The Berezinskii-Kosterlitz-Thouless transitions of the six-state clock model on the square lattice are investigated by means of the corner-transfer matrix renormalization group method.A classical analog of the entanglement entropy S(L, T ) is calculated for L × L square system up to L = 129, as a function of temperature T .The entropy exhibits a peak at T = T * (L), where the temperature depends on both L and the boundary conditions.Applying the finite-size scaling to T * (L) and assuming presence of the Berezinskii-Kosterlitz-Thouless transitions, the two distinct phase-transition temperatures are estimated to be T1 = 0.70 and T2 = 0.88.The results are in agreement with earlier studies.It should be noted that no thermodynamic functions have been used in this study.
The Berezinskii-Kosterlitz-Thouless transitions of the six-state clock model on the square lattice are investigated by means of the corner-transfer matrix renormalization group method.A classical analog of the entanglement entropy S(L, T ) is calculated for L × L square system up to L = 129, as a function of temperature T .The entropy exhibits a peak at T = T * (L), where the temperature depends on both L and the boundary conditions.Applying the finite-size scaling to T * (L) and assuming presence of the Berezinskii-Kosterlitz-Thouless transitions, the two distinct phase-transition temperatures are estimated to be T1 = 0.70 and T2 = 0.88.The results are in agreement with earlier studies.It should be noted that no thermodynamic functions have been used in this study.
We investigate the Berezinskii-Kosterlitz-Thouless transitions for the square-lattice six-state clock model with the corner-transfer matrix renormalization group (CTMRG). Scaling analyses for effective correlation length, magnetization, and entanglement entropy with respect to the cutoff dimension m at the fixed point of the CTMRG provide transition temperatures consistent with a variety of recent numerical studies. We also reveal that the fixed-point spectrum of the corner-transfer matrix in the critical intermediate phase of the six-state clock model is characterized by the scaling dimension consistent with the c=1 boundary conformal field theory associated with the effective Z_{6} dual sine-Gordon model.
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.
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.
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.
We have examined the critical properties of the q=6 clock (vector Potts) model in two dimensions through Monte Carlo simulations. The model was investigated on L\ifmmode\times\else\texttimes\fi{}L square lattices of size L=4 to L=72 with periodic boundary conditions. We found an intermediate XY-like phase between a low-temperature ordered phase and a high-temperature disordered phase. The phase transitions occur at kBT1/J=0.68\ifmmode\pm\else\textpm\fi{}0.02 and kBT2/J=0.92\ifmmode\pm\else\textpm\fi{}0.01 and are of the Kosterlitz-Thouless type. The susceptibility diverges in the intermediate phase and the exponent \ensuremath{\eta} varies between 0.100 at T1 and 0.275 at T2.