Numerical study of the critical phases of the frustrated Z(5) model

The phase diagram of the $\mathbb{Z}(5)$ spin model is studied numerically on the square lattice by means of the {\sl Density Matrix Renormalization Group}. In the regime where the two nearest-neighbor couplings have opposite signs, a critical phase, not observed in earlier Monte Carlo simulations, is identified. The new phase diagram is in agreement with predictions made by M. den Nijs [{\sl Phys. Rev. B} {\bf 31}, 266 (1985)] but for the $\mathbb{Z}(7)$ model rather than the $\mathbb{Z}(5)$ one. All critical phases are shown to be compatible with a central charge $c=1$. The magnetization scaling dimension displays however a different behavior in the different critical phases.