Response to a twist in systems with Zp symmetry: The two-dimensional p-state clock model

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.