Tensor Network Based Finite-Size Scaling for Two-Dimensional Classical Models

Abstract

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 $T_c$ and the critical exponent $\nu$, $\beta$ are at the order of $10^{-7},10^{-5}$, $10^{-4}$ 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 cut-off in bond dimension and elucidate its physical meaning in this context.

References in corpus (25)

• Coarse-graining renormalization by higher-order singular value decomposition
• Tensor Network Renormalization
• Theory of finite-entanglement scaling at one-dimensional quantum critical points
• Quantum criticality with two length scales
• Renormalization of tensor-network states
• Loop optimization for tensor network renormalization
• Matrix product states for critical spin chains: finite size scaling versus finite entanglement scaling
• Real Space Renormalization in Statistical Mechanics
• Finite Correlation Length Scaling in Lorentz-Invariant Gapless iPEPS Wave Functions
• Bayesian Inference in the Scaling Analysis of Critical Phenomena
• Finite correlation length scaling with infinite projected entangled-pair states
• Precise extrapolation of the correlation function asymptotics in uniform tensor network states with application to the Bose-Hubbard and XXZ models
• Conformal Data from Finite Entanglement Scaling
• A scaling hypothesis for matrix product states
• Finite correlation length scaling with infinite projected entangled pair states at finite temperature
• Phase transition of the q-state clock model: duality and tensor renormalization
• Calculation of higher-order moments by higher-order tensor renormalization group
• Doubling of Entanglement Spectrum in Tensor Renormalization Group
• Resolving the Berezinskii-Kosterlitz-Thouless transition in the 2D XY model with tensor-network based level spectroscopy
• Finite-$m$ scaling analysis of Berezinskii-Kosterlitz-Thouless phase transitions and entanglement spectrum for the six-state clock model
• Logarithmic finite-size scaling correction to the leading Fisher zeros in the p-state clock model: A higher-order tensor renormalization group study
• Critical behavior of the two-dimensional icosahedron model
• Tensor network calculation of the logarithmic correction exponent in the XY model
• Finite-size scaling analysis of two-dimensional deformed Affleck-Kennedy-Lieb-Tasaki states
• A Monte Carlo method for critical systems in infinite volume: the planar Ising model

Cited by in corpus (5)

• Finite-size and finite bond dimension effects of tensor network renormalization
• 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 scaling on the torus with periodic projected entangled-pair states
• Spectroscopy with the tensor renormalization group method