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

Wenhan Guo, Tzu-Chieh Wei|arXiv (Cornell University)|May 17, 2023

Theoretical and Computational Physics4 citations

TL;DR

This paper presents a tensor network method to extract conformal field theory (CFT) data—such as scaling dimensions and OPE coefficients—from fixed-point tensors of the 2D classical Ising model using linearized tensor renormalization group (lTRG) and graph-independent local truncation (GILT). It demonstrates that inserting defects into the lattice generates conformal states matching lTRG eigenvectors, and that HOTRG+GILT enables accurate two- and four-point functions under controlled conditions, with improved stability via minimal canonical form.

ABSTRACT

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.

Motivation & Objective

• To develop a method for extracting CFT data, including scaling dimensions and OPE coefficients, from fixed-point tensors of critical lattice models.
• To investigate how point-like defects in the lattice affect tensor network coarse-graining and whether they can be used to construct conformal states.
• To evaluate the performance and limitations of HOTRG and GILT in preserving correlation functions and RG flow stability under defect presence.
• To establish a correspondence between coarse-grained defect tensors and conformal states derived from lTRG fixed-point equations.
• To explore the feasibility of constructing fixed-point tensors from CFT data as a reverse bootstrap approach.

Proposed method

• The study uses the linearized tensor renormalization group (lTRG) to extract scaling dimensions and OPE coefficients from the fixed-point tensor of the 2D Ising model near criticality.
• Defects are introduced on the lattice by modifying local tensors to simulate insertions of conformal operators, enabling the construction of conformal states.
• Graph-independent local truncation (GILT) is applied to remove spurious correlations (CDL tensors) and stabilize the RG flow during coarse-graining.
• Higher-order tensor renormalization group (HOTRG) is used for efficient coarse-graining of the lattice, with truncation and filtering applied at each step.
• The eigenvectors of the lTRG fixed-point equation are compared with coarse-grained defect tensors to verify correspondence with conformal states.
• A minimal canonical form is employed to improve the stability of the RG flow and reduce numerical artifacts.

Figure 1 : (color online) Illustration of the renormalization group flow (top) and the changes in tensors during the RG process.

Experimental results

Research questions

• RQ1 Can scaling dimensions and OPE coefficients be accurately extracted from the fixed-point tensor of the 2D Ising model using lTRG and GILT?
• RQ2 Is there a direct correspondence between coarse-grained defect tensors and conformal states obtained from the lTRG fixed-point equation?
• RQ3 How do point-like defects behave under HOTRG and GILT coarse-graining, and what is the impact on correlation functions?
• RQ4 To what extent can GILT+HOTRG preserve two- and four-point correlation functions in the presence of defects?
• RQ5 Can the minimal canonical form enhance the stability and accuracy of the tensor network RG flow for defect systems?

Key findings

• Scaling dimensions for conformal states with Δ < 3 in the critical 2D Ising model were extracted with high accuracy using both lTRG and transfer matrix methods, matching analytical results.
• The OPE coefficient Cσσε was successfully computed from the lTRG eigenvectors, confirming consistency with known CFT data.
• Two-point spin-spin correlation functions were computed near criticality, and the extracted exponent matched the scaling dimension with high precision.
• Four-point correlation functions were computed, and while the quality was lower, they still enabled extraction of scaling dimensions and OPE coefficients.
• Defects were found to be smeared into 'particle clouds' during coarse-graining, with stronger smearing at corners and edges, but this effect could be mitigated to recover accurate correlation functions.
• The eigenvectors of the lTRG fixed-point equation were shown to correspond precisely to coarse-grained defect tensors, validating the construction of conformal states from lattice operator insertions.

Figure 2 : (color online) Local Entanglement resides on the boundary and Long Range Entanglement in the bulk between two rectangular blocks of lattice sites.