We calculate the ground state energy density $ϵ(g)$ 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 $ϵ″(g)$ or $ϵ(g)$. 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 $\alpha =0.34\pm 0.01$ 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 $ϵ(g)$ behaves as $A{e}^{-\frac{c}{|g_c-g{|}^{\sigma }}}+analyticterms$ 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 $N>9$. The phase transitions exhibited in these models should be generalizations of Kosterlitz-Thouless transition, which has $\sigma =0.5$.

中文翻译：

---

一维量子时钟模型的摄动研究

我们计算基态能量密度 $ϵ（G）$ 对于18阶以下的一维N态量子时钟模型，其中 $G$ 是耦合和 $ñ=3，4，5，。。。，10，20$。使用基于Pad'e逼近的方法，我们提取了奇异结构$ϵ\mathrm{\text{'}\text{'}}（G）$ 要么 $ϵ（G）$。它们对应于经典2D时钟模型的比热和自由能。我们发现，$ñ=3，4$，只有一个临界点 $G_C=\mathrm{1个}$相应的2D经典模型的热容指数为 $\alpha =0.34\pm 0.01$ 对于 $ñ=3$和 $\alpha =-0.01\pm 0.01$ 对于 $ñ=4$。对于$ñ>4$，有两个指数奇点与 $G_{C\mathrm{1个}}=\mathrm{1个}/G_{C2}$和 $ϵ（G）$ 表现为 $\mathrm{一种}{Ë}^{-\frac{C}{|G_C-G{|}^{\sigma }}}+\mathrm{一种}ñ\mathrm{一种}升ÿŤ\mathrm{一世}CŤË\mathrm{[R}米s$ 近 $G_C$。指数$\sigma$ 从逐渐增长 $0.2$ 至 $0.5$ 当N从5增加到9时，稳定在0.5时 $ñ>9$。这些模型中显示的相变应归纳为Kosterlitz-Thouless转换的概$\sigma =0.5$。