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.
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.
Beyond the usual ferromagnetic and paramagnetic phases present in spin systems, the usual q-state clock model, presents an intermediate vortex state when the number of possible orientations q for the system is equal to 5 or larger. Such vortex states give rise to the Berezinskii-Kosterlitz-Thouless (BKT) phase present up to the XY model in the limit q→∞. Based on information theory, we present here an analysis of the classical order parameters plus new short-range parameters defined here. Thus, we show that even using the first nearest neighbors spin-spin correlations only, it is possible to distinguish the two transitions presented by this system for q greater than or equal to 5. Moreover, the appearance at relatively low temperature and disappearance of the BKT phase at a rather fix higher temperature is univocally determined by the short-range interactions recognized by the information content of classical and new parameters.
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.
In this paper, we revisit the q-states clock model for small systems. We present results for the thermodynamics of the q-states clock model from q=2 to q=20 for small square lattices L×L, with L ranging from L=3 to L=64 with free-boundary conditions. Energy, specific heat, entropy and magnetization are measured. We found that the Berezinskii-Kosterlitz-Thouless (BKT)-like transition appears for q>5 regardless of lattice size, while the transition at q=5 is lost for L<10; for q≤4 the BKT transition is never present. We report the phase diagram in terms of q showing the transition from the ferromagnetic (FM) to the paramagnetic (PM) phases at a critical temperature T1 for small systems which turns into a transition from the FM to the BKT phase for larger systems, while a second phase transition between the BKT and the PM phases occurs at T2. We also show that the magnetic phases are well characterized by the two dimensional (2D) distribution of the magnetization values. We make use of this opportunity to do an information theory analysis of the time series obtained from the Monte Carlo simulations. In particular, we calculate the phenomenological mutability and diversity functions. Diversity characterizes the phase transitions, but the phases are less detectable as q increases. Free boundary conditions are used to better mimic the reality of small systems (far from any thermodynamic limit). The role of size is discussed.
Abstract We study q-state clock models of regular and Villain types with using cluster-spin updates and observed double transitions in each model. We calculate the correlation ratio and size-dependent correlation length as quantities for characterizing the existence of Berezinskii–Kosterlitz–Thouless (BKT) phase and its transitions by large-scale Monte Carlo simulations. We discuss the advantage of correlation ratio in comparison to other commonly used quantities in probing BKT transition. Using finite size scaling of BKT type transition, we estimate transition temperatures and corresponding exponents . The comparison between the results from both types revealed that the existing transitions belong to BKT universality.