  The recent advances in creating nearly degenerate quantum dipolar gases in
optical lattices are opening the doors for the exploration of equilibrium
physics of quantum systems with anisotropic and long-range dipolar
interactions. In this paper we study the zero- and finite-temperature phase
diagrams of a system of hard-core dipolar bosons at half-filling, trapped in a
two-dimensional optical lattice. The dipoles are aligned parallel to one
another and tilted out of the optical lattice plane by means of an external
electric field. At zero-temperature, the system is a superfluid at all tilt
angles $\theta$ provided that the strength of dipolar interaction is below a
critical value $V_c(\theta)$. Upon increasing the interaction strength while
keeping $\theta$ fixed, the superfluid phase is destabilized in favor of a
checkerboard or a stripe solid depending on the tilt angle. We explore the
nature of the phase transition between the two solid phases and find evidence
of a micro-emulsion phase, following the Spivak-Kivelson scenario, separating
these two solid phases. Additionally, we study the stability of these quantum
phases against thermal fluctuations and find that the stripe solid is the most
robust, making it the best candidate for experimental observation.
