Instability of paramagnetic state toward glassy state in random Ising antiferromagnet on tetrahedron cactus lattices

Ising antiferromagnet on tetrahedron cactus lattices with randomness in the exchange interactions is studied. Instability line of the paramagnetic state, beyond which glassy or antiferromagnetic state is (meta)stable is obtained. The model is investigated by the replica method. Instability toward antiferromagnetic state does not occur for M≤4 where M is the number of corner sharing tetrahedra for the cactus lattices. Instability toward glassy state occurs at as weak randomness as J∕(−J0)≃0.056,0.020, and ≤10−4 for M=2,3, and 4, respectively, where J0 and J2 are the mean and variance of the Gaussian random exchange interaction, respectively.