Third-order transitions in Ising and Potts models on Watts--Strogatz small-world networks

Abstract

We study third-order transitions in the two-dimensional Ising and Potts model on regular lattices and Watts--Strogatz small-world networks. Cluster observables are used to track post-critical boundary reorganization and pre-critical cluster breakup. For the Ising model, the critical temperature $T_c$ is calibrated independently from Binder-cumulant crossings and susceptibility peaks, whereas for the Potts model on small-world networks it is identified operationally from the dominant critical peak of $\mathrm d\langle P\rangle/\mathrm dT$. The independent and dependent third-order transitions are identified from the isolated-spin peak and the post-critical structural extremum, respectively. For both lattice and small-world topologies, we find the robust ordering $T_{\mathrm{ind}}<T_c<T_{\mathrm{dep}}$. Increasing the rewiring probability shifts all three characteristic temperatures upward and enhances the visibility of the post-critical transition. The effect is especially clear in the Potts model, where perimeter-based observables are more sensitive to multistate boundary fluctuations. The systematic persistence of the characteristic temperature hierarchy across topologies and finite sizes argues against interpreting these features as incidental finite-size irregularities. Instead, our results support their interpretation as genuine third-order transitions whose structural detectability can be amplified by network topology.

Figures (9)

• Figure 1: Illustration of the cluster observables used in the analysis. The cluster size $A(C)$ counts the number of sites in a connected equal-spin cluster, the perimeter $P(C)$ counts the number of boundary edges, and $n_{\mathrm{iso}}$ counts isolated spins.
• Figure 2: Dependent third-order transitions on the regular square lattice. The figure compares the structurally most informative post-critical observables in the Ising and three-state Potts models. In the Ising model, the dependent transition is more clearly resolved in the area-based derivative, whereas in the Potts model it is more clearly resolved in the perimeter-based derivative. Gray dashed lines mark the critical temperature $T_c$, and pink dashed lines indicate the post-critical temperature $T_{\mathrm{dep}}$.
• Figure 3: Independent third-order transition on the regular square lattice. (a),(c) Largest-cluster size $A_{\max}$ for the Ising and three-state Potts models. (b),(d) Number of isolated spins $n_{\mathrm{iso}}$ for the Ising and three-state Potts models. In both models, the low-temperature-side precursor is most clearly identified by the maximum of $\langle n_{\mathrm{iso}}\rangle$, while $A_{\max}$ provides supporting information on the breakup of large domains. Gray dashed lines mark the critical temperature $T_c$, blue dashed lines mark the independent third-order temperature $T_{\mathrm{ind}}$, and pink dashed lines indicate the post-critical temperature $T_{\mathrm{dep}}$ for reference.
• Figure 4: Calibration of the conventional critical temperature on Watts--Strogatz small-world networks for the Ising model at rewiring probability $p=0.1$. Panels (a)--(c) show the absolute magnetization, magnetic susceptibility, and fourth-order Binder cumulant. The susceptibility peak and Binder-cumulant crossing provide mutually consistent estimates of $T_c$.
• Figure 5: Topology dependence of third-order precursor transitions in the Ising model on Watts--Strogatz small-world networks. Panels (a),(b), (c),(d), and (e),(f) correspond to rewiring probabilities $p=0.1$, $0.3$, and $1.0$, respectively. The top row shows the perimeter-based post-critical signal $\mathrm{d}\langle P\rangle/\mathrm{d}T$, and the bottom row shows the isolated-spin indicator $\langle n_{\mathrm{iso}}\rangle$. Increasing rewiring shifts the full hierarchy of characteristic temperatures upward and makes the post-critical structural precursor easier to separate from the dominant critical peak. Gray dashed lines mark $T_c$, blue dashed lines mark $T_{\mathrm{ind}}$, and pink dashed lines mark $T_{\mathrm{dep}}$.