Change in the Order of a Phase Transition in the 2D Potts Model with Equivalent Neighbours

TL;DR

This study investigates how finite-range interactions alter the order of the phase transition in the 2D $q=3$ Potts model. It combines a Fukui-Todo cluster Monte Carlo algorithm with an extended FK representation and a reweighting-based partition-function zeros analysis to locate Fisher zeros and extract critical exponents from zero scaling and density. The results indicate a marginal interaction range in the range $z_c \in [80,84]$, with exponents gradually shifting from second-order values toward first-order behavior as $z$ increases, though strong scaling corrections necessitate careful extrapolation. Overall, the work provides a precise approach to identifying order changes in finite-range Potts systems and aligns with prior findings while highlighting the role of scaling corrections and symmetry-preserving discretization in determining $z_c$.

Abstract

Two dimensional Potts model is a classical example where the symmetry of the order parameter controls the order of a phase transition: on a square lattice with nearest-neighbours interaction, when the number of states $q$ is less than or equal to 4, the second-order phase transition is observed, while for $q>4$ the first-order phase transition occurs. Recent research shows that even when the number of states is fixed, increasing the interaction range allows one to reach the point where the order of the phase transition changes. We focus on a $q=3$ 2D Potts model and, from the analysis of the partition function zeros, locate the number of interacting neighbours that change the order of the phase transition.

Figures (6)

• Figure 1: Magnetisation as a function of inverse temperature $\beta$ of a $q=3$ 2D Potts model with $z=88$ interacting neighbours in a narrow region around the pseudo critical point. Each line represents a different system size in a range from $L_{min}=32$ to $L_{max}=256$. As the system size increases, the curve becomes steeper.
• Figure 2: Specific heat (a) and magnetic susceptibility (b) as functions of inverse temperature $\beta$ of a $q=3$ 2D Potts model with $z=88$ interacting neighbours near the critical point. Each line represents a different system size in a range from $L_{min}=32$ to $L_{max}=256$. As the system size increases, the peaks of both observables get higher and more narrow.
• Figure 3: Coordinates of the first five Fisher zeros for the model with $z=68$. Different colours represent different sequential orders of a zero $j$. For a fixed $j$, the larger the system size $L$, the closer the zero lies towards the critical point.
• Figure 4: (a): dependency of the imaginary part of Fisher zeros on the logarithm of a system size. Different colours represent different sequential numbers of zeros as shown in the legend. The closest to the real axis zero is shown in blue. The dependency is qualitatively as expected from Eq. (\ref{['ansatz']}). (b): logarithm of the partition function zeros density and its fit according to Eq. (\ref{['density_scaling']}).
• Figure 5: Critical exponents values obtained from the partition function zeros analysis. As expected, when the fit is based on larger system sizes only, the result is shifted closer to the expected value. (a): correlation length critical exponent $\nu$ as a function of the minimal system size taken into consideration, obtained from the scaling of the closest zero. (b): specific heat critical exponent $\alpha$ as a function of the minimal system size taken into consideration, obtained from the zeros density analysis.