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Physical Review Research

Tensor-network renormalization approach to the q -state clock model

Guanrong Li, Kwok Ho Pai, Zheng‐Cheng Gu

Published May 31, 2022• 25 citations
Publisher

Abstract

We simulate the phase diagram and critical behavior of the $q$-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 $R$ of the compactified boson theories at both transition points with high precision. In particular, the radius $R$ 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 $q$ and the corresponding operator product expansion (OPE) coefficient of the compactified boson theory can also be read out directly from the fixed-point tensor.

DOI
10.1103/physrevresearch.4.023159

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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≥5q\ensuremath{\ge}5q\ensuremath≥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 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.
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 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 \leq 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.
In this thesis, we present a novel method combining energy-based finite-size scaling with tensor network renormalization (TNR) to study phase transitions in lattice models. This approach effectively calculates running coupling constants and reduces the numerical errors typically associated with TNR, thus requiring fewer renormalization group (RG) steps and less computational resources. Our methodology, contrasting with traditional methods, doesn't depend on large system sizes, making it efficient and robust against simulation scale challenges. We also explore the origins of numerical errors in TNR from a field-theoretical perspective, focusing on how these errors scale with the approximation parameter DDD. This understanding is crucial for error management in simulations. Moreover, we investigate the tensor structure of fixed points in lattice models, addressing challenges from finite bond dimensions using an analytical approach involving conformal mappings. We discover that the tensor elements of fixed-point tensors align with four-point functions of primary operators in conformal field theory (CFT), demonstrating a significant link between CFT and lattice models. This finding underscores the universality of non-trivial infrared physics at the lattice level, bridging theoretical concepts with practical computations in lattice models, and offering deeper insights into the universal aspects of critical phenomena.