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arXiv:2605.20990

Competing anisotropies and phase transitions in the $q$-state clock model with a $p$-fold crystalline field

Milan ˇ Zukovič

Published May 20, 2026• 0 citations
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Abstract

We study the two-dimensional $q$-state clock model in the presence of an additional $p$-fold symmetry-breaking crystalline field using Monte Carlo simulations. While the pure clock model exhibits Berezinskii--Kosterlitz--Thouless (BKT) transitions for sufficiently large $q$, the effect of competing discrete anisotropies on this topological phase remains nontrivial. We show that even weak crystalline fields qualitatively modify the phase diagram by suppressing the BKT phase and inducing transitions to states with true long-range order. The resulting behavior depends sensitively on the interplay between the intrinsic $\mathbb{Z}_q$ symmetry and the imposed $\mathbb{Z}_p$ anisotropy. In particular, in the six-state clock model for $p=2$ we observe qualitatively different scenarios depending on the sign of the field: a single transition for $h_2>0$ and a two-step ordering process for $h_2<0$ with an intermediate ordered phase. For $p=3$, the system exhibits a direct transition consistent with three-state Potts criticality. These results demonstrate that the phase structure cannot be inferred from symmetry considerations alone, but is governed by the competition between distinct locking mechanisms. Our findings provide a discrete counterpart to the multi-frequency sine-Gordon description of generalized $XY$ models and illustrate how additional anisotropies reshape topological phase transitions in two dimensions.

