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arXiv:1706.03455 · Chinese Physics Letters

Phase Transition of the q -State Clock Model: Duality and Tensor Renormalization

Jing Chen, Haijun Liao, Haidong Xie, Xing-Jie Han, Rui-Zhen Huang, Song Cheng, Zhongchao Wei, Z. Y. Xie, Tao Xiang

Published May 1, 2017• 46 citations
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Abstract

We investigate the critical behavior and the duality property of the ferromagnetic $q$-state clock model on the square lattice based on the tensor-network formalism. From the entanglement spectra of local tensors defined in the original and dual lattices, we obtain the exact self-dual points for the model with $q \leq 5 $ and approximate self-dual points for $q \geq 6$. We calculate accurately the lower and upper critical temperatures for the six-state clock model from the fixed-point tensors determined using the higher-order tensor renormalization group method and compare with other numerical results.

DOI
10.1088/0256-307x/34/5/050503

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We investigate the critical behavior and the duality property of the ferromagnetic qqq-state clock model on the square lattice based on the tensor-network formalism. From the entanglement spectra of local tensors defined in the original and dual lattices, we obtain the exact self-dual points for the model with q≤5q \leq 5 q≤5 and approximate self-dual points for q≥6q \geq 6q≥6. We calculate accurately the lower and upper critical temperatures for the six-state clock model from the fixed-point tensors determined using the higher-order tensor renormalization group method and compare with other numerical results.
Using the tensor renormalization group method based on the higher-order singular value decomposition, we have studied the phase transitions of the five-state clock model on the square lattice. The temperature dependence of the specific heat indicates the system has two phase transitions, as verified clearly by the correlation function at three representative temperatures. By calculating the magnetic susceptibility, we obtained only the upper critical temperature as T c2 = 0.9565(7). Investigating the fixed-point tensor, we precisely locate the transition temperatures at T c1 = 0.9029(1) and T c2 = 0.9520(1), consistent well with the Monte Carlo and the density matrix renormalization group results.
We simulate the phase diagram and critical behavior of the qqq-state clock model on the square lattice by using the state-of-the-art loop optimization for tensor-network renormalization (loop-TNR) algorithm. The two phase transition points for $q\ensuremath{\ge}5$ are determined with very high accuracy. Furthermore, by computing the conformal scaling dimensions for both transition points, we are able to determine the radius RRR of the compactified boson theories at both transition points with high precision. In particular, the radius RRR at higher temperature phase transition point is precisely the same as the one predicted by Berezinskii-Kosterlitz-Thouless (BKT) transition. Moreover, we find that the fixed-point tensors at higher temperature transition point also converge to the same one approximately for large enough qqq and the corresponding operator product expansion (OPE) coefficient of the compactified boson theory can also be read out directly from the fixed-point tensor.
We accurately simulate the phase diagram and critical behavior of the qqq-state clock model on the square lattice by using the state-of-the-art loop optimization for tensor network renormalzation(loop-TNR) algorithm. The two phase transition points for q≥5q \geq 5q≥5 are determined with very high accuracy. Furthermore, by computing the conformal scaling dimensions, we are able to accurately determine the compactification radius RRR of the compactified boson theories at both phase transition points. In particular, the compactification radius RRR at high-temperature critical point is precisely the same as the predicted RRR for Berezinskii-Kosterlitz-Thouless (BKT) transition. Moreover, we find that the fixed point tensors at high-temperature critical point also converge(up to numerical errors) to the same one for large enough qqq and the corresponding operator product expansion(OPE) coefficient of the compactified boson theory can also be read out directly from the fixed point tensor.
The permutation model is a classical spin system where elements of the symmetric group interact with one another. The partition function of this model is directly related to the entanglement structure of random quantum circuits and random tensor networks. In these contexts, the entanglement entropy undergoes a transition between area-law and volume-law scaling, depending on the model parameters. This transition point has attracted considerable attention. In the present work, we investigate the ferromagnetic-paramagnetic phase transition of the permutation model, which corresponds to the entanglement entropy transition. Using exact real-space renormalization group calculations on self-dual hierarchical lattices, we numerically determine finite-replica critical points for (q=mn=2,...,6). We compare the results with the duality prediction based on the Fourier transform of the symmetric group and then extrapolate the b=3 data toward the replica limit mn→0mn\to0mn→0, where the effective dimension is two. The comparison supports the duality-based estimate while also clarifying the systematic uncertainty associated with the extrapolation formula.
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.
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.