Scaling dimensions from linearized tensor renormalization group transformations

TL;DR

This paper introduces a tensor RG method to extract scaling dimensions by linearizing around fixed points, extending tensor network techniques to compute critical properties without relying on conformal field theory, and benchmarks it on Ising models.

Contribution

It develops a canonical tensor RG approach for calculating scaling dimensions, which was not previously implemented for general tensor-network RG schemes.

Findings

Successfully benchmarked on 1D and 2D Ising models.

Potential applicability to 3D systems where traditional methods fail.

Provides a new tensor-based framework for critical property extraction.

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.