Tensor-network study of a quantum phase transition on the Sierpiński fractal

The transverse-field Ising model on the Sierpiński fractal, which is characterized by the fractal dimension ${log}_2^{}3\approx 1.585$, is studied by a tensor-network method known as the higher-order tensor renormalization group. We analyze the ground-state energy and the spontaneous magnetization in the thermodynamic limit. The system exhibits the second-order phase transition at the critical transverse field $h_{\mathrm{c}}^{}=1.865$. The critical exponents $\beta \approx 0.198$ and $\delta \approx 8.7$ are obtained. Complementary to the tensor-network method, we make use of the real-space renormalization group and improved mean-field approximations for comparison.

Figure 1

Structure of the Sierpiński fractal. (a) The smallest unit consists of a site shown by the black circle, from which the three bonds $i,j$, and $k$ emerge. (b)–(d) Connecting the three units, one can iteratively expand the size of the unit. The vertical dotted lines correspond to the imaginary-time evolution, which is considered in Sec. 3.

Figure 2

Graphical representation of the processes in the HOTRG method in Eqs. (13a)–(13c). We use the large symbols to indicate the tensors $T_{}^{\left(\ell \right)}$ (circles), $A_{}^{\left(\ell \right)}$ (diamonds), $B_{}^{\left(\ell \right)}$ (squares), and the projectors $U,U^{\prime }$, and $U^{″}$ (triangles).

Figure 3

Ground-state energy per site $E_0^{}$ versus transverse field $h_x^{}$ at $h_z^{}=0$. The data obtained by the HOTRG method are shown by the thick full lines. The mean-field approximations ${\mathrm{MFA}}_1^{}$, ${\mathrm{MFA}}_3^{}$, and ${\mathrm{MFA}}_9^{}$ are, respectively, shown by dashed, dotted, and dot-dashed lines. The top inset shows $E_0^{}$ around the transition point $h_{\mathrm{c}}^{}$. The bottom inset shows $\langle {\sigma }_{}^z\rangle$.

Figure 4

Ground-state energies per site with respect to $h_x^{}$ calculated by the HOTRG method $\left(D=8\right)$ and by the RSRG method ($D=24$) when $h_z^{}=0$. The inset shows their difference.

Figure 5

Spontaneous magnetization $\langle {\sigma }^z\rangle$ and the induced polarization $\langle {\sigma }^x\rangle$ with respect to $h_x^{}$. The full lines show the calculated result by the HOTRG method ($D=8$) and the dashed ones by the RSRG method ($D=24$). The inset shows the susceptibility $\chi$ defined by Eq. (15).

Figure 6

Linear behavior in ${\langle {\sigma }^z\rangle }^{1/0.20}$ below $h_{\mathrm{c}}^{}$, which means $\beta =0.20$. The inset shows linear behavior in ${\langle {\sigma }^z\rangle }^{8.7}$ with respect to the longitudinal field $h_z^{}\to 0$ at the criticality $h_x^{}=h_{\mathrm{c}}^{}$; the linear behavior occurs at $\delta =8.7$. These values are obtained by the HOTRG method at $D=20$.

Figure 7

Graphical representations of Eqs. (A2) and (A6) with the planar (top) and the vertical (bottom) constructions of the $M$ tensors, respectively. These tensors can be interpreted as rectangular matrices of the size $D^2\times 2^2{D}^4$ (top) and $2^2\times 2^2{D}^4$ (bottom), respectively, and the singular value decomposition in Eqs. (A3) and (A7) is applied.

Figure 8

Entanglement entropies ${\varepsilon }_{\mathrm{planar}}^{\left(n\right)}$ and ${\varepsilon }_{\mathrm{vertical}}^{\left(n\right)}$ with respect to the number of stacking $n$, at $h_x^{}=0$ (circles), $h_x^{}=h_{\mathrm{c}}^{}$ (triangles), and $h_x^{}=3$ (squares) for $D=2$, when $\mathrm{\Delta }\tau =0.01$ and $h_z^{}=0$.

Figure 9

Entanglement entropies at $h_x^{}=h_{\mathrm{c}}^{}$ for the three selected imaginary-time steps $\mathrm{\Delta }\tau =0.1$ (circles), $\mathrm{\Delta }\tau =0.01$ (triangles), and $\mathrm{\Delta }\tau =0.001$ (squares).