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 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.
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.
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.
Tensor renormalization group, originally devised as a numerical technique, is emerging as a rigorous analytical framework for studying lattice models in statistical physics. Here we introduce a new renormalization map - the 2x1 map - which coarse-grains the lattice anisotropically by a factor of two in one direction followed by a 90-degree rotation. We develop a novel graphical language that translates the action of the 2x1 map into a system of inequalities on tensor components, with rigorous estimates in the Hilbert-Schmidt norm. We define a finite-dimensional "bounding box" called the hat-tensor, and a master function governing its RG flow. Iterating this function numerically, we establish convergence to the high-temperature fixed point for tensors lying within a quantifiable neighborhood. Our main theorem shows that tensors with deviations bounded by 0.02 in 63 orthogonal sectors flow to the fixed point. We also apply the method to specific models - the 2D Ising and XY models - obtaining explicit bounds on their high-temperature phase. This work brings the Tensor RG program closer towards a rigorous, computer-assisted construction of critical fixed points.
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 D. 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.
Tensor networks are a class of methods for studying many-body systems. They give a geometrical description of the internal structure of many-body states, operators, and partition functions, that can be used to implement efficient algorithms to simulate them numerically. In this thesis, after providing a very brief overview of the field, we present new tensor network methods for three different use cases. \n \nFirst, we present a new real-space renormalization group algorithm for tensor networks. Compared to existing methods, its advantages are its low computational cost, simplicity of implementation, and applicability to any network. We benchmark it on the 2D classical Ising model and find accuracy comparable with the best existing tensor network methods. Due to its simplicity and generalizability, we consider our algorithm to be an excellent candidate for implementation of real-space renormalization in higher dimensions, and discuss some of the details and the remaining challenges in 3D. \n \nSecond, we show how certain topological conformal defects of critical lattice theories can be represented as tensor networks, using the 2D classical Ising model as an example. Furthermore, we show how such tensor network descriptions, combined with a renormalization group algorithm, can be used to obtain accurate estimates of the universal properties of these defects. We also show how coarse-graining of defects can be applied to any conformal defect (i.e. not just topological ones), and yields a set of associated scaling dimensions. \n \nFinally, we leave behind the focus on renormalization group methods, and present a method for spatially resolving the overlap between two tensor network states, to obtain localized information about the similarities and differences between them. For a given region, the similarity of two states in this region can be quantified by the Uhlmann fidelity of their reduced density matrices, and we show how such fidelities can be efficiently computed in many cases when the two states are represented as tensor networks. We demonstrate the usefulness of evaluating such subsystem fidelities with three example applications: studying local quenches, comparing critical and non-critical states, and quantifying convergence in tensor network simulations.