Scaling dimensions from linearized tensor renormalization group transformations

Abstract

We show a way to perform the canonical renormalization group (RG) prescription in tensor space: write down the tensor RG equation, linearize it around a fixed-point tensor, and diagonalize the resulting linearized RG equation to obtain scaling dimensions. The tensor RG methods have had a great success in producing accurate free energy compared with the conventional real-space RG schemes. However, the above-mentioned canonical procedure has not been implemented for general tensor-network-based RG schemes. We extend the success of the tensor methods further to extraction of scaling dimensions through the canonical RG prescription, without explicitly using the conformal field theory. This approach is benchmarked in the context of the Ising models in 1D and 2D. Based on a pure RG argument, the proposed method has potential applications to 3D systems, where the existing bread-and-butter method is inapplicable.

17 pages, 11 figures; move most technical details to appendices to make the essential idea clear

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Cited by in corpus (11)

• Developments in the Tensor Network -- from Statistical Mechanics to Quantum Entanglement
• Tensor RG approach to high-temperature fixed point
• Tensor Renormalization Group at Low Temperatures: Discontinuity Fixed Point
• Rotations, Negative Eigenvalues, and Newton Method in Tensor Network Renormalization Group
• Three-dimensional real space renormalization group with well-controlled approximations
• Transfer Matrix and Lattice Dilatation Operator for High-Quality Fixed Points in Tensor Network Renormalization Group
• Essential difference between 2D and 3D from the perspective of real-space renormalization group
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• Lattice-reflection symmetry in tensor-network renormalization group with entanglement filtering in two and three dimensions
• Variational boundary based tensor network renormalization group
• Lattice and PT symmetries in tensor-network renormalization group: Case study of a hard-square lattice gas model