Laboratoire de Physique Théorique
Toulouse - UMR 5152

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, even when the discreteness is irrelevant in the thermodynamic limit. Instead, 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 $p>4$.

Ref : Y. Kumano, K. Hukushima, Y. Tomita, and M. O., Phys. Rev. B 88, 104427 (2013)