Unconventional U(1) to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>Z</mml:mi><mml:mi>q</mml:mi></mml:msub></mml:math> crossover in quantum and classical <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>q</mml:mi></mml:math> -state clock models

We consider two-dimensional $q$-state quantum clock models with fluctuations connecting states all-to-all transitions different choices for the matrix elements. study phase in these using Monte Carlo simulations and finite-size scaling, aim of characterizing crossover from emergent U(1) symmetry at transition (for $q\ensuremath{\ge}4$) to ${Z}_{q}$ ordered state. also classical three-dimensional spatial anisotropy corresponding space-time systems. The all systems is governed by a so-called dangerously irrelevant operator. specifically $q=5$ $q=6$ forms anisotropies models. In cases, we find expected XY critical exponents scaling dimensions ${y}_{q}$ fields. However, initial weak violation phase, characterized symmetric order parameter ${\ensuremath{\phi}}_{q}$, scales an unexpected way. As function system size (length) $L$, close temperature ${\ensuremath{\phi}}_{q}\ensuremath{\propto}{L}^{p}$, where known value exponent $p=2$ isotropic model. contrast, strongly anisotropic models, $p=3$. For weakly observe $p=3$ scaling. $p$ directly impacts ${\ensuremath{\nu}}^{\ensuremath{'}}$ governing divergence length scale ${\ensuremath{\xi}}^{\ensuremath{'}}$ thermodynamic limit, according relationship ${\ensuremath{\nu}}^{\ensuremath{'}}=\ensuremath{\nu}(1+|{y}_{q}|/p)$, $\ensuremath{\nu}$ conventional correlation exponent. present phenomenological argument based on anomalous renormalization field presence anisotropy, possibly as consequence topological (vortex) line defects. Thus, our points intriguing interplay between perturbations, which may affect other symmetries.