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arXiv:2409.13012 · Physical review. B./Physical review. B

Transfer matrix and lattice dilatation operator for high-quality fixed points in tensor network renormalization group

Nikolay Ebel, Tom Kennedy, Slava Rychkov

Published Aug 6, 2025• 4 citations
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Abstract

Tensor network renormalization group maps study critical points of 2D lattice models like the Ising model by finding the fixed point of the RG map. In a prior work [Phys. Rev. X 15, 031047 (2025)]. we showed that by adding a rotation to the RG map, the Newton method could be implemented to find an extremely accurate fixed point. For a particular RG map (Gilt-TNR), we studied the spectrum of the Jacobian of the RG map at the fixed point and found good agreement between the eigenvalues corresponding to relevant and marginal operators and their known exact values. In this companion work, we use two further methods to extract many more scaling dimensions from this Newton method fixed point, and compare the numerical results with the predictions of conformal field theory (CFT). The first method is the well-known transfer matrix (TM), while the second method we refer to as the lattice dilatation operator (LDO). We introduce some extensions of these methods that also provide spins of the CFT operators, modulo an integer. With comparable computing resources, the TM and LDO methods perform equally well. The agreement for the scaling dimensions and spins is excellent up to $\mathrm{\ensuremath{\Delta}}={4}^{1}{/}_{8}$, and reasonably good up to 2 units higher. Some of the eigenvalues of the Jacobian of the RG map can come from perturbations associated with total derivative interactions and so are not universal. In some past studies [Phys. Rev. Res. 3, 023048 (2021); Phys. Rev. E 109, 034111 (2024)] such nonuniversal eigenvalues did not appear in the Jacobian. We explain this surprising result by showing that their RG map has the unusual property that the Jacobian is equivalent to the LDO operator.

DOI
10.1103/y5fk-l98w

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Tensor network renormalization group maps study critical points of 2D lattice models like the Ising model by finding the fixed point of the RG map. In a prior work [Phys. Rev. X 15, 031047 (2025)]. we showed that by adding a rotation to the RG map, the Newton method could be implemented to find an extremely accurate fixed point. For a particular RG map (Gilt-TNR), we studied the spectrum of the Jacobian of the RG map at the fixed point and found good agreement between the eigenvalues corresponding to relevant and marginal operators and their known exact values. In this companion work, we use two further methods to extract many more scaling dimensions from this Newton method fixed point, and compare the numerical results with the predictions of conformal field theory (CFT). The first method is the well-known transfer matrix (TM), while the second method we refer to as the lattice dilatation operator (LDO). We introduce some extensions of these methods that also provide spins of the CFT operators, modulo an integer. With comparable computing resources, the TM and LDO methods perform equally well. The agreement for the scaling dimensions and spins is excellent up to \ensuremathΔ=41/8\mathrm{\ensuremath{\Delta}}={4}^{1}{/}_{8}\ensuremathΔ=41/8​, and reasonably good up to 2 units higher. Some of the eigenvalues of the Jacobian of the RG map can come from perturbations associated with total derivative interactions and so are not universal. In some past studies [Phys. Rev. Res. 3, 023048 (2021); Phys. Rev. E 109, 034111 (2024)] such nonuniversal eigenvalues did not appear in the Jacobian. We explain this surprising result by showing that their RG map has the unusual property that the Jacobian is equivalent to the LDO operator.
In the tensor network approach to statistical physics, properties of the critical point of a 2D lattice model are encoded by a four-legged tensor which is a fixed point of a renormalization group (RG) map. The traditional way to find the fixed-point tensor consists in iterating the RG map after having tuned the temperature to criticality. Here, we develop a different and more direct technique, which solves the fixed-point equation via the Newton method. This is challenging due to the existence of marginal deformations—linear transformations of the coordinate frame, which parametrize a two-dimensional family of fixed points. We address this challenge by including a 90° rotation into the RG map. This flips the sign of the problematic marginal eigenvalues, rendering the fixed point isolated and accessible via the Newton method. We demonstrate the power of this technique via explicit computations for the 2D Ising and three-state Potts models. Using the Gilt-TNR algorithm at bond dimension χ = 30 , we find the fixed-point tensors with 10 − 9 accuracy, much higher than what was previously achieved.
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.
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 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.
On the lattice, a renormalization group (RG) flow for two-dimensional partition functions expressed as a tensor network can be obtained using the tensor network renormalization (TNR) algorithm [G. Evenbly, G. Vidal, Phys. Rev. Lett. 115 (18), 180405 (2015)]. In this work we explain how to extend TNR to field theories in the continuum. First, a short-distance length scale 1/Λ1/Λ1/Λ is introduced in the continuum partition function by smearing the fields. The resulting object is still defined in the continuum but has no fluctuations at distances shorter than 1/Λ1/Λ1/Λ. An infinitesimal coarse-graining step is then generated by the combined action of a rescalingrescalingrescaling operator LLL and a disentanglingdisentanglingdisentangling operator KKK that implements a quasi-local field redefinition. As demonstrated for a free boson in two dimensions, continuous TNR exactly preserves translation and rotation symmetries and can generate a proper RG flow. Moreover, from a critical fixed point of this RG flow one can then extract the conformal data of the underlying conformal field theory.
Consider the partition function of a classical system in two spatial dimensions, or the Euclidean path integral of a quantum system in two space-time dimensions, both on a lattice. We show that the tensor network renormalization algorithm [G. Evenbly and G. Vidal Phys. Rev. Lett. 115, 180405 (2015)] can be used to implement local scale transformations on these objects, namely, a lattice version of conformal maps. Specifically, we explain how to implement the lattice equivalent of the logarithmic conformal map that transforms the Euclidean plane into a cylinder. As an application, and with the 2D critical Ising model as a concrete example, we use this map to build a lattice version of the scaling operators of the underlying conformal field theory, from which one can extract their scaling dimensions and operator product expansion coefficients.