an interstitial liquid (a liquid in which a small fraction of
vortex lines remain localized at the pinning centers) in two steps, so that the
Bragg glass and the liquid are separated by a narrow phase that we identify
from analysis of its density structure as a polycrystalline Bose glass. Both
the Bragg glass to Bose glass and the Bose glass to interstitial liquid
transitions are first-order. We also find that a local melting temperature
defined using a criterion based on the degree of localization of the vortex
lines exhibits spatial variations similar to those observed in recent
experiments.
