Three-state random energy model

We introduce a spin-1 version of the random energy model with a crystal field. The crystal field controls the density of 0 spins in the system. We solve the model in the microcanonincal ensemble. The model has a spin-glass transition at a finite temperature for all strengths of the crystal field. By introducing the magnetic field we also obtain the de Almeida-Thouless line for the model. The spin-glass transition persists in the presence of an external field. We also find that the magnetization shows nonmonotonic behavior for high positive crystal field strengths. The zero magnetic field specific heat and magnetic susceptibility also exhibit a cusp beyond a threshold value of the crystal field.