  We study the phase diagram of Q-state Potts models, for Q=4 cos^2(PI/p) a
Beraha number (p>2 integer), in the complex-temperature plane. The models are
defined on L x N strips of the square or triangular lattice, with boundary
conditions on the Potts spins that are periodic in the longitudinal (N)
direction and free or fixed in the transverse (L) direction. The relevant
partition functions can then be computed as sums over partition functions of an
A\_{p-1} type RSOS model, thus making contact with the theory of quantum
groups. We compute the accumulation sets, as N -> infinity, of partition
function zeros for p=4,5,6,infinity and L=2,3,4 and study selected features for
p>6 and/or L>4. This information enables us to formulate several conjectures
about the thermodynamic limit, L -> infinity, of these accumulation sets. The
resulting phase diagrams are quite different from those of the generic case
(irrational p). For free transverse boundary conditions, the partition function
zeros are found to be dense in large parts of the complex plane, even for the
Ising model (p=4). We show how this feature is modified by taking fixed
transverse boundary conditions.
