论文标题
非常规的u（1）至$ \ mathbf {z_q} $在量子和经典$ {\ bf q} $中交叉 - 状态时钟模型
Unconventional U(1) to $\mathbf{Z_q}$ cross-over in quantum and classical ${\bf q}$-state clock models
论文作者
论文摘要
我们考虑二维$ q $ - 状态量子时钟模型，其量子波动与矩阵元素不同选择的时钟过渡将状态连接起来。我们使用量子蒙特卡洛模拟研究了这些模型中的量子相变，目的是表征从跃迁时出现的u（1）对称的交叉（对于$ q \ ge 4 $）到订购状态的$ z_q $对称性。我们还研究了与量子系统时空各向异性相对应的空间各向异性的经典三维时钟模型。在所有这些系统中，u（1）至$ {z_q} $对称交叉均由危险的无关操作员支配。我们专门研究$ Q = 5 $和$ Q = 6 $模型，具有不同形式的量子波动和经典模型中不同的各向异性。我们找到了时钟字段的预期经典XY关键指数和缩放尺寸$ y_q $。但是，在顺序的阶段，最初对U（1）对称的违规行为，其特征在于$ ZQ $对称顺序参数$ ϕ_Q $，以出乎意料的方式缩放。作为系统尺寸$ l $的函数，靠近关键温度$ ϕ_q \ propto l^p $，其中指数的已知值为$ p = 2 $在经典的各向同性时钟模型中。相比之下，对于强烈的各向异性古典模型和量子模型，我们发现$ p = 3 $。对于微弱的各向异性古典模型，我们观察到跨越$ p = 2 $到$ p = 3 $缩放的交叉。指数$ p $直接影响指数$ν'$，从而在热力学限制中u（1）to u（1）to $ z_q $ cross-over长度比例$ coss-over长度比例$ n =ν（1+| y_q |/p）$，其中$ c相关长度是传统的。我们提出了一个基于各向异性存在的时钟场异常重归其化的现象学论点，这可能是由于拓扑（涡旋）线缺陷的结果。
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.
