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 $\left\{ {-1, 0, 1} \right\}$. We consider the external magnetic field $h$ to be positive, and let the chemical potential $\lambda$ 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 $\lambda$: 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 $\lambda$; 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.