Global optimization of tensor renormalization group using the corner transfer matrix

A tensor network renormalization algorithm with global optimization based on the corner transfer matrix is proposed. Since the environment is updated by the corner transfer matrix renormalization group method, the forward-backward iteration is unnecessary, which is a time-consuming part of other methods with global optimization. In addition, a further approximation reducing the order of the computational cost of contraction for the calculation of the coarse-grained tensor is proposed. The computational time of our algorithm in two dimensions scales as the sixth power of the bond dimension, while the higher-order tensor renormalization group and the higher-order second renormalization group methods have the seventh power. We perform benchmark calculations in the Ising model on the square lattice and show that the time-to-solution of the proposed algorithm is faster than that of other methods.

Figure 1

Graphical representations of the HOTRG algorithm. (a) The bond-merging operators in HOTRG are given as a solution of the local optimization problem to minimize this norm. (b) A coarse-grained tensor $T^{\prime }$ is obtained by contraction.

Figure 2

More accurate bond-merging operators $P$ and $Q$ are obtained as a solution of the optimization problem with larger tensor networks. Here, the cost function of the optimization problem with $6\times 6$ clusters is shown.

Figure 3

The bond-merging operators in CTM-TRG are obtained as the oblique projector between the left- and right-half networks.

Figure 4

Graphical representation of the additional approximation, which reduces the computational cost of contraction for the coarse-graining tensor. The red diagonal lines indicate that their bond dimension is ${\chi }^{\prime }$. (a) The local tensor is decomposed into two three-index tensors. The truncation operators $p$ and $q$ reduce the bond dimension from ${\chi }^2$ to ${\chi }^{\prime }$ (see, also, Fig. 5). (b) A coarse-grained tensor in CTM-TRG is obtained by the simultaneous scale transformation along the horizontal and vertical directions.

Figure 5

(a) The truncation operators $p$ and $q$ in Fig. 4 are calculated as the oblique projector which minimizes the difference between the two networks. (b) The corner matrix $C_{\text{TL}}$ is decomposed into $c_{\text{TL}}$ and $c_{\text{TL}}^{\prime }$ by using the singular-value decomposition.

Figure 6

Comparison of relative errors in the free energy. The vertical dashed line indicates the critical temperature.

Figure 7

Elapsed time of 20 renormalization steps as a function of the bond dimension. The dashed lines proportional to ${\chi }^6$ and ${\chi }^7$ are guides for the eye.

Figure 8

Relative error in the free energy at (a) $T=T_c/1.01$ and (b) $T_c/0.99$ as a function of the elapsed time of 20 steps.

Figure 9

Relative error in the free energy at criticality as a function of (a) the elapsed time and (b) the bond dimension. The dashed line shows a fitting result.