Tensor Renormalization Group Algorithms with a Projective Truncation Method

TL;DR

This paper introduces a projective truncation method for tensor renormalization group algorithms that reduces computational cost while maintaining high accuracy in calculating the free energy of the Ising model.

Contribution

The paper proposes a novel application of projective truncation to TRG, lowering computational complexity and demonstrating accurate results with minimal iterations.

Findings

Computational cost reduced from O(χ^6) to O(χ^5)

High accuracy in free energy at critical temperature for χ=32, 48, 64

Systematic errors are manageable with a few iteration steps

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.