Entanglement-based tensor-network strong-disorder renormalization group

We propose an entanglement-based algorithm of the tensor-network strong-disorder renormalization group (tSDRG) method for quantum spin systems with quenched randomness. In contrast to the previous tSDRG algorithm based on the energy spectrum of renormalized block Hamiltonians, we directly utilize the entanglement structure associated with the blocks to be renormalized. We examine accuracy of the algorithm for the random antiferromagnetic Heisenberg models on one-dimensional, triangular, and square lattices. We then find that the entanglement-based tSDRG achieves better accuracy than the previous one for the square-lattice model with weak randomness, while it is less efficient for the one-dimensional and triangular-lattice models particularly in the strong-randomness region. The theoretical background and possible improvements of the algorithm are also discussed.

Figure 1

Schematic picture of the tree-tensor network constructed in the tSDRG. Circles on the bottom and triangles in the middle layers respectively represent the original spins and the renormalization matrices $V$ or $U$. The semicircle at the top represents the ground-state wave function in the truncated basis at the final step of tSDRG. See the text in Sec. 2 for the details.

Figure 2

(a) The system after a certain number of H-tSDRG iterations and (b) the corresponding block representation of the system. Here, we suppose that the link between the blocks $(R,R^{\prime })=(5,7)$ is the “strongest” (i.e., has the largest gap $\mathrm{\Delta }$) so that the (5,7) blocks are renormalized in this H-tSDRG step. The system after the renormalization process is depicted in (c) and (d).

Figure 3

The blocks involved in the rDM ${\rho }_{r,r^{\prime }}\left(r^{\prime \prime }\right)$ [Eq. (13)] with $(r,r^{\prime })=(5,7)$ and $r^{\prime \prime }=8$. The calculation of ${\tilde{\rho }}_{r,r^{\prime }}$ [Eq. (14)] for $(r,r^{\prime })=(5,7)$ requires ${\rho }_{5,7}\left(r^{\prime \prime }\right)$ with $r^{\prime \prime }=1,2,3,4,6,8,9,10$.

Figure 4

Schematic graph of the new-block bases in the E-tSDRG algorithm: (a) $n\ge \chi$ (usual case) and (b) $n<\chi$ (the case of the small-rank problem). Solid lines (black) represent the eigenstates of the mixed-state rDM ${\tilde{\rho }}_{R,R^{\prime }}$ and the dotted lines (red) represent the eigenstates of the block-pair Hamiltonian in the projected space, $P^{†}{\mathcal{H}}_{R,R^{\prime }}^{\mathrm{P}}P$. The dashed line (blue) denotes the cutoff line.

Figure 5

Random averages of the errors of the ground-state energy per spin, $\overline{\delta e}$, as functions of $\delta$ for (a) a 1D chain with $N=24$ spins, (b) a triangular lattice with $N=24$ spins, and (c) a square lattice with $N=36$ spins. Solid squares and circles represent respectively the results of E-tSDRG with and without the additional treatment for the small-rank problem (see text in Sec. 2b). Open triangles and diamonds show the results of the H-tSDRG using ${\mathrm{\Delta }}_{\max }^{\mathrm{I}}$ and ${\mathrm{\Delta }}_{\mathrm{gs}}^{\mathrm{P}}$, respectively. The bond dimension used in the calculations is $\chi =80$. The data in (a) are plotted in a logarithmic scale while the data in (b) and (c) are in a linear scale.

Figure 6

Random-averaged ground-state energy per spin, $\overline{E}$, as functions of $\delta$ for the triangular lattice with $N=36$ spins. Solid squares and circles represent respectively the results of E-tSDRG with and without the additional treatment for the small-rank problem (see text in Sec. 2b). Open triangles and diamonds show the results of the H-tSDRG using ${\mathrm{\Delta }}_{\max }^{\mathrm{I}}$ and ${\mathrm{\Delta }}_{\mathrm{gs}}^{\mathrm{P}}$, respectively. The bond dimension used in the calculations is $\chi =80$. Inset shows the energy gain compared to the data of the H-tSDRG with ${\mathrm{\Delta }}_{\mathrm{gs}}^{\mathrm{P}}, \mathrm{\Delta }\overline{E}=\overline{E}-{\overline{E}}_{{\mathrm{\Delta }}_{\mathrm{gs}}^{\mathrm{P}}}$.

Figure 7

Random averages of the errors of the ground-state correlation functions, $\overline{\delta g}$, as functions of $\delta$ for (a) a 1D chain with $N=24$ spins, (b) a triangular lattice with $N=24$ spins, and (c) a square lattice with $N=36$ spins. Solid squares and circles represent respectively the results of E-tSDRG with and without the additional treatment for the small-rank problem (see text in Sec. 2b). Open triangles and diamonds show the results of the H-tSDRG using ${\mathrm{\Delta }}_{\max }^{\mathrm{I}}$ and ${\mathrm{\Delta }}_{\mathrm{gs}}^{\mathrm{P}}$, respectively. The bond dimension used in the calculations is $\chi =80$. The data in (a) are plotted in a logarithmic scale while the data in (b) and (c) are in a linear scale.

Figure 8

Random-averaged static spin structure factors obtained by the E-tSDRG (with the additional treatment) with $\chi =80$ for (a) a triangular lattice with $N=24$ and $\delta =0.25$, (b) a triangular lattice with $N=24$ and $\delta =1.00$, (c) a square lattice with $N=36$ and $\delta =0.25$, and (d) a square lattice with $N=36$ and $\delta =1.00$.