  We study circumbinary accretion discs in the framework of the restricted
three-body problem (R3Bp) and via numerically solving the height-integrated
equations of viscous hydrodynamics. Varying the mass ratio of the binary, we
find a pronounced change in the behaviour of the disc near mass ratio $q \equiv
M_s/M_p \sim 0.04$. For mass ratios above $q=0.04$, solutions for the
hydrodynamic flow transition from steady, to strongly-fluctuating; a narrow
annular gap in the surface density around the secondary's orbit changes to a
hollow central cavity; and a spatial symmetry is lost, resulting in a lopsided
disc. This phase transition is coincident with the mass ratio above which
stable orbits do not exist around the L4 and L5 equilibrium points of the R3B
problem. Using the DISCO code, we find that for thin discs, for which a gap or
cavity can remain open, the mass ratio of the transition is relatively
insensitive to disc viscosity and pressure. The $q=0.04$ transition has
relevance for the evolution of massive black hole binary+disc systems at the
centers of galactic nuclei, as well as for young stellar binaries and possibly
planets around brown dwarfs.
