Stability of a two-sublattice spin-glass model

We study the stability of the replica-symmetric solution of a two-sublattice infinite-range spin-glass model, which can describe the transition from an antiferromagnetic to a spin-glass state. The eigenvalues associated with replica-symmetric perturbations are in general complex. The natural generalization of the usual stability condition is to require the real part of these eigenvalues to be positive. The necessary and sufficient conditions for all the roots of the secular equation to have positive real parts is given by the Hurwitz criterion. The generalized stability condition allows a consistent analysis of the phase diagram within the replica-symmetric approximation.