The Berezinskii-Kosterlitz-Thouless (BKT) transition, the transition of the two-dimensional sine-Gordon model, plays an important role in low-dimensional physics. We relate the operator content of the BKT transition to that of the SU (2) Wess-Zumino-Witten model, using twisted boundary conditions. With this method, in order k - 1 to determine the BKT critical point, we can use the level crossing of the lower excitations instead of those for the periodic boundary case, thus the convergence to the transition point is highly improved. We verify the efficiency of this method by applying it to the S = 1, 2 spin chains.
The Berezinskii-Kosterlitz-Thouless (BKT) transition, the transition of the two-dimensional sine-Gordon model, plays an important role in low-dimensional physics. We relate the operator content of the BKT transition to that of the SU (2) Wess-Zumino-Witten model, using twisted boundary conditions. With this method, in order k - 1 to determine the BKT critical point, we can use the level crossing of the lower excitations instead of those for the periodic boundary case, thus the convergence to the transition point is highly improved. We verify the efficiency of this method by applying it to the S = 1, 2 spin chains.
A number of two-dimensional (2D) critical phenomena can be described in terms of the 2D sine-Gordon model. With bosonization, several 1D quantum systems can be transformed to the same model. However, the transition of the 2D sine-Gordon model, the Berezinskii-Kosterlitz-Thouless (BKT) transition, is essentially different from a second-order transition. The divergence of the correlation length is more rapid than any power law, and there are logarithmic corrections. These pathological features make it difficult to determine the BKT transition point and critical indices from finite-size calculations. In this paper we calculate correlation functions of this model using a real-space renormalization technique. It is found that several correlation functions, or eigenvalues of the corresponding transfer matrix for a finite system, become degenerate on the BKT line, including the logarithmic corrections. By the use of this degeneracy, which reflects the hidden SU(2) symmetry on the BKT line, it is possible to determine the BKT critical line with high precision from a small amount of data and to identify the universality class. In addition, new universal relations are found. These results shed light on the relation between Abelian and non-Abelian bosonization.
The Berezinskii-Kosterlitz-Thouless (BKT) phase transition is considered in the condition of lowest temperatures, when thermal fluctuations give place to quantum ones. For this goal, the critical dynamic of the Sine-Gordon model near the quantum critical point is considered. The approach based on the Keldysh-Schwinger technique of non-equilibrium dynamics description is used, as well as the method of taking into account the crossover from the thermal fluctuation regime to the quantum one in the renormalization group technique. For the system with low vortex concentration, it is shown that the BKT transition unavoidably occurs at a temperature above the crossover temperature from the thermal fluctuation regime to quantum one. As a result at small temperatures the critical exponent of the BKT transition, η=1/4, remains unchanged, however, the universal relation of the superfluid density jump adjusts a little from the well-known relation by Nelson and Kosterlitz.
The sine–Gordon model describes topological phase transitions in 1 + 1 systems, referred to as Berezinskii–Kosterlitz–Thouless transitions (BKTT), such as the superfluid–normal fluid, plasma–dielectric and roughening transitions. Analysing the sine–Gordon model in detail using boundary conformal field theory, we develop a new method for treating the BKTT under open boundary conditions (OBC). In particular we find that the chiral parts of the vertex operators are useful for considering the BKTT under OBC.
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 locality of field theories strongly constrains the possible behaviors of symmetry-twisted partition functions, and thus they serve as order parameters to detect low-energy realizations of global symmetries, such as spontaneous symmetry breaking (SSB). We demonstrate that the tensor renormalization group (TRG) offers an efficient framework to compute the symmetry-twisted partition functions, which enables us to detect the symmetry-breaking transition and also to study associated critical phenomena. As concrete examples of SSB, we investigate the two-dimensional (2D) classical Ising model and the three-dimensional (3D) classical O ( 2 ) nonlinear sigma model, and we identify their critical points solely from the twisted partition function. By employing the finite-size scaling argument, we find the critical temperature T c = 2.2017 ( 2 ) with the critical exponent ν = 0.663 ( 33 ) for the 3D O ( 2 ) model. In addition, we also study the Berezinskii–Kosterlitz–Thouless (BKT) criticality of the 2D classical O ( 2 ) model by extracting the helicity modulus from the twisted partition functions, and we obtain the BKT transition temperature, T BKT = 0.8928 ( 2 ) .