Real-space renormalization-group study of theZ(5)model

The phase diagram of the nearest-neighbor $Z\left(5\right)\mathrm{}$ model on the square and triangular lattices is investigated using techniques of the real-space renormalization group. We use the recently introduced technique of Nienhuis, Berker, Riedel, and Schick, which yields the expected first-order transition on the Potts subspace of the $Z\left(5\right)\mathrm{}$ model on the triangular lattice. For the general $Z\left(5\right)\mathrm{}$ model we obtain a line of first-order transitions which crosses over into a line of second-order transitions. The same technique when applied to the square lattice is found to show this crossover behavior only in combination with a Migdal procedure.