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arXiv:2205.02642 · Physical review. E

Monte Carlo study of duality and the Berezinskii-Kosterlitz-Thouless phase transitions of the two-dimensional q -state clock model in flow representations

Hao Chen, Pengcheng Hou, Sheng Fang, Youjin Deng

Published Aug 5, 2022• 22 citations
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Abstract

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.

DOI
10.1103/physreve.106.024106

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The two-dimensional qqq-state clock model for q\ensuremath≥5q\ensuremath{\ge}5q\ensuremath≥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 \ensuremathη1=1/4{\ensuremath{\eta}}_{1}=1/4\ensuremathη1​=1/4 and \ensuremathη2=4/q2{\ensuremath{\eta}}_{2}=4/{q}^{2}\ensuremathη2​=4/q2 at the two BKT transitions. Further, the approximate self-dual points \ensuremathβsd(L){\ensuremath{\beta}}_{\mathrm{sd}}(L)\ensuremathβsd​(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 \ensuremathβsd\ensuremath≃q/2\ensuremathπ{\ensuremath{\beta}}_{\mathrm{sd}}\ensuremath{\simeq}q/2\ensuremath{\pi}\ensuremathβsd​\ensuremath≃q/2\ensuremathπ asymptotically at the large-qqq limit. The exponent \ensuremathη\ensuremath{\eta}\ensuremathη at \ensuremathβsd{\ensuremath{\beta}}_{\mathrm{sd}}\ensuremathβsd​ is consistent with 1/q1/q1/q within statistical error as long as q\ensuremath≥5q\ensuremath{\ge}5q\ensuremath≥5. Based on this, we further conjecture that \ensuremathη(\ensuremathβsd)=1/q\ensuremath{\eta}({\ensuremath{\beta}}_{\mathrm{sd}})=1/q\ensuremathη(\ensuremathβsd​)=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.
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.
Quantum phase transitions are studied in the nonchiral ppp-clock chain, and a new explicitly U(1)-symmetric clock model, by monitoring the ground-state fidelity susceptibility. For $p\ensuremath{\ge}5$, the self-dual Zp{\mathbb{Z}}_{p}Zp​-symmetric chain displays a double-hump structure in the fidelity susceptibility with both peak positions and heights scaling logarithmically to their corresponding thermodynamic values. This scaling is precisely as expected for two Beresinskii-Kosterlitz-Thouless (BKT) transitions located symmetrically about the self-dual point, and so confirms numerically the theoretical scenario that sets p=5p=5p=5 as the lowest ppp supporting BKT transitions in Zp{\mathbb{Z}}_{p}Zp​-symmetric clock models. For our U(1)-symmetric, non-self-dual minimal modification of the ppp-clock model we find that the phase diagram depends strongly on the parity of ppp and only one BKT transition survives for $p\ensuremath{\ge}5$. Using asymptotic calculus we map the self-dual clock model exactly, in the large ppp limit, to the quantum O(2)O(2)O(2) rotor chain. Finally, using bond-algebraic dualities we estimate the critical BKT transition temperatures of the classical planar ppp-clock models defined on square lattices, in the limit of extreme spatial anisotropy. Our values agree remarkably well with those determined via classical Monte Carlo for isotropic lattices. This work highlights the power of the fidelity susceptibility as a tool for diagnosing the BKT transitions even when only discrete symmetries are present.
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.
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=2q=2q=2 to q=20q=20q=20 for small square lattices L×LL \times LL×L, with L ranging from L=3L=3L=3 to L=64L=64L=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>5q>5q>5 regardless of lattice size, while the transition at q=5q=5q=5 is lost for L<10L<10L<10; for q≤4q\leq 4q≤4 the BKT transition is never present. We report the phase diagram in terms of qqq showing the transition from the ferromagnetic (FM) to the paramagnetic (PM) phases at a critical temperature T1_11​ 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_22​. 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 qqq 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 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.