Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2302.02585 · Physical review. B./Physical review. B

Tensor network based finite-size scaling for two-dimensional Ising model

Ching-Yu Huang, Sing-Hong Chan, Ying-Jer Kao, Pochung Chen

Published May 11, 2023• 13 citations
arXivHTMLPDF

Abstract

We propose a scheme to perform tensor network based finite-size scaling analysis for two-dimensional classical models. In the tensor network representation of the partition function, we use higher-order tensor renormalization group (HOTRG) method to coarse grain the weight tensor. The renormalized tensor is then used to construct the approximated transfer matrix of an infinite strip of finite width. By diagonalizing the transfer matrix we obtain the correlation length, the magnetization, and the energy density, which are used in finite-size scaling analysis to determine the critical temperature and the critical exponents. As a benchmark we study the two-dimensional classical Ising model. We show that the critical temperature and the critical exponents can be accurately determined. With HOTRG bond dimension $D=70$, the absolute errors of the critical temperature ${T}_{c}$ and the critical exponent $\ensuremath{\nu}$, $\ensuremath{\beta}$ are at the order of ${10}^{\ensuremath{-}7},{10}^{\ensuremath{-}5}$, ${10}^{\ensuremath{-}4}$, respectively. Furthermore, the results can be systematically improved by increasing the bond dimension of the HOTRG method. Finally, we study the length scale induced by the finite cutoff in bond dimension and elucidate its physical meaning in this context.

DOI
10.1103/physrevb.107.205123

Related papers

Tensor network finite-size scaling for two-dimensional 3-state clock model

Debasmita Maiti, Sing-Hong Chan, Pochung Chen

Abstract We benchmark recently proposed tensor network based finite-size scaling analysis in (Phys. Rev. B 2023 107 205123) against two-dimensional classical 3-state clock model. Due to the higher complexity of the model, more complicated crossover behavior is observed. We advocate that the crossover behavior can be understood from the perspective of finite bond dimension inducing relevant perturbation. This leads to a general strategy to best estimate the critical properties for a given set of control parameters. For the critical temperature T c , the relative error at the order of 10−7 can be reached with bond dimension D = 70. On the other hand, with bond dimension D = 60, the relative errors of the critical exponents ν , β , α are at the order of 10−2. Increasing the bond dimension to D = 90, these relative errors can be reduced at least to the order of 10−3. In all cases our results indicate that the errors can be systematically reduced by increasing the bond dimension and the stacking number.

arXiv:2312.14002•New Journal of Physics•Apr 24, 2025•1 citation

Tensor Network Methods for Extracting CFT Data from Fixed-Point Tensors and Defect Coarse Graining

Wenhan Guo, Tzu-Chieh Wei

We present a comprehensive study on the extraction of CFT data using tensor network methods, specially, from the fixed-point tensor of the linearized tensor renormalization group (lTRG) for the 2D classical Ising model near the critical temperature. Utilizing two different methods, we extract operator scaling dimensions and operator-product-expansion (OPE) coefficients by introducing defects on the lattice and by employing the fixed-point tensor. We also explore the effects of point-like defects in the lattice on the coarse-graining process. We find that there is a correspondence between coarse-grained defect tensors and conformal states obtained from lTRG fixed-point equation. We also analyze the capabilities and limitations of our proposed coarse-graining scheme for tensor networks with point-like defects, which includes graph independent local truncation (GILT) and higher-order tensor renormalization group (HOTRG). Our results provide a better understanding of the capacity and limitations of the tenor renormalization group scheme in coarse-graining defect tensors, and we show that GILT+HOTRG can be used to give accurate two- and four-point functions under specific conditions. We also find that employing the minimal canonical form further improves the stability of the RG flow.

arXiv:2305.09899•May 17, 2023•4 citations

Tensor network methods for extracting conformal field theory data from fixed-point tensors and defect coarse graining

Wenhan Guo, Tzu-Chieh Wei

We present a comprehensive study on extracting conformal field theory data using tensor network methods, especially, from the fixed-point tensor of the linearized tensor renormalization group (lTRG) for the classical two-dimensional Ising model near the critical temperature. Utilizing two different methods, we extract operator scaling dimensions and operator product expansion coefficients by introducing defects on the lattice and by employing the fixed-point tensor. We also explore the effects of pointlike defects in the lattice on the coarse-graining process. We find that there is a correspondence between coarse-grained defect tensors and conformal states obtained from the lTRG fixed-point equation. We also analyze the capabilities and limitations of our proposed coarse-graining scheme for tensor networks with pointlike defects, including graph-independent local truncation (GILT) and higher-order tensor renormalization group (HOTRG). Our results provide a better understanding of the capacity and limitations of the tensor renormalization group scheme in coarse-graining defect tensors, and we show that GILT+HOTRG can be used to give accurate two- and four-point functions under specific conditions. We also find that employing the minimal canonical form further improves the stability of the RG flow.

