We study the phase diagram, both at zero and finite temperature, in a class of $\mathbb{Z}_q$ models with infinite range interactions. We are able to identify the transitions between a symmetry-breaking and a trivial phase by using a mean-field approach and a perturbative expansion. We perform our analysis on a Hamiltonian with $2p$-body interactions and we find first-order transitions for any $p>1$; in the case $p=1$, the transitions are first-order for $q=3$ and second-order otherwise. In the infinite-range case there is no trace of gapless incommensurate phase but, when the transverse field is maximally chiral, the model is in a symmetry-breaking phase for arbitrarily large fields. We analytically study the transtion in the limit of infinite $q$, where the model possesses a continuous $U(1)$ symmetry.

中文翻译：

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具有无限范围相互作用的量子时钟模型

我们在具有无限范围相互作用的一类 $\mathbb{Z}_q$ 模型中研究零温度和有限温度下的相图。我们能够通过使用平均场方法和微扰扩展来识别对称破坏和平凡阶段之间的转变。我们对具有 $2p$-body 相互作用的哈密顿量进行分析，我们发现任何 $p>1$ 的一阶跃迁；在 $p=1$ 的情况下，$q=3$ 的转换为一阶，否则为二阶。在无限范围的情况下，没有无间隙不公称相位的痕迹，但是，当横向场是最大手征时，对于任意大的场，模型处于对称破坏阶段。我们分析研究无限$q$ 极限中的转换，其中模型具有连续的$U(1)$ 对称性。