  We construct a two-dimensional tight-binding model of an optical lattice,
where the low energy excitations should be described by the spin-1 Maxwell
equations in the Hamiltonian form, and such linear dispersion excitations with
pesudospin-1 are so called as the Maxwell quasiparticles. The system has rich
topological features, for examples, the threefold degeneracy points called
Maxwell points may have nontrivial $\pm 2\pi$ Berry phases and the anomalous
quantum Hall effect with spin-momentum locking may appear in topological
Maxwell insulators. We propose realistic schemes for realizing the Maxwell
metals/insulators and detecting the intrinsic properties of the topological
Maxwell quasiparticles with ultracold atoms in optical lattices.