Physical review. E•Mar 7, 2024•9 citations

Finite-size and finite bond dimension effects of tensor network renormalization

Atsushi Ueda, Masaki Oshikawa

We propose a general procedure for extracting the running coupling constants of the underlying field theory of a given classical statistical model on a two-dimensional lattice, combining tensor network renormalization (TNR) and the finite-size scaling theory of conformal field theory. By tracking the coupling constants at each scale, we are able to visualize the renormalization group flow and demonstrate it with the classical Ising and three-state Potts models. Furthermore, utilizing this methodology, we reveal the limitations due to finite bond dimension $D$ on TNR applied to critical systems. We find that a finite correlation length is imposed by the finite bond dimension in TNR, and it can be attributed to an emergent relevant perturbation that respects the symmetries of the system. The correlation length shows the same power-law dependence on $D$ as the ``finite entanglement scaling'' of the matrix product states.

arXiv:2302.06632•Physical review. B./Physical review. B•Jul 14, 2023•30 citations

Tensor network algorithm by coarse-graining tensor renormalization on finite periodic lattices

Hui‐Hai Zhao, Z. Y. Xie, Tao Xiang +1 more

We develop coarse-graining tensor renormalization group algorithms to compute physical properties of two-dimensional lattice models on finite periodic lattices. Two different coarse-graining strategies, one based on the tensor renormalization group and the other based on the higher-order tensor renormalization group, are introduced. In order to optimize the tensor network model globally, a sweeping scheme is proposed to account for the renormalization effect from the environment tensors under the framework of second renormalization group. We demonstrate the algorithms by the classical Ising model on the square lattice and the Kitaev model on the honeycomb lattice, and show that the finite-size algorithms achieve substantially more accurate results than the corresponding infinite-size ones.

arXiv:1510.03333•Physical review. B./Physical review. B•Mar 9, 2016•40 citations

Multi-impurity method for the bond-weighted tensor renormalization group

Satoshi Morita, Naoki Kawashima

We propose a multi-impurity method for the bond-weighted tensor renormalization group (BWTRG) to compute the higher-order moment of physical quantities in a two-dimensional system. The replacement of the bond weight with an impurity matrix in a bond-weighted triad tensor network represents a physical quantity such as the magnetization and the energy. We demonstrate that the accuracy of the proposed method is much higher than the conventional tensor renormalization group for the Ising model and the five-state Potts model. Furthermore, we perform the finite-size scaling analysis and observe that the dimensionless quantity characterizing the structure of the fixed point tensor satisfies the same scaling relation in the critical region as the Binder parameter. The estimated critical temperature dependence on the bond dimension indicates that the exponent relating the correlation length to the bond dimension varies continuously with respect to the BWTRG hyperparameter. We find that BWTRG with the optimal hyperparameter is more efficient in terms of computational time than alternative approaches based on the matrix product state in estimating the critical temperature.

