Renormalization of tensor-network states

We have discussed the tensor-network representation of classical statistical or interacting quantum lattice models, and given a comprehensive introduction to the numerical methods we recently proposed for studying the tensor-network states/models in two dimensions. A second renormalization scheme is introduced to take into account the environment contribution in the calculation of the partition function of classical tensor-network models or the expectation values of quantum tensor-network states. It improves significantly the accuracy of the coarse-grained tensor renormalization-group method. In the study of the quantum tensor-network states, we point out that the renormalization effect of the environment can be efficiently and accurately described by the bond vector. This, combined with the imaginary-time evolution of the wave function, provides an accurate projection method to determine the tensor-network wave function. It reduces significantly the truncation error and enables a tensor-network state with a large bond dimension, which is difficult to be accessed by other methods, to be accurately determined.

Figure 1

(Color online) Schematic representation of the partition function of a classical system defined by Eq. (8) on the dual lattice of the triangular lattice, namely, the honeycomb lattices.

Figure 2

(Color online) A unit cell of a square lattice and its corresponding dual variables.

Figure 3

(Color online) Rewiring of a honeycomb lattice by the singular-value decomposition in the TRG iteration.

Figure 4

(Color online) Schematic representation of Eqs. (31, 32).

Figure 5

(Color online) Schematic representation of Eq. (35). A local tensor $T^a$ is defined by contracting the three internal bonds on each triangle.

Figure 6

The last six sites in the TRG iteration. A central symmetric boundary condition is assumed.

Figure 7

(Color online) (a) Configuration of a system and (b) its corresponding environment lattice.

Figure 8

(Color online) Schematic representation of the iterative renormalization to the singular values $\Lambda$ of $M$. $\tilde{M}$ is obtained from Eq. (38) by taking a mean-field approximated $M^e$ defined by Eq. (45).

Figure 9

(Color online) Configurations of a finite honeycomb lattice with 6, 10, 16, or 24 sites. The corresponding environment lattice, by excluding the two red vertices, contains 4, 8, 14, and 22 sites, respectively.

Figure 10

(Color online) Relative errors of the free energy for the Ising model on a triangular lattice obtained by considering the second renormalization effect from four finite environment lattices which contains 4, 8, 14, and 22 sites, respectively. The configurations of these environments are shown in Fig. 9. The TRG result is also shown for comparison.

Figure 11

(Color online) One step of the forward iteration of the environment lattice.

Figure 12

(Color online) Comparison of the relative error of the free energy for the Ising model on triangular lattices obtained using TRG (red), the mean-field approximated SRG (blue), and the SRG (black) methods with $D_{cut}=24$, respectively. The critical temperature is $T_c=4/\text{ln}3$.

Figure 13

(Color online) The relative error of the free energy as a function of the truncation dimension $D_{cut}$ for the Ising model on triangular lattices obtained using the TRG (black) and SRG (blue), respectively. $T=3.2$.

Figure 14

(Color online) To convert (a) a square lattice to (b) a honeycomb lattice by singular decomposing the fourth-order tensor $T$ at each point into two separated third-order tensors along the direction indicated by the red dashed lines.

Figure 15

(Color online) Comparison of the relative errors of the free energy for the Ising model on a square lattice obtained by the SRG with those obtained by the TRG. $D=24$.

Figure 16

(Color online) To convert (a) a Kagome lattice to (c) a honeycomb lattice by rewiring with singular-value decompositions and contraction.

Figure 17

(Color online) Upper panel: comparison of the ground-state energy of the 1D Heisenberg model as a function of iteration number obtained using the projection method with and without taking canonical transformation for the matrix-product wave functions, $E_{can}$ and $E_{MF}$. Lower panel: the energy difference $E_{MF}-E_{can}$. The bond dimension $D=10$.

Figure 18

Graphical representation of the spin-conservation law defined by Eqs. (61, 62). The bond spin takes a positive/negative sign if the bond arrow points toward/outward the vertex.

Figure 19

(Color online) The SRG result of the ground-state energy as a function of the truncation dimension $D_{cut}$ for the Heisenberg model on a honeycomb lattice. $D$ is the bond dimension of the wave function.

Figure 20

The ground-state energy of the Heisenberg model on a honeycomb lattice as a function of the bond dimension $D$ obtained by the SRG with $D_{cut}=130$.

Figure 21

The staggered magnetization as a function of $D$ for the Heisenberg model on a honeycomb lattice.

Figure 22

(Color online) The field dependence of the staggered magnetization $M_s$ for the Heisenberg model on a honeycomb lattice. The inset shows the $h_s^{1/2}$ dependence of $M_s$.

Figure 23

(Color online) The staggered magnetization as a function of $1/D$ for the Heisenberg model on a honeycomb lattice.