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arXiv:2411.00423 · Physical review. E

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

Wei Liu, Xin Zhang, Lei Shi, Kai Qi, Xiang Li, Fangfang Wang, Zengru Di

Published May 20, 2025• 2 citations
arXivHTMLPDF

Abstract

Within the canonical ensemble framework, this paper investigates the presence of higher-order transition signals in the q-state Potts model (for q≥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 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.

DOI
10.1103/physreve.111.054128

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Within the canonical ensemble framework, this paper investigates the presence of higher-order transition signals in the q-state Potts model (for q≥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 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.
We employ the microcanonical inflection-point analysis method, developed for the systematic identification and classification of phase transitions in systems of any size, to study the two-dimensional Ising model at various lattice sizes and in the thermodynamic limit. Exact results for the density of states, which were obtained by exact algorithmic computation, provide evidence for higher-order transitions in addition to the well-studied second-order ferromagnetic-paramagnetic phase transition. An independent third-order phase transition is identified in the ferromagnetic phase, whereas another third-order transition resides in the paramagnetic phase. The latter is a dependent transition, i.e., it is inevitably associated with the critical transition, but it remains separate from the critical point in the thermodynamic limit. For a deeper insight into the nature of these additional transitions, a detailed analysis of spin clusters is performed.
This article investigates the pseudo transitions of the Blume-Capel model on two-dimensional finite-size lattices. By employing the Wang-Landau sampling method and microcanonical inflection point analysis, we identified the positions of phase transitions as well as higher-order phase transitions. Through Metropolis sampling and canonical ensemble analysis, we determined the geometric characteristics of the system at these transition points. When the crystal field parameter DDD exceeds 1.965, crossing the tricritical point, no third-order dependent phase transition is observed. However, a fourth-order independent transition was identified in the high-temperature region, and through Metropolis sampling analysis, a phase transition from the ordered paramagnetic phase to the disordered paramagnetic phase was confirmed, enhancing the phase diagram. Furthermore, the positions of the third-order phase transition obtained from both microcanonical and canonical analyses are consistent and mutually corroborative. We speculate that third-order dependent transitions vanish in the presence of strong first-order phase transitions.
Abstract We study third-order transitions in the Ising and Potts models on regular two-dimensional lattices and on 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 TcT_cTc​ 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 d⟨P⟩/dT\mathrm d\langle P\rangle/\mathrm dTd⟨P⟩/dT and independently checked against the specific-heat peak. 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 Tind＜Tc＜TdepT_{\mathrm{ind}}＜T_c＜T_{\mathrm{dep}}Tind​＜Tc​＜Tdep​. 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 robust third-order precursor transitions whose structural detectability can be amplified by network topology.
It is known that finite size scaling theory is well established for the systems undergoing a strong first-order transition [1-3]. However, if the correlation length is larger than the system size, methods using finite size scaling theory may fail to give indication of the first-order transition. The aim in this work is to observe how the system approaches to the first-order regime (where the correlation length is smaller but comparable to the size of the system) in two-dimensional 7-state Potts model which is known to exhibit first-order transition [4,5]. Cluster algorithm [6,7] is used to calculate commonly used equilibrium averages and the energy histogram, as well as cluster size related measurables like the average cluster size, fluctuations in the cluster size and the distribution of the cluster sizes [8,9]. The behavior of these measurables on different sizes varying from 8 \times 8 to 64 \times 64 is observed and the effect of this behavior on the order of the transition is studied.
Abstract The two-dimensional Potts Model with 2 to 10 states is studied using a cluster algorithm to calculate fluctuations in cluster size, distributions of cluster sizes as well as commonly used quantities like equilibrium averages and the histograms for energy and the order parameter. Results provide information about the variation of cluster sizes depending on the temperature and the number of states. They also give evidence for first-order transition when energy and the order parameter related measurables are inconclusive on small size lattices.
We investigate the first-order phase transitions of the qqq-state Potts models with q=5,6,7q = 5, 6, 7q=5,6,7, and 888 on the two-dimensional square lattice, using Monte Carlo simulations. At the very weakly first-order transition of the q=5q=5q=5 system, the standard data-collapse procedure for the order parameter, carried out with results for a broad range of system sizes, works deceptively well and produces non-trivial critical exponents different from the trivial values expected for first-order transitions. However, a more systematic study reveals significant drifts in the `pseudo-critical' exponents as a function of the system size. For this purpose, we employ two methods of analysis: the data-collapse procedure with narrow range of the system size, and the Binder-cumulant crossing technique for pairs of system sizes. In both methods, the estimates start to drift toward the trivial values as the system size used in the analysis exceeds a certain `cross-over' length scale. This length scale is remarkably smaller than the correlation length at the transition point for weakly first-order transitions, e.g., less than one tenth for q=5q=5q=5, in contrast to the naive expectation that the system size has to be comparable to or larger than the correlation length to observe the correct behavior. The results overall show that proper care is indispensable to diagnose the nature of a phase transition with limited system sizes.