We investigate both analytically and numerically the renormalization group equations in two-dimensional (2D) $Z\left(N\right)$ vector models. The position of the critical points of the two phase transitions for $N>4$ is established and the critical index $\nu$ is computed. For $N=7$ and 17 the critical points are located by Monte Carlo simulations, and some of the corresponding critical indices are determined. The behavior of the helicity modulus is studied for $N=5$, 7, and 17. Using these and other available Monte Carlo data we discuss the scaling of the critical points with $N$ and some other open theoretical problems.

• Received 17 December 2011

DOI:https://doi.org/10.1103/PhysRevE.85.021114