Nucleation of a three-state spin model on complex networks

• Nucleation of a three-state spin model on complex networks is studied.
• Heterogeneous mean-field theory is developed for evaluating free-energy barrier.
• Four regions are distinguished as distinct nucleation scenarios.
• A maximal nucleation rate is found at the two-step nucleation region.

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}{−1,0,1}. We consider the external magnetic field hh to be positive, and let the chemical potential λλ vary between −h−h and hh in a low temperature, such that the 11 configuration is stable, and −1−1 configuration and/or 00 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 hh and λλ: the system undergoes a two-step nucleation process from −1−1 configuration to 00 configuration and then to 11 configuration (region I); nucleation becomes a one-step process without an intermediate metastable configuration directly from −1−1 configuration to 11 configuration (region II(1)) or directly from 00 configuration to 11 configuration (region II(2)) depending on the sign of λλ; 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.