  We numerically simulate strongly correlated ultracold bosons coupled to a
high-finesse cavity field, pumped by a laser beam in the transverse direction.
Assuming a weak classical optical lattice added in the cavity direction, we
model this system by a generalized Bose-Hubbard model, which is solved by means
of bosonic dynamical mean-field theory. The complete phase diagram is
established, which contains two novel self-organized quantum phases, lattice
supersolid and checkerboard solid, in addition to conventional phases such as
superfluid and Mott insulator. At finite but low temperature, thermal
fluctuations are found to enhance the buildup of the self-organized phases. We
demonstrate that cavity-mediated long-range interactions can give rise to
stable lattice supersolid and checkerboard solid phases even in the regime of
strong s-wave scattering. In the presence of a harmonic trap, we discuss
coexistence of these self-organized phases, as relevant to experiments.
