Unconventional U(1) to Zq crossover in quantum and classical q -state clock models
Files
First author draft
Date
2021-02-10
Authors
Patil, Pranay
Shao, Hui
Sandvik, Anders W.
Version
First author draft
OA Version
Citation
Pranay Patil, Hui Shao, Anders W Sandvik. "Unconventional U(1) to Zq crossover in quantum and classical q -state clock models." Physical Review B, Volume 103, 2021. Issue 5, https://doi.org/10.1103/physrevb.103.054418
Abstract
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≥4) to Zq 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 Zq 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 yq of the clock fields. However, the initial weak violation of the U(1) symmetry in the ordered phase, characterized by a Zq symmetric order parameter ϕq, scales in an unexpected way. As a function of the system size (length) L, close to the critical temperature ϕq∝Lp, 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 ν′ governing the divergence of the U(1) to Zq crossover length scale ξ′ in the thermodynamic limit, according to the relationship ν′=ν(1+|yq|/p), where ν 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.