Phase transition of the Ising model on a fractal lattice

The phase transition of the Ising model is investigated on a planar lattice that has a fractal structure. On the lattice, the number of bonds that cross the border of a finite area is doubled when the linear size of the area is extended by a factor of 4. The free energy and the spontaneous magnetization of the system are obtained by means of the higher-order tensor renormalization group method. The system exhibits the order-disorder phase transition, where the critical indices are different from those of the square-lattice Ising model. An exponential decay is observed in the density-matrix spectrum even at the critical point. It is possible to interpret that the system is less entangled because of the fractal geometry.

Figure 1

Composition of the fractal lattice. Upper left: A local vertex around an Ising spin shown by the open dot. Middle: The basic cluster which contains $N_1^{}=12$ vertices. Lower right: The extended cluster which contains $N_2^{}={12}_{}^2$ vertices. In each step of the system extension, the linear size of the system increases by a factor of 4, where only 12 units are linked, and where four units at the corners are missing, if it is compared with a $4\times 4$ square cluster.

Figure 2

The specific heat $c\left(T\right)$ per site in Eq. (19). Inset: Numerical derivative of $c\left(T\right)$ with respect to temperature; a sharp peak is observed at $T_{\mathrm{c}}^{}\approx 1.317$.

Figure 3

The spontaneous magnetization per site $m\left(T\right)$. Inset: The power-law behavior below $T_{\mathrm{c}}^{}=1.31716$.

Figure 4

Decay of the singular values after $n=8$ extensions.