Response to a twist in systems withZpsymmetry: The two-dimensionalp-state clock model

Abstract

We study response to a twist in the two-dimensional $p$-state clock model, which has the discrete ${Z}_{p}$ symmetry. The response is measured in terms of helicity modulus, which is usually defined with respect to an infinitesimal twist. However, we demonstrate that such a definition is inappropriate for the clock model. The helicity modulus must be defined with respect to a finite, quantized twist which matches the discrete ${Z}_{p}$ symmetry of the model. Numerical study of the appropriately defined helicity modulus resolves controversy over the clock model, showing the existence of two Berezinskii-Kosterlitz-Thouless transitions for $p>4$.