Unconventional U(1) to $\mathbf{Z_q}$ cross-over in quantum and classical ${\bf q}$-state clock models

• Main Authors: Patil, Pranay, Shao, Hui, Sandvik, Anders W
• Format: Journal Article
• Language: English
• Published: 07-09-2020
• Subjects: Physics - Statistical Mechanics; Physics - Strongly Correlated Electrons
• DOI: 10.48550/arxiv.2009.03249

Phys. Rev. B 103, 054418 (2021) 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 $\phi_q$, scales in an
unexpected way. As a function of the system size $L$, close to the critical
temperature $\phi_q \propto L^p$, 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 $\nu'$ governing the
divergence of the U(1) to $Z_q$ cross-over length scale $\xi'$ in the
thermodynamic limit, according to the relationship $\nu'=\nu(1+|y_q|/p)$, where
$\nu$ 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.