The p-state clock model in two dimensions is a system of discrete rotors with a quasiliquid phase in a region T1<T<T2 for p>4. We show that, for p>4 and above a temperature T(eu), all macroscopic thermal averages become identical to those of the continuous rotor (p=infinity). This collapse of thermodynamic observables creates a regime of extended universality in the phase diagram and an emergent symmetry, not present in the Hamiltonian. For p> or =8, the collapse starts in the quasiliquid phase and makes the transition at T2 identical to the Berezinskii-Kosterlitz-Thouless (BKT) transition of the continuous rotor. For p< or =6, the transition at T2 is below T(eu) and no longer a BKT transition. The results generate a range of experimental predictions, such as the motion of magnetic domain walls, and limits on macroscopic distinguishability of different microscopic interactions.
The p-state clock model in two dimensions is a system of discrete rotors with a quasiliquid phase in a region T1<T<T2 for p>4. We show that, for p>4 and above a temperature T(eu), all macroscopic thermal averages become identical to those of the continuous rotor (p=infinity). This collapse of thermodynamic observables creates a regime of extended universality in the phase diagram and an emergent symmetry, not present in the Hamiltonian. For p> or =8, the collapse starts in the quasiliquid phase and makes the transition at T2 identical to the Berezinskii-Kosterlitz-Thouless (BKT) transition of the continuous rotor. For p< or =6, the transition at T2 is below T(eu) and no longer a BKT transition. The results generate a range of experimental predictions, such as the motion of magnetic domain walls, and limits on macroscopic distinguishability of different microscopic interactions.
Nature uses phase transitions as powerful regulators of processes ranging from climate to the alteration of phase behavior of cell membranes, building on the fact that thermodynamic properties of a solid, liquid, or gas are sensitive fingerprints of intermolecular interactions. The only known exceptions from this sensitivity are critical points, where two phases become indistinguishable and thermodynamic properties exhibit universal behavior: systems with widely different intermolecular interactions behave identically. Here we report a new, stronger form of universality, in which different members of a family of two-dimensional systems—the discrete p-state clock model—behave identically both near and away from critical points, if the temperature exceeds a value Teu ('extended universality). We show that all thermal averages are identical to those of the continuous planar rotor model (p = ∞) above Teu, that phase transitions above Teu are identical to the Berezinskii-Kosterlita-Thouless (BKT) transition, and that transitions below Teu are distinctly non-BKT. The results generate a comprehensive map of the three phases of the model and, by virtue of the discrete rotors behaving like continuous rotors, an emergent symmetry, not present in the Hamiltonian. This symmetry, or many-to-one map of intermolecular interactions onto thermodynamic states, demonstrates previously unknown limits for macroscopic distinguishability of different microscopic interactions.
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.
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 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.
The Berezinskii-Kosterlitz-Thouless (BKT) transition is the paradigmatic example of a topological phase transition without symmetry breaking, where a quasiordered phase, characterized by a power-law scaling of the correlation functions at low temperature, is disrupted by the proliferation of topological excitations above the critical temperature T_{BKT}. In this Letter, we consider the effect of long-range decaying couplings ∼r^{-2-σ} on the BKT transition. After pointing out the relevance of this nontrivial problem, we discuss the phase diagram, which is far richer than the corresponding short-range one. It features-for 7/4<σ<2-a quasiordered phase in a finite temperature range T_{c}<T<T_{BKT}, which occurs between a symmetry broken phase for T<T_{c} and a disordered phase for T>T_{BKT}. The transition temperature T_{c} displays unique universal features quite different from those of the traditional, short-range XY model. Given the universal nature of our findings, they may be observed in current experimental realizations in 2D atomic, molecular, and optical quantum systems.