Tensor Renormalization Group Centered About a Core Tensor

Abstract

We propose a modified form of a tensor renormalization group algorithm for evaluating partition functions of classical statistical mechanical models on 2D lattices. This algorithm coarse-grains only the rows and columns of the lattice adjacent to a single core tensor at each step, such that the lattice size shrinks linearly with the number of coarse-graining steps as opposed to shrinking exponentially as in the usual tensor renormalization group (TRG). However, the cost of this new approach only scales as $O({\chi }^4)$ in terms of the bond dimension $\chi$, significantly cheaper than the $O({\chi }^6)$ cost scaling of TRG, whereas numerical benchmarking indicates that both approaches have comparable accuracy for the same bond dimension $\chi$. It follows that the new algorithm can allow for more accurate investigation of classical models on modestly sized lattices than is possible with standard TRG.

arXiv comment7 pages, 7 figures. Appendices: 1 figure

References in corpus (34)

• The density-matrix renormalization group in the age of matrix product states
• Entanglement in quantum critical phenomena
• A Practical Introduction to Tensor Networks: Matrix Product States and Projected Entangled Pair States
• Matrix Product States, Projected Entangled Pair States, and variational renormalization group methods for quantum spin systems
• Thermodynamic limit of the density matrix renormalization for the spin-1 Heisenberg chain
• Tensor-Entanglement-Filtering Renormalization Approach and Symmetry Protected Topological Order
• A class of quantum many-body states that can be efficiently simulated
• Classical simulation of infinite-size quantum lattice systems in two spatial dimensions
• Tensor renormalization group approach to 2D classical lattice models
• Classical simulation of quantum many-body systems with a tree tensor network
• Renormalization algorithms for Quantum-Many Body Systems in two and higher dimensions
• Accurate determination of tensor network state of quantum lattice models in two dimensions
• Hand-waving and Interpretive Dance: An Introductory Course on Tensor Networks
• Coarse-graining renormalization by higher-order singular value decomposition
• Renormalization and tensor product states in spin chains and lattices
• Tensor Network Renormalization
• Tensor-entanglement renormalization group approach to 2D quantum systems
• Simulation of two-dimensional quantum systems using a tree tensor network that exploits the entropic area law
• Second Renormalization of Tensor-Network States
• Renormalization of tensor-network states
• Loop optimization for tensor network renormalization
• Entanglement renormalization in two spatial dimensions
• Ground State and Finite Temperature Lanczos Methods
• Multi-scale Entanglement Renormalization Ansatz in Two Dimensions: Quantum Ising Model
• Tensor network renormalization yields the multi-scale entanglement renormalization ansatz
• Renormalization of tensor networks using graph independent local truncations
• Coarse graining methods for spin net and spin foam models
• Algorithms for tensor network renormalization
• Anisotropic Tensor Renormalization Group
• Gauge fixing, canonical forms and optimal truncations in tensor networks with closed loops
• Renormalization group flows of Hamiltonians using tensor networks
• Renormalization group contraction of tensor networks in three dimensions
• Local scale transformations on the lattice with tensor network renormalization
• Tensor Renormalization Group Algorithms with a Projective Truncation Method

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• All-mode Renormalization for Tensor Network with Stochastic Noise
• A Practical Introduction to Tensor Network Renormalization with TNRKit.jl