Tensor-network strong-disorder renormalization groups for random quantum spin systems in two dimensions

Novel randomness-induced disordered ground states in two-dimensional (2D) quantum spin systems have been attracting much interest. For quantitative analysis of such random quantum spin systems, one of the most promising numerical approaches is the tensor-network strong-disorder renormalization group (tSDRG), which was basically established for one-dimensional (1D) systems. In this paper, we propose a possible improvement of its algorithm toward 2D random spin systems, focusing on a generating process of the tree network structure of tensors, and precisely examine their performances for the random antiferromagnetic Heisenberg model not only on the 1D chain but also on the square- and triangular-lattices. On the basis of comparison with the exact numerical results up to 36 site systems, we demonstrate that accuracy of the optimal tSDRG algorithm is significantly improved for the 1D and 2D systems in the strong-randomness regime.

Figure 1

Schematic diagrams of the renormalization procedure of the tSDRG. (a) The system after a certain number of tSDRG iterations, composed of blocks (green rectangles) of original spins (red circles). (b) The effective Hamiltonian [Eq. (4)] consisting of the intrablock Hamiltonians ${\mathcal{H}}_r^{\mathrm{B}}$ and interblock Hamiltonians ${\mathcal{H}}_{r,r^{\prime }}^{\mathrm{I}}$. (c) The effective Hamiltonian after renormalizing the blocks of 1 and 2 into a new block “$1+2$.”

Figure 2

Schematic tree-tensor-network representation of the ground-state wavefunction obtained by the tSDRG. Circles (red) represent the original spins, and triangles (green) represent the renormalization-group transformation matrices. Semicircle (blue) at the top of the network represents the ground-state eigenvector of the effective Hamiltonian (4) at the final step of the tSDRG.

Figure 3

Schematic diagram of the energy gaps $\{{\mathrm{\Delta }}_{\max }^{\mathrm{I}},{\mathrm{\Delta }}_{\mathrm{gs}}^{\mathrm{I}},{\mathrm{\Delta }}_{\max }^{\mathrm{P}},{\mathrm{\Delta }}_{\mathrm{gs}}^{\mathrm{P}}, {\mathrm{\Delta }}_{\mathrm{cut}}^{\mathrm{P}}\}$ introduced in the tSDRG algorithms. Eigenenergy spectrum $\left\{{\varepsilon }_l\right\}$ of the interblock Hamiltonian ${\mathcal{H}}_{r,r^{\prime }}^{\mathrm{I}}$ is represented as horizontal lines in the left panel. The spectrum $\left\{{\widehat{\varepsilon }}_l\right\}$ of the block-pair Hamiltonian ${\mathcal{H}}_{r+r^{\prime }}^{\mathrm{P}}={\mathcal{H}}_r^{\mathrm{B}}+{\mathcal{H}}_{r^{\prime }}^{\mathrm{B}}+{\mathcal{H}}_{r,r^{\prime }}^{\mathrm{I}}$ is also depicted in the right panel.

Figure 4

$\chi$ dependencies of accuracy of the tSDRGs with various $\mathrm{\Delta }$ for the random AF Heisenberg chain of $\delta =0.50$ and $N=24$: (a) error of the ground-state energy, ${\overline{\delta e}}_{\mathrm{\Delta }}$, and (b) error of the ground-state correlation function, ${\overline{\delta g}}_{\mathrm{\Delta }}$. Solid symbols represent the results of the tSDRGs with ${\mathrm{\Delta }}_{\max }^{\mathrm{I}}$ and ${\mathrm{\Delta }}_{\mathrm{gs}}^{\mathrm{I}}$, while open symbols indicate the results for ${\mathrm{\Delta }}_{\max }^{\mathrm{P}}, {\mathrm{\Delta }}_{\mathrm{gs}}^{\mathrm{P}}$, and ${\mathrm{\Delta }}_{\mathrm{cut}}^{\mathrm{P}}$. Error bars due to the random average are negligible compared with the symbols.

Figure 5

$\delta$ dependencies of accuracy of the tSDRGs with various $\mathrm{\Delta }$ at $\chi =80$ for the random AF Heisenberg chain of $N=24$: (a) Error of the ground-state energy, ${\overline{\delta e}}_{\mathrm{\Delta }}$, and (b) error of the ground-state correlation function, ${\overline{\delta g}}_{\mathrm{\Delta }}$. Solid symbols represent the results of the tSDRGs with ${\mathrm{\Delta }}_{\max }^{\mathrm{I}}$ and ${\mathrm{\Delta }}_{\mathrm{gs}}^{\mathrm{I}}$, while open symbols indicate the results for ${\mathrm{\Delta }}_{\max }^{\mathrm{P}}, {\mathrm{\Delta }}_{\mathrm{gs}}^{\mathrm{P}}$, and ${\mathrm{\Delta }}_{\mathrm{cut}}^{\mathrm{P}}$. Error bars are negligible compared with the symbols.

