量子与经典q态时钟模型中非常规的U(1)到$\boldsymbol{Z_q}$渡越
Unconventional U(1) to $\mathbf{Z_q}$ cross-over in quantum and classical ${\bf q}$-state clock models
AI总结：
本研究通过量子蒙特卡洛模拟探究二维q态量子时钟模型及对应三维经典各向异性时钟模型的U(1)到Z_q对称性渡越，发现了反常的序参量标度指数p并给出唯象解释。
摘要：
我们研究了具有量子涨落的二维q态量子时钟模型，其中量子涨落通过不同矩阵元选择的时钟跃迁连接不同态。我们采用量子蒙特卡洛模拟研究这些模型中的量子相变，旨在表征从相变处（对于$q \ge 4$）涌现的U(1)对称性到有序态的$Z_q$对称性的渡越行为。我们还研究了具有空间各向异性的三维经典时钟模型，其各向异性对应于量子系统的时空各向异性。所有这些系统中的U(1)到$Z_q$对称性渡越均由一个危险无关算符主导。\n我们专门研究了具有不同形式量子涨落的$q=5$和$q=6$量子模型，以及具有不同各向异性的经典模型。我们得到了预期的经典XY临界指数，以及时钟场的标度维度$y_q$。然而，有序相中由$Z_q$对称序参量$\phi_q$表征的U(1)对称性初始弱破缺，呈现出出乎意料的标度行为。在临界温度附近，作为系统尺寸$L$的函数，$\phi_q \propto L^p$；在经典各向同性时钟模型中，该指数的已知值为$p=2$。相比之下，对于强各向异性经典模型和量子模型，我们发现$p=3$。对于弱各向异性经典模型，我们观察到从$p=2$到$p=3$的标度渡越。\n指数$p$直接影响热力学极限下支配U(1)到$Z_q$渡越长度标度$\xi'$发散行为的指数$\nu'$，二者满足关系$\nu'=\nu(1+|y_q|/p)$，其中$\nu$是常规的关联长度指数。我们基于各向异性存在下时钟场的反常重整化（可能源于拓扑（涡旋）线缺陷），提出了一种唯象解释。
英文摘要：
We consider two-dimensional $q$-state quantum clock models with quantum fluctuations connecting states with clock transitions with different choices for matrix elements. We study the quantum phase transitions in these models using quantum Monte Carlo simulations, with the aim of characterizing the cross-over from emergent U(1) symmetry at the transition (for $q \ge 4$) to $Z_q$ symmetry of the ordered state. We also study classical three-dimensional clock models with spatial anisotropy corresponding to the space-time anisotropy of the quantum systems. The U(1) to ${Z_q}$ symmetry cross-over in all these systems is governed by a dangerously irrelevant operator. We specifically study $q=5$ and $q=6$ models with different forms of the quantum fluctuations and different anisotropies in the classical models. We find the expected classical XY critical exponents and scaling dimensions $y_q$ of the clock fields. However, the initial weak violation of the U(1) symmetry in the ordered phase, characterized by a $Z_q$ symmetric order parameter $ϕ_q$, scales in an unexpected way. As a function of the system size $L$, close to the critical temperature $ϕ_q \propto L^p$, where the known value of the exponent is $p=2$ in the classical isotropic clock model. In contrast, for strongly anisotropic classical models and the quantum models we find $p=3$. For weakly anisotropic classical models we observe a cross-over from $p=2$ to $p=3$ scaling. The exponent $p$ directly impacts the exponent $ν'$ governing the divergence of the U(1) to $Z_q$ cross-over length scale $ξ'$ in the thermodynamic limit, according to the relationship $ν'=ν(1+|y_q|/p)$, where $ν$ is the conventional correlation length exponent. We present a phenomenological argument based on an anomalous renormalization of the clock field in the presence of anisotropy, possibly as a consequence of topological (vortex) line defects.
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