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Berezinskii-Kosterlitz-Thouless transitions in the six-state clock model

Haruhiko Matsuo, K. Nomura

Published Jan 1, 2008• 10 citations
OpenAlex

Abstract

Classical 2D clock model is known to have a critical phase with Berezinskii-Kosterlitz-Thouless(BKT) transitions. These transitions have logarithmic corrections which make numerical analysis difficult. In order to resolve this difficulty, one of the authors has proposed the method called level spectroscopy, which is based on the conformal field theory. We extend this method to the multi-degenerated case. As an example, we study the classical 2D 6-clock model which can be mapped to the quantum self-dual 1D 6-clock model. Additionally, we confirm that the self-dual point has a precise numerical agreement with the analytical result, and we argue the degeneracy of the excitation states at the self-dual point from the effective field theoretical point of view. 1

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Berezinskii–Kosterlitz–Thouless transitions in the six-state clock model

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A classical 2D clock model is known to have a critical phase with Berezinskii–Kosterlitz–Thouless (BKT) transitions. These transitions have logarithmic corrections which make numerical analysis difficult. In order to resolve this difficulty, one of the authors has proposed a method called 'level spectroscopy', which is based on the conformal field theory. We extend this method to the multi-degenerated case. As an example, we study the classical 2D six-clock model which can be mapped to the quantum self-dual 1D six-clock model. Additionally, we confirm that the self-dual point has a precise numerical agreement with the analytical result, and we argue the degeneracy of the excitation states at the self-dual point from the effective field theoretical point of view.

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Berezinskii-Kosterlitz-Thouless (BKT) transition is one of the instability mechanisms for the Tomonaga-Luttinger liquid. But in the BKT transition, there are logarithmic-correction problems, which make it very difficult to treat BKT transitions numerically. We propose a method, ``level spectroscopy'', to overcome such difficulties, based on the renormalization group analysis and the symmetry consideration.

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BKT transitions of the XY and six-state clock models on the various two-dimensional lattices

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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.

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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.

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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.

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The two-dimensional $q$-state clock model for $q\ensuremath{\ge}5$ undergoes two Berezinskii-Kosterlitz-Thouless (BKT) phase transitions as temperature decreases. Here we report an extensive worm-type simulation of the square-lattice clock model for $q=5$--9 in a pair of flow representations, from high- and low-temperature expansions, respectively. By finite-size scaling analysis of susceptibilitylike quantities, we determine the critical points with a precision improving over the existing results. Due to the dual flow representations, each point in the critical region is observed to simultaneously exhibit a pair of anomalous dimensions, which are ${\ensuremath{\eta}}_{1}=1/4$ and ${\ensuremath{\eta}}_{2}=4/{q}^{2}$ at the two BKT transitions. Further, the approximate self-dual points ${\ensuremath{\beta}}_{\mathrm{sd}}(L)$, defined by the stringent condition that the susceptibilitylike quantities in both flow representations are identical, are found to be nearly independent of system size $L$ and behave as ${\ensuremath{\beta}}_{\mathrm{sd}}\ensuremath{\simeq}q/2\ensuremath{\pi}$ asymptotically at the large-$q$ limit. The exponent $\ensuremath{\eta}$ at ${\ensuremath{\beta}}_{\mathrm{sd}}$ is consistent with $1/q$ within statistical error as long as $q\ensuremath{\ge}5$. Based on this, we further conjecture that $\ensuremath{\eta}({\ensuremath{\beta}}_{\mathrm{sd}})=1/q$ holds exactly and is universal for systems in the $q$-state clock universality class. Our work provides a vivid demonstration of rich phenomena associated with the duality and self-duality of the clock model in two dimensions.

arXiv:2205.02642•Physical review. E•Aug 5, 2022•22 citations
Classical 2D clock model is known to have a critical phase with Berezinskii-Kosterlitz-Thouless(BKT) transitions. These transitions have logarithmic corrections which make numerical analysis difficult. In order to resolve this difficulty, one of the authors has proposed the method called level spectroscopy, which is based on the conformal field theory. We extend this method to the multi-degenerated case. As an example, we study the classical 2D 6-clock model which can be mapped to the quantum self-dual 1D 6-clock model. Additionally, we confirm that the self-dual point has a precise numerical agreement with the analytical result, and we argue the degeneracy of the excitation states at the self-dual point from the effective field theoretical point of view. 1
A classical 2D clock model is known to have a critical phase with Berezinskii–Kosterlitz–Thouless (BKT) transitions. These transitions have logarithmic corrections which make numerical analysis difficult. In order to resolve this difficulty, one of the authors has proposed a method called 'level spectroscopy', which is based on the conformal field theory. We extend this method to the multi-degenerated case. As an example, we study the classical 2D six-clock model which can be mapped to the quantum self-dual 1D six-clock model. Additionally, we confirm that the self-dual point has a precise numerical agreement with the analytical result, and we argue the degeneracy of the excitation states at the self-dual point from the effective field theoretical point of view.
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.
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.
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.
The two-dimensional qqq-state clock model for $q\ensuremath{\ge}5$ undergoes two Berezinskii-Kosterlitz-Thouless (BKT) phase transitions as temperature decreases. Here we report an extensive worm-type simulation of the square-lattice clock model for q=5q=5q=5--9 in a pair of flow representations, from high- and low-temperature expansions, respectively. By finite-size scaling analysis of susceptibilitylike quantities, we determine the critical points with a precision improving over the existing results. Due to the dual flow representations, each point in the critical region is observed to simultaneously exhibit a pair of anomalous dimensions, which are ${\ensuremath{\eta}}_{1}=1/4$ and ${\ensuremath{\eta}}_{2}=4/{q}^{2}$ at the two BKT transitions. Further, the approximate self-dual points ${\ensuremath{\beta}}_{\mathrm{sd}}(L)$, defined by the stringent condition that the susceptibilitylike quantities in both flow representations are identical, are found to be nearly independent of system size LLL and behave as ${\ensuremath{\beta}}_{\mathrm{sd}}\ensuremath{\simeq}q/2\ensuremath{\pi}$ asymptotically at the large-qqq limit. The exponent $\ensuremath{\eta}$ at ${\ensuremath{\beta}}_{\mathrm{sd}}$ is consistent with 1/q1/q1/q within statistical error as long as $q\ensuremath{\ge}5$. Based on this, we further conjecture that $\ensuremath{\eta}({\ensuremath{\beta}}_{\mathrm{sd}})=1/q$ holds exactly and is universal for systems in the qqq-state clock universality class. Our work provides a vivid demonstration of rich phenomena associated with the duality and self-duality of the clock model in two dimensions.