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 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.
Through coarse-graining, tensor network representations of a two-dimensional critical lattice model flow to a universal four-leg tensor, corresponding to a conformal field theory (CFT) fixed-point. We computed explicit elements of the critical fixed-point tensor, which we identify as the CFT four-point function. This allows us to directly extract the operator product expansion coefficients of the CFT from these tensor elements. Combined with the scaling dimensions obtained from the transfer matrix, we determine the complete set of the CFT data from the fixed-point tensor for any critical unitary lattice model.
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 Tc 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.
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.
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.
The critical two-dimensional classical Ising model on the square lattice has two topological conformal defects: the Z2 symmetry defect ${D}_{\ensuremath{\epsilon}}$ and the Kramers-Wannier duality defect ${D}_{\ensuremath{\sigma}}$. These two defects implement antiperiodic boundary conditions and a more exotic form of twisted boundary conditions, respectively. On the torus, the partition function ZD of the critical Ising model in the presence of a topological conformal defect D is expressed in terms of the scaling dimensions ${\mathrm{\ensuremath{\Delta}}}_{\ensuremath{\alpha}}$ and conformal spins ${s}_{\ensuremath{\alpha}}$ of a distinct set of primary fields (and their descendants, or conformal towers) of the Ising conformal field theory. This characteristic conformal data ${{{\mathrm{\ensuremath{\Delta}}}_{\ensuremath{\alpha}},{s}_{\ensuremath{\alpha}}}}_{D}$ can be extracted from the eigenvalue spectrum of a transfer matrix MD for the partition function ZD. In this paper, we investigate the use of tensor network techniques to both represent and coarse grain the partition functions ${Z}_{{D}_{\ensuremath{\epsilon}}}$ and ${Z}_{{D}_{\ensuremath{\sigma}}}$ of the critical Ising model with either a symmetry defect ${D}_{\ensuremath{\epsilon}}$ or a duality defect ${D}_{\ensuremath{\sigma}}$. We also explain how to coarse grain the corresponding transfer matrices ${M}_{{D}_{\ensuremath{\epsilon}}}$ and ${M}_{{D}_{\ensuremath{\sigma}}}$, from which we can extract accurate numerical estimates of ${{{\mathrm{\ensuremath{\Delta}}}_{\ensuremath{\alpha}},{s}_{\ensuremath{\alpha}}}}_{{D}_{\ensuremath{\epsilon}}}$ and ${{{\mathrm{\ensuremath{\Delta}}}_{\ensuremath{\alpha}},{s}_{\ensuremath{\alpha}}}}_{{D}_{\ensuremath{\sigma}}}$. Two key ingredients of our approach are (i) coarse graining of the defect D, which applies to any (i.e., not just topological) conformal defect and yields a set of associated scaling dimensions ${\mathrm{\ensuremath{\Delta}}}_{\ensuremath{\alpha}}$, and (ii) construction and coarse graining of a generalized translation operator using a local unitary transformation that moves the defect, which only exist for topological conformal defects and yields the corresponding conformal spins ${s}_{\ensuremath{\alpha}}$.