Exact results for Ising models on the triangular Kagomé lattice

A two-dimensional Ising model on a particular topological structure, a “triangles-in-triangles” Kagomé structure, which is called a triangular-Kagomé lattice, is investigated. In this model, we consider two interaction parameters J_aa and J_ab , corresponding to the interactions among spins on the inner sites and between spins on the inner and vertex sites of the triangular in the Kagomé lattice, respectively. By summing over all spins at inner sites in the partition function, we arrive at the partition function of the Kagomé lattice. The effective interaction of the corresponding Kagomé lattice is always ferromagnetic, even for antiferromagnetic J_aa . The critical properties of the system depend only on J_aa/∣J_ab∣ . When J_aa/∣J_ab∣>-1 , the system has long range order at low temperature. However, when J_aa/∣J_ab∣<-1 , the partition function of the triangular-Kagomé lattice can be related to that of the Kagomé lattice with effective interaction at a temperature higher than its critical temperature. Therefore, the system will be in paramagnetic phase at all temperatures for J_aa/∣J_ab∣<-1 . The phase diagram for J_aa/∣J_ab∣ is given exactly.