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arXiv:2403.17309 · Physical review. B./Physical review. B

Tensor network renormalization approach to antiferromagnetic 6-state clock model on the Union Jack lattice

Kenji Homma, Satoshi Morita, Naoki Kawashima

Published Apr 17, 2025• 2 citations
arXivHTMLPDF

Abstract

Using the nuclear norm regularization techniques on tensor network renormalization algorithm, we study the phase diagram, the critical behavior, and the duality property of the antiferromagnetic 6-state clock model on the Union Jack lattice. We find that this model undergoes multiple phase transitions; there is the Berezinskii-Kosterlitz-Thouless, ${Z}_{6}$ symmetry breaking, and chiral transition with decreasing temperature. Furthermore, we provide convincing numerical evidence that its quasi-long-range order is well explained by the compactified boson conformal field theory (CFT) and the chiral transition is in perfect agreement with the Ising CFT, including central charge, scaling dimension spectrum, and operator product expansion coefficients.

DOI
10.1103/physrevb.111.134427

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Using the nuclear norm regularization techniques on tensor network renormalization algorithm, we study the phase diagram, the critical behavior, and the duality property of the antiferromagnetic 6-state clock model on the Union Jack lattice. We find that this model undergoes multiple phase transitions; there is the Berezinskii-Kosterlitz-Thouless, Z6{Z}_{6}Z6​ symmetry breaking, and chiral transition with decreasing temperature. Furthermore, we provide convincing numerical evidence that its quasi-long-range order is well explained by the compactified boson conformal field theory (CFT) and the chiral transition is in perfect agreement with the Ising CFT, including central charge, scaling dimension spectrum, and operator product expansion coefficients.
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.
We investigate the triangular-lattice antiferromagnetic Ising model with a spatially anisotropic next-nearest-neighbor ferromagnetic coupling, which was first discussed by Kitatani and Oguchi. By employing the effective geometric factor, we analyze the scaling dimensions of the operators around the Berezinskii-Kosterlitz-Thouless (BKT) transition lines, and determine the global phase diagram. Our numerical data exhibit that two types of BKT-transition lines separate the intermediate critical region from the ordered and disordered phases, and they do not merge into a single curve in the antiferromagnetic region. We also estimate the central charge and perform some consistency checks among scaling dimensions in order to provide the evidence of the six-state clock universality. Further, we provide an analysis of the shapes of boundaries based on the crossover argument.
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.
The q-state clock model, sometimes called the discrete XY model, is known to show a second-order (symmetry-breaking) phase transition in two dimensions for q≤4 (q=2 corresponds to the Ising model). On the other hand, the q→∞ limit of the model corresponds to the XY model, which shows the infinite order (non-symmetry-breaking) Berezinskii-Kosterlitz-Thouless (BKT) phase transition in two dimensions. Interestingly, the two-dimensional clock model with q≥5 is predicted to show three different phases and two associated phase transitions. There are varying opinions about the actual characters of phases and the associated transitions. In this work, we develop the basic and higher-order mean-field (MF) theories to study the q-state clock model systematically. Our MF calculations reaffirm that, for large q, there are three phases: (broken) Z_{q} symmetric ferromagnetic phase at the low temperature, emergent U(1) symmetric BKT phase at the intermediate temperature, and paramagnetic (disordered) phase at the high temperature. The phase transition at the higher temperature is found to be of the BKT type, and the other transition at the lower temperature is argued to be a large-order spontaneous symmetry-breaking type (the largeness of transition order yields the possibility of having some of the numerical characteristics of a BKT transition). The higher-order MF theory developed here better characterizes phases by estimating the spin-spin correlation between two neighbors.