arXiv:2411.13998•Physical review. B./Physical review. B•Feb 24, 2025•9 citations
We propose a scheme to perform tensor network based finite-size scaling analysis for two-dimensional classical models. In the tensor network representation of the partition function, we use higher-order tensor renormalization group (HOTRG) method to coarse grain the weight tensor. The renormalized tensor is then used to construct the approximated transfer matrix of an infinite strip of finite width. By diagonalizing the transfer matrix we obtain the correlation length, the magnetization, and the energy density, which are used in finite-size scaling analysis to determine the critical temperature and the critical exponents. As a benchmark we study the two-dimensional classical Ising model. We show that the critical temperature and the critical exponents can be accurately determined. With HOTRG bond dimension D=70D=70D=70, the absolute errors of the critical temperature Tc{T}_{c}Tc​ and the critical exponent \ensuremathν\ensuremath{\nu}\ensuremathν, \ensuremathβ\ensuremath{\beta}\ensuremathβ are at the order of 10\ensuremath−7,10\ensuremath−5{10}^{\ensuremath{-}7},{10}^{\ensuremath{-}5}10\ensuremath−7,10\ensuremath−5, 10\ensuremath−4{10}^{\ensuremath{-}4}10\ensuremath−4, respectively. Furthermore, the results can be systematically improved by increasing the bond dimension of the HOTRG method. Finally, we study the length scale induced by the finite cutoff in bond dimension and elucidate its physical meaning in this context.
Abstract We benchmark recently proposed tensor network based finite-size scaling analysis in (Phys. Rev. B 2023 107 205123) against two-dimensional classical 3-state clock model. Due to the higher complexity of the model, more complicated crossover behavior is observed. We advocate that the crossover behavior can be understood from the perspective of finite bond dimension inducing relevant perturbation. This leads to a general strategy to best estimate the critical properties for a given set of control parameters. For the critical temperature T c , the relative error at the order of 10−7 can be reached with bond dimension D = 70. On the other hand, with bond dimension D = 60, the relative errors of the critical exponents ν , β , α are at the order of 10−2. Increasing the bond dimension to D = 90, these relative errors can be reduced at least to the order of 10−3. In all cases our results indicate that the errors can be systematically reduced by increasing the bond dimension and the stacking number.
We present a comprehensive study on the extraction of CFT data using tensor network methods, specially, from the fixed-point tensor of the linearized tensor renormalization group (lTRG) for the 2D classical Ising model near the critical temperature. Utilizing two different methods, we extract operator scaling dimensions and operator-product-expansion (OPE) coefficients by introducing defects on the lattice and by employing the fixed-point tensor. We also explore the effects of point-like defects in the lattice on the coarse-graining process. We find that there is a correspondence between coarse-grained defect tensors and conformal states obtained from lTRG fixed-point equation. We also analyze the capabilities and limitations of our proposed coarse-graining scheme for tensor networks with point-like defects, which includes graph independent local truncation (GILT) and higher-order tensor renormalization group (HOTRG). Our results provide a better understanding of the capacity and limitations of the tenor renormalization group scheme in coarse-graining defect tensors, and we show that GILT+HOTRG can be used to give accurate two- and four-point functions under specific conditions. We also find that employing the minimal canonical form further improves the stability of the RG flow.
We present a comprehensive study on extracting conformal field theory data using tensor network methods, especially, from the fixed-point tensor of the linearized tensor renormalization group (lTRG) for the classical two-dimensional Ising model near the critical temperature. Utilizing two different methods, we extract operator scaling dimensions and operator product expansion coefficients by introducing defects on the lattice and by employing the fixed-point tensor. We also explore the effects of pointlike defects in the lattice on the coarse-graining process. We find that there is a correspondence between coarse-grained defect tensors and conformal states obtained from the lTRG fixed-point equation. We also analyze the capabilities and limitations of our proposed coarse-graining scheme for tensor networks with pointlike defects, including graph-independent local truncation (GILT) and higher-order tensor renormalization group (HOTRG). Our results provide a better understanding of the capacity and limitations of the tensor renormalization group scheme in coarse-graining defect tensors, and we show that GILT+HOTRG can be used to give accurate two- and four-point functions under specific conditions. We also find that employing the minimal canonical form further improves the stability of the RG flow.
We propose a general procedure for extracting the running coupling constants of the underlying field theory of a given classical statistical model on a two-dimensional lattice, combining tensor network renormalization (TNR) and the finite-size scaling theory of conformal field theory. By tracking the coupling constants at each scale, we are able to visualize the renormalization group flow and demonstrate it with the classical Ising and three-state Potts models. Furthermore, utilizing this methodology, we reveal the limitations due to finite bond dimension DDD on TNR applied to critical systems. We find that a finite correlation length is imposed by the finite bond dimension in TNR, and it can be attributed to an emergent relevant perturbation that respects the symmetries of the system. The correlation length shows the same power-law dependence on DDD as the ``finite entanglement scaling'' of the matrix product states.
We develop coarse-graining tensor renormalization group algorithms to compute physical properties of two-dimensional lattice models on finite periodic lattices. Two different coarse-graining strategies, one based on the tensor renormalization group and the other based on the higher-order tensor renormalization group, are introduced. In order to optimize the tensor network model globally, a sweeping scheme is proposed to account for the renormalization effect from the environment tensors under the framework of second renormalization group. We demonstrate the algorithms by the classical Ising model on the square lattice and the Kitaev model on the honeycomb lattice, and show that the finite-size algorithms achieve substantially more accurate results than the corresponding infinite-size ones.
We propose a multi-impurity method for the bond-weighted tensor renormalization group (BWTRG) to compute the higher-order moment of physical quantities in a two-dimensional system. The replacement of the bond weight with an impurity matrix in a bond-weighted triad tensor network represents a physical quantity such as the magnetization and the energy. We demonstrate that the accuracy of the proposed method is much higher than the conventional tensor renormalization group for the Ising model and the five-state Potts model. Furthermore, we perform the finite-size scaling analysis and observe that the dimensionless quantity characterizing the structure of the fixed point tensor satisfies the same scaling relation in the critical region as the Binder parameter. The estimated critical temperature dependence on the bond dimension indicates that the exponent relating the correlation length to the bond dimension varies continuously with respect to the BWTRG hyperparameter. We find that BWTRG with the optimal hyperparameter is more efficient in terms of computational time than alternative approaches based on the matrix product state in estimating the critical temperature.