In this paper, we will investigate critical phenomena by considering a model spin-glass on scale-free networks. For this purpose, we consider the Ghatak-Sherrington (GS) model, a spin-1 spin-glass model with a crystal field, instead of the usual Isin
g-type model. Scale-free networks on which the GS model is placed are constructed from the static model, in which the number of vertices is fixed from the beginning. On the basis of the replica-symmetric solution, we obtain the analytical solutions, i.e., free energy and order parameters, and we derive the various phase diagrams consisting of the paramagnetic, ferromagnetic, and spin glass phases as functions of temperature $T$, the degree exponent $lambda$, the mean degree $K$, and the fraction of the ferromagnetic interactions $rho$. Since the present model is based on the GS model, which considers the three states ($S=0, pm 1$), the $S=0$ state plays a crucial role in the $lambda$-dependent critical behavior: glass transition temperature $T_{g}$ has a finite value, even when $2 < lambda < 3$. In addition, when the crystal field becomes nonzero, the present model clearly exhibits three types of inverse transitions, which occur when an ordered phase is more entropic than a disordered one.
We propose an expanded spin-glass model, called the quantum Ghatak-Sherrington model, which considers spin-1 quantum spin operators in a crystal field and in a transverse field. The analytic solutions and phase diagrams of this model are obtained by
using the one-step replica symmetry-breaking ansatz under the static approximation. Our results represent the splitting within one spin-glass (SG) phase depending on the values of crystal and transverse fields. The two separated SG phases, characterized by a density of filled states, show certain differences in their shapes and phase boundaries. Such SG splitting becomes more distinctive when the degeneracy of the empty states of spins is larger than one of their filled states.
