Numerical study of the critical phases of the frustrated $Z\left(5\right)$ model

The phase diagram of the $Z\left(5\right)$ spin model is studied numerically on the square lattice by means of the 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 den Nijs (1985 Phys. Rev. B 31 266) but for the $Z\left(7\right)$ model rather than the $Z\left(5\right)$ 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. (paper: disordered systems, classical and quantum)