Loop optimization for tensor network renormalization (loop-TNR) is a real-space renormalization group algorithm suitable for studying 1+1D critical systems. While the original proposal by Yang et al. focused on classical models, we extend this algorithm with new techniques to enable accurate and efficient extraction of conformal data from critical quantum models. Benchmark results are provided for a number of quantum models, including ones described by non-minimal or non-unitary conformal field theories, showcasing both the strengths and limitations of loop-TNR. We discuss the subtle issue of non-analytic finite size effect in quantum lattice models and its impact on loop-TNR, and propose the use of virtual-space transfer-matrix to circumvent it, using the XY model as a demonstration. We then generalize loop-TNR to fermionic systems by incorporating Grassmann numbers, and benchmark the generalized algorithm on the t-V model. \n \nNext, we demonstrate a non-trivial application of loop-TNR by studying the 1D domain wall between untwisted and twisted 2D lattice gauge theories of finite groups G. We numerically study such domain walls for G = Z_N (with N<6) using loop-TNR, and discover a large class of gapless models. We also study the physical mechanism for these gapless domain walls and propose quantum field theory descriptions that agree perfectly with our numerical results. By taking advantage of the classification and construction of twisted gauge models using group cohomology theory, we systematically construct general lattice models to realize gapless domain walls for arbitrary finite symmetry group G. Such constructions can be generalized into arbitrary dimensions and might provide us a systematical way to study gapless domain walls and topological quantum phase transitions.
Loop optimization for tensor network renormalization (loop-TNR) is a real-space renormalization group algorithm suitable for studying 1+1D critical systems. While the original proposal by Yang et al. focused on classical models, we extend this algorithm with new techniques to enable accurate and efficient extraction of conformal data from critical quantum models. Benchmark results are provided for a number of quantum models, including ones described by non-minimal or non-unitary conformal field theories, showcasing both the strengths and limitations of loop-TNR. We discuss the subtle issue of non-analytic finite size effect in quantum lattice models and its impact on loop-TNR, and propose the use of virtual-space transfer-matrix to circumvent it, using the XY model as a demonstration. We then generalize loop-TNR to fermionic systems by incorporating Grassmann numbers, and benchmark the generalized algorithm on the t-V model. \n \nNext, we demonstrate a non-trivial application of loop-TNR by studying the 1D domain wall between untwisted and twisted 2D lattice gauge theories of finite groups G. We numerically study such domain walls for G = Z_N (with N<6) using loop-TNR, and discover a large class of gapless models. We also study the physical mechanism for these gapless domain walls and propose quantum field theory descriptions that agree perfectly with our numerical results. By taking advantage of the classification and construction of twisted gauge models using group cohomology theory, we systematically construct general lattice models to realize gapless domain walls for arbitrary finite symmetry group G. Such constructions can be generalized into arbitrary dimensions and might provide us a systematical way to study gapless domain walls and topological quantum phase transitions.
In recent years, tensor network renormalization (TNR) has emerged as an efficient and accurate method for studying (1+1)D quantum systems or 2D classical systems using real-space renormalization group (RG) techniques. One notable application of TNR is its ability to extract central charge and conformal scaling dimensions for critical systems. In this paper, we present the implementation of the Loop-TNR algorithm, which allows for the computation of dynamical correlation functions. Our algorithm goes beyond traditional approaches by not only calculating correlations in the spatial direction, where the separation is an integer, but also in the temporal direction, where the time difference can contain decimal values. Our algorithm is designed to handle both imaginary-time and real-time correlations, utilizing a tensor network representation constructed from a path-integral formalism. Additionally, we highlight that the Loop-TNR algorithm can also be applied to investigate critical properties of non-Hermitian systems, an area that was previously inaccessible using density matrix renormalization group(DMRG) and matrix product state(MPS) based algorithms.
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.
Emergent generalized symmetries and the holographic principle play very important roles in understanding quantum phase transitions. In recent years, tensor-network renormalization with generalized symmetry and the corresponding fixed-point tensor formulation have been proposed as a discrete spacetime framework for reformulating conformal field theory. Nevertheless, a complete holographic formulation within this framework is still lacking, and the current formulation does not directly accommodate sector partition functions that are modular covariant in general. In this work, we formulate R-TNR, which builds the categorical constraints of a fusion category R into local tensor spaces and preserves these constraints under Loop-TNR. For suitable choices of R, the initial tensors constructed here flow under R-Loop-TNR to stable gapless fixed points. In the Ising, tricritical Ising, and three-state Potts conformal field theory examples, a broad range of initial parameters leads to the same gapless fixed point in each example. The method resolves conformal towers by anyon sector and yields scaling dimensions, conformal spins, and magnitudes of sector-resolved structure constants with extremely high accuracy compared to previous methods. These results realize a topological bootstrap based on stable symmetry-topological-order in which finite categorical and geometric input, together with an initial tensor that need not be fine-tuned, suffices to recover the infrared conformal field theory and its sector-resolved conformal data.
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/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/Λ. An infinitesimal coarse-graining step is then generated by the combined action of a rescaling operator L and a disentangling operator K 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.
We present TNRKit, an open-source Julia package for Tensor Network Renormalization (TNR) of two- and three-dimensional classical statistical models and Euclidean lattice field theories. Built on top of TensorKit, it provides a symmetry-aware framework for constructing tensor-network representations of partition functions and coarse-graining them using methods such as TRG, HOTRG, and LoopTNR. Beyond thermodynamic quantities, the package enables the extraction of universal conformal data -- including scaling dimensions and the central charge -- directly from fixed-point tensors. TNRKit is designed with both usability and extensibility in mind, offering a practical platform for applying, benchmarking, and developing modern tensor renormalization algorithms. This paper also serves as a self-contained introduction to the TNR framework.
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.