Tensor Renormalization Group Algorithms with a Projective Truncation Method

Abstract

We apply the projective truncation technique to the tensor renormalization group (TRG) algorithm in order to reduce the computational cost from $O({\chi }^6)$ to $O({\chi }^5)$, where $\chi$ is the bond dimension, and propose three kinds of algorithms for demonstration. On the other hand, the technique causes a systematic error due to the incompleteness of a projector composed of isometries, and in addition requires iteration steps to determine the isometries. Nevertheless, we find that the accuracy of the free energy for the Ising model on a square lattice is recovered to the level of TRG with a few iteration steps even at the critical temperature for $\chi$ = 32, 48, and 64.

arXiv comment11 pages, 17 figures

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• Examples of symmetry-preserving truncations in tensor field theory
• Discrete aspects of continuous symmetries in the tensorial formulation of Abelian gauge theories
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• All-mode Renormalization for Tensor Network with Stochastic Noise
• Forward-mode automatic differentiation for the tensor renormalization group and its relation to the impurity method