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.
Highlights
The renormalization group (RG) is a powerful technique for studying physical systems where fluctuations in all scales of length are important [1]; the most famous example in statistical mechanics is critical phenomena
We show the RG flow of A(n+1) − A(n) without sign fixing in Fig. 9(c); the sign ambiguities in Eq (36) prevent us from achieving a manifestly fixed-point tensor, except at RG step n = 22, where the tensor happens to have all signs correct by accident
We develop the canonical RG prescription in tensor space using the higher-order tensor renormalization group (HOTRG)-like scheme in this paper in order to prepare for the further applications to 3D systems
Summary
The renormalization group (RG) is a powerful technique for studying physical systems where fluctuations in all scales of length are important [1]; the most famous example in statistical mechanics is critical phenomena. And Vidal used the tensor network renormalization (TNR) [25,26] to implement local scale transformation that maps a plane to a cylinder [29]; the spectrum of eigenvalues of a transfer matrix on the cylinder gives scaling dimensions These two methods have been applied to extract scaling dimensions from on, while the canonical RG prescription in tensor space has never been followed up. The higher-order tensor renormalization group (HOTRG) [14] is combined with a recently developed technique, graphindependent local truncation (GILT) [30] in Sec. II C, to generate correct tensor RG flows that will go to a critical fixed point at a general bond dimension. The method might be relevant in three dimensions (3D), where Gu and Wen’s method is inapplicable and Evenbly and Vidal’s local-scaling-transformation idea is nontrivial to implement
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