Phase diagram and exotic spin-spin correlations of anisotropic Ising model on the Sierpi\'nski gasket

The anisotropic antiferromagnetic Ising model on the fractal Sierpi\'{n}ski gasket is intensively studied, and a number of exotic properties are disclosed. The ground state phase diagram in the plane of magnetic field-interaction of the system is obtained. The thermodynamic properties of the three plateau phases are probed by exploring the temperature-dependence of magnetization, specific heat, susceptibility and spin-spin correlations. No phase transitions are observed in this model. In the absence of a magnetic field, the unusual temperature dependence of the spin correlation length is obtained with $0 \leq$J$_b/$J$_a<1$, and an interesting crossover behavior between different phases at J$_b/$J$_a=1$ is unveiled, whose dynamics can be described by the J$_b/$J$_a$-dependence of the specific heat, susceptibility and spin correlation functions. The exotic spin-spin correlation patterns that share the same special rotational symmetry as that of the Sierpi\'{n}ski gasket are obtained in both the $1/3$ plateau disordered phase and the $5/9$ plateau partially ordered ferrimagnetic phase. Moreover, a quantum scheme is formulated to study the thermodynamics of the fractal Sierpi\'{n}ski gasket with Heisenberg interactions. We find that the unusual temperature dependence of the correlation length remains intact in a small quantum fluctuation.