Nucleation of a three-state spin model on complex networks

We study the metastability and nucleation of the Blume–Capel model on complex networks, in which each node can take one of three possible spin variables {−1,0,1}. We consider the external magnetic field h to be positive, and let the chemical potential λ vary between −h and h in a low temperature, such that the 1 configuration is stable, and −1 configuration and/or 0 configuration are metastable. Combining the heterogeneous mean-field theory with simulations, we show that there exist four regions with distinct nucleation scenarios depending on the values of h and λ: the system undergoes a two-step nucleation process from −1 configuration to 0 configuration and then to 1 configuration (region I); nucleation becomes a one-step process without an intermediate metastable configuration directly from −1 configuration to 1 configuration (region II(1)) or directly from 0 configuration to 1 configuration (region II(2)) depending on the sign of λ; the metastability of the system vanishes and nucleation is thus irrelevant (region III). Furthermore, we show that in the region I nucleation rates for each step intersect that results in the occurrence of a maximum in the total nucleation rate.