Figure 6

$\chi$ dependencies of accuracy of the tSDRG algorithms with various $\mathrm{\Delta }$ for the random AF Heisenberg model on the $N=36$ square lattice: (a) error of the ground-state energy, ${\overline{\delta e}}_{\mathrm{\Delta }}$, and (b) error of the ground-state correlation function, ${\overline{\delta g}}_{\mathrm{\Delta }}$, for $\delta =0.25$. Panels (c) and (d) respectively represent ${\overline{\delta e}}_{\mathrm{\Delta }}$ and ${\overline{\delta g}}_{\mathrm{\Delta }}$ for $\delta =1.00$. Error bars are negligible compared with the symbols.

Figure 7

Intensity plots of the static spin structure factor $\overline{S}\left(\mathit{q}\right)$ for the square-lattice Heisenberg model of $N=36$: (a) QMC result and (b) tSDRG result with ${\mathrm{\Delta }}_{\max }^{\mathrm{I}}$ of $\chi =80$, for $\delta =0.25$. (c) and (d) respectively show QMC result and tSDRG result of $\chi =80$ with ${\mathrm{\Delta }}_{\max }^{\mathrm{I}}$ for $\delta =1.00$. The green line shows the boundary of the first Brillouin zone of the square lattice.

Figure 8

$\delta$ dependencies of accuracy of the tSDRGs with various $\mathrm{\Delta }$ at $\chi =80$ for the random AF Heisenberg model on the $N=36$ square lattice: (a) Error of the ground-state energy, ${\overline{\delta e}}_{\mathrm{\Delta }}$, and (b) error of the ground-state correlation function, ${\overline{\delta g}}_{\mathrm{\Delta }}$. Solid symbols represent the tSDRG results with ${\mathrm{\Delta }}_{\max }^{\mathrm{I}}$ and ${\mathrm{\Delta }}_{\mathrm{gs}}^{\mathrm{I}}$, while open symbols indicate the results for ${\mathrm{\Delta }}_{\max }^{\mathrm{P}}, {\mathrm{\Delta }}_{\mathrm{gs}}^{\mathrm{P}}$, and ${\mathrm{\Delta }}_{\mathrm{cut}}^{\mathrm{P}}$. Error bars are negligible compared with the symbols.

Figure 9

$\chi$ dependencies of accuracy of the tSDRG algorithms with various $\mathrm{\Delta }$ for the random AF Heisenberg model on the $N=24$ triangular lattice: (a) error of the ground-state energy, ${\overline{\delta e}}_{\mathrm{\Delta }}$, and (b) error of the ground-state correlation function, ${\overline{\delta g}}_{\mathrm{\Delta }}$, for $\delta =0.25$. (c) and (d) respectively represent ${\overline{\delta e}}_{\mathrm{\Delta }}$ and ${\overline{\delta g}}_{\mathrm{\Delta }}$ for $\delta =1.00$. Error bars are negligible compared with the symbols.

Figure 10

Intensity plots of the static spin structure factor $\overline{S}\left(\mathit{q}\right)$ for the $N=24$ triangular-lattice Heisenberg model: (a) ED result and (b) tSDRG result of $\chi =80$ with ${\mathrm{\Delta }}_{\max }^{\mathrm{I}}$ for $\delta =0.25$. (c) and (d) respectively show ED result and tSDRG result of $\chi =80$ with ${\mathrm{\Delta }}_{\max }^{\mathrm{I}}$ for $\delta =1.00$. The green line shows the boundary of the first Brillouin zone of the triangular lattice.

Figure 11

$\delta$ dependencies of accuracy of the tSDRGs with various $\mathrm{\Delta }$ at $\chi =80$ for the random AF Heisenberg model on triangular lattices: (a) Error of the ground-state energy, ${\overline{\delta e}}_{\mathrm{\Delta }}$, and (b) error of the ground-state correlation function, ${\overline{\delta g}}_{\mathrm{\Delta }}$, for $N=24$. Solid symbols represent the results of the tSDRGs with ${\mathrm{\Delta }}_{\max }^{\mathrm{I}}$ and ${\mathrm{\Delta }}_{\mathrm{gs}}^{\mathrm{I}}$, while open symbols indicate the results for ${\mathrm{\Delta }}_{\max }^{\mathrm{P}}, {\mathrm{\Delta }}_{\mathrm{gs}}^{\mathrm{P}}$, and ${\mathrm{\Delta }}_{\mathrm{cut}}^{\mathrm{P}}$. (c) shows $\delta$ dependencies of the ground-state energy ${\overline{E}}_{\mathrm{\Delta }}$ for $N=36$ and $\chi =80$. The inset shows the $\delta$ dependence of ${\overline{\delta e^{\prime }}}_{\mathrm{\Delta }}$, which is the energy gain from ${\mathrm{\Delta }}_{\mathrm{cut}}^{\mathrm{P}}$ defined by Eq. (18). Error bars are negligible compared with the symbols.