  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 four. 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 that 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.
