Coarse-graining renormalization by higher-order singular value decomposition

We propose a novel coarse-graining tensor renormalization group method based on the higher-order singular value decomposition. This method provides an accurate but low computational cost technique for studying both classical and quantum lattice models in two or three dimensions. We have demonstrated this method using the Ising model on the square and cubic lattices. By keeping up to 16 bond basis states, we obtain by far the most accurate numerical renormalization group results for the three-dimensional Ising model. We have also applied the method to study the ground state as well as finite temperature properties for the two-dimensional quantum transverse Ising model and obtain the results which are consistent with published data.

Figure 1

(a) A HOTRG contraction of the tensor-network state along the $y$ axis on the square lattice. (b) Steps of contraction and renormalization of two local tensors. The initial tensor $T^{\left(0\right)}=T$.

Figure 2

Graphical representation of Eq. (13) for determining the environment tensor $E_{ka{j}_1{i}_1}^{\left(n\right)}$ from $E_{ijkl}^{(n+1)}$ in the backward iteration.

Figure 3

Graphical representation of Eq. (14) for determining the bond density matrix ${\rho }_{zw,xy}^{\left(n\right)}$ through $E_{ijkl}^{(n+2)}$.

Figure 4

Comparison of the relative errors of free energy with respect to the exact results for the 2D Ising model obtained by various methods with $D=24$. The critical temperature $T_c=2/\ln (1+\sqrt{2})$.

Figure 5

(a) A HOTRG coarse-graining step along the $z$ axis on the cubic lattice. (b) Steps of contraction and renormalization of two local tensors.

Figure 6

Graphical representation for the determination of the environment tensor $E_{mnjiuk}^{\left(n\right)}$ from $E_{lrfbud}^{(n+1)}$ in three dimensions.

Figure 7

Graphical representation for the determination of the bond density matrix ${\rho }_{zw,xy}^{\left(n\right)}$ from the environment tensor $E_{lrfbud}^{(n+2)}$ in three dimensions.

Figure 8

The internal energy and the specific heat for the 3D Ising model obtained by the HOTRG with $D$ = 14. The Monte Carlo result (black curve) obtained from an empirical fit formula given in Ref. 27 is shown for comparison.

Figure 9

The internal energy ($D=14$) and its fitting curves with Eq. (17) around the critical point for the 3D Ising model. $\alpha$ is the critical exponent for the specific heat.

Figure 10

Temperature dependence of the magnetization for the 3D Ising model (D $=$ 14). The Monte Carlo result is from Ref. 35. (Inset) Logarithmic plot of the magnetization around the critical point. The slope of the fitting curve gives the critical exponent of the magnetization, $\gamma =0.3295$.

Figure 11

The critical temperature $T_c$ as a function of the bond dimension $D$ for the 3D Ising model obtained from the internal energy ($U$) and magnetization ($M$), respectively.

Figure 12

The ground-state energy $E_0$, the magnetization $M_x\equiv \langle {\sigma }_x\rangle$, and $M_z\equiv \langle {\sigma }_z\rangle$ versus the applied field $h$ for the 2D quantum Ising model obtained by the HOTRG with $D=14$.

Figure 13

Temperature dependence of the internal energy in three different fields. The solid dots and open circles are obtained with the 3D coarse-graining HOTRG with $D=14$ and the imaginary time evolution approach with $D=24$, respectively.

Figure 14

The specific heat versus temperature obtained by the imaginary time evolution approach with $D=24$ for the 2D quantum Ising model.

Figure 15

Transverse magnetization as a function of temperature for the 2D quantum Ising model. The solid dots and open circles are obtained with the 3D coarse-graining HOTRG with $D=14$ and the imaginary time evolution approach with $D=24$, respectively.

Figure 16

Longitudinal magnetization as a function of temperature for the 2D quantum Ising model. The solid dots and open circles are obtained with the 3D coarse-graining HOTRG with $D=14$ and the imaginary time evolution approach with $D=24$, respectively.