Exact renormalization-group approach to critical phenomena in an Ising system with long-range interactions on the Sierpiński-gasket lattice

Abstract

An Ising spin system with two-point long-range interactions on the Sierpiński-gasket lattice is considered. The interactions are assigned along edges of all equilateral triangles being part of the self-similar structure. In addition, the long-range couplings, i.e., all couplings besides nearest-neighbor interactions K, are assumed to be independent of the distance, i.e., Q 1 = Q, l = 1, 2, …, ∞, where Q 1 denotes an interaction between the two spins on the distance 2' a with a being a lattice constant, and Q a positive parameter. Using an exact renormalization-group method, based upon the decimation transformation, the system is shown to display a line of nontrivial fixed points K ∗ = K ∗(Q) . Each point of this line corresponds to a phase transition of a rather high order. The critical behavior of the system near every nontrivial fixed point is claimed to be characterized by two dimensions, viz., the fractal dimension and the so-called critical dimension, which reflects some topological properties of the system. The critical dimension is defined through considering the decay of an effective two-point correlation function, and depends on Q.