  We study in detail the active Ising model, a stochastic lattice gas where
collective motion emerges from the spontaneous breaking of a discrete symmetry.
On a 2d lattice, active particles undergo a diffusion biased in one of two
possible directions (left and right) and align ferromagnetically their
direction of motion, hence yielding a minimal flocking model with discrete
rotational symmetry. We show that the transition to collective motion amounts
in this model to a bona fide liquid-gas phase transition in the canonical
ensemble. The phase diagram in the density/velocity parameter plane has a
critical point at zero velocity which belongs to the Ising universality class.
In the density/temperature "canonical" ensemble, the usual critical point of
the equilibrium liquid-gas transition is sent to infinite density because the
different symmetries between liquid and gas phases preclude a supercritical
region. We build a continuum theory which reproduces qualitatively the behavior
of the microscopic model. In particular we predict analytically the shapes of
the phase diagrams in the vicinity of the critical points, the binodal and
spinodal densities at coexistence, and the speeds and shapes of the
phase-separated profiles.
