Finite-size and finite bond dimension effects of tensor network renormalization

Abstract

We propose a general procedure for extracting the running coupling constants of the underlying field theory of a given classical statistical model on a two-dimensional lattice, combining tensor network renormalization (TNR) and the finite-size scaling theory of conformal field theory. By tracking the coupling constants at each scale, we are able to visualize the renormalization group (RG) flow and demonstrate it with the classical Ising and 3-state Potts models. Furthermore, utilizing the new methodology, we reveal the limitations due to finite bond dimension D on TNR applied to critical systems. We find that a finite correlation length is imposed by the finite bond dimension in TNR, and it can be attributed to an emergent relevant perturbation that respects the symmetries of the system. The correlation length shows the same power-law dependence on D as the "finite entanglement scaling" of the Matrix Product States.

References in corpus (11)

• Tensor renormalization group approach to 2D classical lattice models
• Entanglement spectrum in one-dimensional systems
• Scaling of entanglement support for Matrix Product States
• Matrix product states for critical spin chains: finite size scaling versus finite entanglement scaling
• Closure of the Operator Product Expansion in the Non-Unitary Bootstrap
• Resolving the Berezinskii-Kosterlitz-Thouless transition in the 2D XY model with tensor-network based level spectroscopy
• Operator fusion from wavefunction overlaps: Universal finite-size corrections and application to Haagerup model
• Critical behavior of the two-dimensional icosahedron model
• Universal information of critical quantum spin chains from wavefunction overlaps
• Multi-boundary generalization of thermofield double states and their realization in critical quantum spin chains
• Tensor Network Based Finite-Size Scaling for Two-Dimensional Classical Models

Cited by in corpus (20)

• Nuclear norm regularized loop optimization for tensor network
• Quantum criticality with emergent symmetry in the extended Shastry-Sutherland model
• Tensor renormalization group study of (1+1)-dimensional U(1) gauge-Higgs model at $\theta =\pi$ with Lüscher's admissibility condition
• Finite-size corrections to the energy spectra of gapless one-dimensional systems in the presence of boundaries
• Rotations, Negative Eigenvalues, and Newton Method in Tensor Network Renormalization Group
• Finite-Entanglement Scaling of 2D Metals
• Transfer Matrix and Lattice Dilatation Operator for High-Quality Fixed Points in Tensor Network Renormalization Group
• Finite-size scaling on the torus with periodic projected entangled-pair states
• Phase structure analysis of 2d lattice CP(1) model with $\theta$ term using tensor renormalization group method
• Interplay of frustration and quantum fluctuations in a spin-1/2 anisotropic square lattice
• Spectroscopy with the tensor renormalization group method
• Tensor network simulations for nonorientable surfaces
• Tensor network renormalization approach to antiferromagnetic 6-state clock model on the Union Jack lattice
• Tensor renormalization group approach to critical phenomena via symmetry-twisted partition functions
• A Practical Introduction to Tensor Network Renormalization with TNRKit.jl
• Chiral Edge States Emerging on Anyon-Net
• Self-congruent point in critical matrix product states: An effective field theory for finite-entanglement scaling
• Forward-mode automatic differentiation for the tensor renormalization group and its relation to the impurity method
• Cheaper and more noise-resilient quantum state preparation using eigenvector continuation
• Application of deep neural networks for computing the renormalization group flow of the two-dimensional phi^4 field theory