Unconventional U(1) to Z q crossover in quantum and classical q -state clock models

We consider two-dimensional $q$-state quantum clock models with quantum fluctuations connecting states with all-to-all clock transitions with different choices for the matrix elements. We study the quantum phase transitions in these models using quantum Monte Carlo simulations and finite-size scaling, with the aim of characterizing the crossover from emergent U(1) symmetry at the transition (for $q\ensuremath{\ge}4$) 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 crossover in all these systems is governed by a so-called 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. In all cases, 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 ${\ensuremath{\phi}}_{q}$, scales in an unexpected way. As a function of the system size (length) $L$, close to the critical temperature ${\ensuremath{\phi}}_{q}\ensuremath{\propto}{L}^{p}$, 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 crossover from $p=2$ to $p=3$ scaling. The exponent $p$ directly impacts the exponent ${\ensuremath{\nu}}^{\ensuremath{'}}$ governing the divergence of the U(1) to ${Z}_{q}$ crossover length scale ${\ensuremath{\xi}}^{\ensuremath{'}}$ in the thermodynamic limit, according to the relationship ${\ensuremath{\nu}}^{\ensuremath{'}}=\ensuremath{\nu}(1+|{y}_{q}|/p)$, where $\ensuremath{\nu}$ is the conventional correlation length exponent. We present a phenomenological argument for $p=3$ based on an anomalous renormalization of the clock field in the presence of anisotropy, possibly as a consequence of topological (vortex) line defects. Thus, our study points to an intriguing interplay between conventional and dangerously irrelevant perturbations, which may also affect other quantum systems with emergent symmetries.