Frustration in a generalized kagomé Ising antiferromagnet: Exact results

We obtain the exact ground-state phase diagram of a generalized kagomé antiferromagnet with both pair and triplet interactions, J_{2} and J_{3}, respectively, in the presence of a magnetic field h appropriately tuned. We find that when the pair interaction J_{2}<0 dominates, the ground state is geometrically frustrated; on the other hand, the ground state is disordered but not frustrated when the triplet interaction J_{3} dominates, the boundaries between the two cases being at J_{3}=±J_{2}. The exact ground-state crossover lines between the two distinct types of disorder remain identifiable crossover curves at finite temperatures. In the frustrated domain, the ground state of the three-parameter model is identical to the ground state of the prototype one-parameter (J_{2}<0) model of geometrical frustration. Towards further understanding the frustration domain of the three-parameter model, a closed-form approximation (exact at zero temperature) determines solutions on a two-parameter subspace for induced magnetization and parallel magnetic susceptibility at finite fields h and temperatures T, the inverse susceptibility showing a Curie-Weiss behavior. We argue that the existence of an exact T=0 threshold magnetic field, below which the magnetization remains zero, indicates the existence of a gapped spectrum attributable to the presence of the triplet interaction J_{3}.