  Recently, diverse phase transition (PT) types have been obtained in multiplex
networks, such as discontinuous, continuous, and mixed-order PTs. However, they
emerge from individual systems, and there is no theoretical understanding of
such PTs in a single framework. Here, we study a spin model called the
Ashkin-Teller (AT) model in a mono-layer scale-free network; this can be
regarded as a model of two species of Ising spin placed on each layer of a
double-layer network. The four-spin interaction in the AT model represents the
inter-layer interaction in the multiplex network. Diverse PTs emerge depending
on the inter-layer coupling strength and network structure. Especially, we find
that mixed-order PTs occur at the critical end points. The origin of such
behavior is explained in the framework of Landau-Ginzburg theory.