DOI
10.48550/arxiv.2605.20990

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Competing anisotropies and phase transitions in the qqq-state clock model with a ppp-fold crystalline field
We study the two-dimensional qqq-state clock model in the presence of an additional ppp-fold symmetry-breaking crystalline field using Monte Carlo simulations. While the pure clock model exhibits Berezinskii--Kosterlitz--Thouless (BKT) transitions for sufficiently large qqq, the effect of competing discrete anisotropies on this topological phase remains nontrivial. We show that even weak crystalline fields qualitatively modify the phase diagram by suppressing the BKT phase and inducing transitions to states with true long-range order. The resulting behavior depends sensitively on the interplay between the intrinsic Zq\mathbb{Z}_qZq​ symmetry and the imposed Zp\mathbb{Z}_pZp​ anisotropy. In particular, in the six-state clock model for p=2p=2p=2 we observe qualitatively different scenarios depending on the sign of the field: a single transition for h2>0h_2>0h2​>0 and a two-step ordering process for h2<0h_2<0h2​<0 with an intermediate ordered phase. For p=3p=3p=3, the system exhibits a direct transition consistent with three-state Potts criticality. These results demonstrate that the phase structure cannot be inferred from symmetry considerations alone, but is governed by the competition between distinct locking mechanisms. Our findings provide a discrete counterpart to the multi-frequency sine-Gordon description of generalized XYXYXY models and illustrate how additional anisotropies reshape topological phase transitions in two dimensions.
The q-state clock model, sometimes called the discrete XY model, is known to show a second-order (symmetry-breaking) phase transition in two dimensions for q≤4 (q=2 corresponds to the Ising model). On the other hand, the q→∞ limit of the model corresponds to the XY model, which shows the infinite order (non-symmetry-breaking) Berezinskii-Kosterlitz-Thouless (BKT) phase transition in two dimensions. Interestingly, the two-dimensional clock model with q≥5 is predicted to show three different phases and two associated phase transitions. There are varying opinions about the actual characters of phases and the associated transitions. In this work, we develop the basic and higher-order mean-field (MF) theories to study the q-state clock model systematically. Our MF calculations reaffirm that, for large q, there are three phases: (broken) Z_{q} symmetric ferromagnetic phase at the low temperature, emergent U(1) symmetric BKT phase at the intermediate temperature, and paramagnetic (disordered) phase at the high temperature. The phase transition at the higher temperature is found to be of the BKT type, and the other transition at the lower temperature is argued to be a large-order spontaneous symmetry-breaking type (the largeness of transition order yields the possibility of having some of the numerical characteristics of a BKT transition). The higher-order MF theory developed here better characterizes phases by estimating the spin-spin correlation between two neighbors.
Abstract In a two-dimensional (2D) spin system, the XY model, characterized by planar rotational symmetry, exhibits a unique phenomenon known as the Berezinskii-Kosterlitz-Thouless (BKT) transition. In contrast, the clock model, which introduces discrete rotational symmetry, exhibits the BKT transition at two different temperatures due to this discreteness. In this study, we numerically investigate the BKT transition for XY and six-state clock models over various two-dimensional lattices. We employ two primary methods: the Monte Carlo method, which analyzes the size dependence of the ratio of the correlation functions for two different distances, and a machine-learning approach to classify the different phases --- namely, the low-temperature ordered phase, the intermediate BKT phase, and the high-temperature disordered phase. We identify the BKT transition temperatures for the XY and six-state models on honeycomb, kagome, and diced lattices. Combined with the previously calculated data for the triangular lattice, we then compare these values with the second-order phase transition temperatures of the 2D Ising model, for which exact solutions are known. Our results indicate that the ratio of the BKT transition temperatures for each lattice relative to the Ising model transition temperatures are close, although the values are not universal.
Abstract In a two-dimensional (2D) spin system, the XY model, characterized by planar rotational symmetry, exhibits a unique phenomenon known as the Berezinskii–Kosterlitz–Thouless (BKT) transition. In contrast, the clock model, which introduces discrete rotational symmetry, exhibits the BKT transition at two different temperatures due to this discreteness. In this study, we numerically investigate the BKT transition for XY and six-state clock models over various 2D lattices. We employ two primary methods: the Monte Carlo method, which analyzes the size dependence of the ratio of the correlation functions for two different distances, and a machine-learning approach to classify the different phases—namely, the low-temperature ordered phase, the intermediate BKT phase, and the high-temperature disordered phase. We identify the BKT transition temperatures for the XY and six-state models on honeycomb, kagome, and diced lattices. Combined with the previously calculated data for the triangular lattice, we then compare these values with the second-order phase transition temperatures of the 2D Ising model, for which exact solutions are known. Our results indicate that the ratio of the BKT transition temperatures for each lattice relative to the Ising model transition temperatures are close, although the values are not universal.
We present a comprehensive study on the frustrated J_{1}-J_{2} classical q-state clock model with even q>4 on a two-dimensional square lattice, revealing a rich ensemble of phases driven by competing interactions. In the unfrustrated regime (J_{1}>2J_{2}), the model reproduces the standard clock model phenomenology: a low-temperature Z_{q}-broken ferromagnet, an intermediate XY-like critical quasi-long-range-ordered phase with emergent U(1) symmetry, and a high-temperature paramagnet. For J_{1}<2J_{2}, frustration stabilizes five distinct regimes: the disordered paramagnet, a stripe-ordered phase breaking Z_{q}×Z_{2} symmetry, two Z_{2}-broken nematic phases (one with and one without quasi-long-range order), and an exotic stripe phase with emergent discrete Z_{q} spin degrees of freedom prohibited in the microscopic Hamiltonian. Remarkably, this seemingly forbidden Z_{q} order emerges via a relevant operator in the infrared long-wavelength limit, rather than from an irrelevant perturbation, highlighting a nonstandard route to emergence. Using large-scale corner transfer matrix renormalization group calculations, complemented by classical Monte Carlo simulations, we map the complete phase diagram and identify Berezinskii-Kosterlitz-Thouless, Ising, first-order, and unconventional Landau-incompatible transitions between different phases. Finally, we propose an effective field-theoretic framework that encompasses these emergent orders and their interwoven transitions.
We study the phases and phase transitions of a disordered one-dimensional quantum qqq-state clock Hamiltonian using large-scale Monte Carlo simulations. Making contact with earlier results, we confirm that the clean, translational invariant version of the model, for q=6q=6q=6, hosts an intermediate emergent quasi-long-range ordered (QLRO) phase between the symmetry-broken true long-range ordered (TLRO) phase and the disordered (paramagnetic) phase. With increasing disorder strength, the quasi-long-range ordered phase shrinks and finally vanishes at a multicritical point, beyond which there is a direct transition from the TLRO phase to the paramagnetic phase. After establishing the phase diagram, we characterize the critical behaviors of the various quantum phase transitions in the model. We find that weak disorder is an irrelevant perturbation of the Berezinskii-Kosterlitz-Thouless transitions that separate the QLRO phase from the TLRO and paramagnetic phases. For stronger disorder, some of the critical exponents become disorder-dependent already before the system reaches the multicritical point. We also show that beyond the multicritical point, the direct transition from the TLRO phase to the paramagnetic phase is governed by an infinite-randomness critical point in line with strong-disorder renormalization group predictions. While our numerical results are for q=6q=6q=6, we expect the qualitative features of the behavior to hold for all q>4q>4q>4.
We perform the state-of-the-art tensor network simulations directly in the thermodynamic limit to clarify the critical properties of the q-state clock model on the square lattice. We determine accurately the two phase transition temperatures through the singularity of the classical analog of the entanglement entropy, and provide extensive numerical evidences to show that both transitions are of the Berezinskii-Kosterlitz-Thouless (BKT) type for q≥5 and that the low-energy physics of this model is well described by the Z_{q}-deformed sine-Gordon theory. We also determine the characteristic conformal parameters, especially the compactification radius, that govern the critical properties of the intermediate BKT phase.