Geometric properties of the additional third-order transitions in the two-dimensional Potts model

TL;DR

This study explores higher-order phase transitions in the 2D Potts model using geometric order parameters, confirming their existence and revealing how different parameters detect various transition types.

Contribution

It introduces geometric order parameters to identify third-order transitions in the Potts model, demonstrating their effectiveness and limitations across different q-values.

Findings

Isolated spins number reliably indicates third-order independent transitions.

Average perimeter of clusters detects third-order dependent transitions, but not in first-order cases.

Results align with microcanonical inflection-point analysis, validating the approach.

Abstract

Within the canonical ensemble framework, this paper investigates the presence of higher-order transition signals in the $q$-state Potts model (for $q \geq 3$), using two geometric order parameters: isolated spins number and the average perimeter of clusters. Our results confirm that higher-order transitions exist in the Potts model, where the number of isolated spins reliably indicates third-order independent transitions. This signal persists regardless of the system's phase transition order, even at higher values of $q$. In contrast, the average perimeter of clusters, used as an order parameter for detecting third-order dependent transitions, shows that for $q = 6$ and $q = 8$, the signal for third-order dependent transitions disappears, indicating its absence in systems undergoing first-order transitions. These findings are consistent with results from microcanonical inflection-point analysis, further validating the robustness of this approach.