Evidences of the Generalizations of BKT Transition in Quantum Clock Model

We calculate the ground state energy density $\epsilon(g)$ for the one dimensional N-state quantum clock model up to order 18, where $g$ is the coupling and $N=3,4,5,...,10,20$. Using methods based on Pad\'e approximation, we extract the singular structure of $\epsilon''(g)$ or $\epsilon(g)$. They correspond to the specific heat and free energy of the classical 2D clock model. We find that, for $N=3,4$, there is a single critical point at $g_c=1$.The heat capacity exponent of the corresponding 2D classical model is $\alpha=0.34\pm0.01$ for $N=3$, and $\alpha=-0.01\pm 0.01$ for $N=4$. For $N>4$, There are two exponential singularities related by $g_{c1}=1/g_{c2}$, and $\epsilon(g)$ behaves as $Ae^{-\frac{c}{|g_c-g|^{\sigma}}}+analytic\ terms$ near $g_c$. The exponent $\sigma$ gradually grows from $0.2$ to $0.5$ as N increases from 5 to 9, and it stabilizes at 0.5 when $N>9$. These phase transitions should be generalizations of Kosterlitz-Thouless transition, which has $\sigma=0.5$. The physical pictures of these phase transitions are still unclear.