Tensor renormalization group algorithms with a projective truncation method

We apply the projective truncation technique to the tensor renormalization group (TRG) algorithm in order to reduce the computational cost from $O({\ensuremath{\chi}}⁶)$ to $O({\ensuremath{\chi}}⁵)$, where $\ensuremath{\chi}$ 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 $\ensuremath{\chi}=32$, 48, and 64.