Disorder dependence of the phase-transition order in the two-dimensional q = 4 and q = 5 Potts models on the hexagonal lattice

The two-dimensional ferromagnetic q-state Potts model on the hexagonal (honeycomb) lattice undergoes a second-order transition for q = 4 (critical temperature fixed by duality, $k_BT_c/J=1/\ln 5\approx 0.6213$) and a first-order transition for q = 5; the Monte Carlo study by Murtazaev and Babaev (JETP 134, 720 (2022)) established both and explicitly deferred the question of how quenched non-magnetic (site) disorder changes the order of these transitions. Later literature has settled part of the deferred question: for q = 4, weak dilution (spin concentration $p\ge 0.80$) leaves the transition second order but produces a disorder-dependent universality class. What remains open is the decisive q = 5 case on this lattice — where the clean transition is first order and dilution is expected to round it — and the strong-dilution regime of both models down to the honeycomb site-percolation threshold. The problem is to determine the transition-order classification, critical line $T_c(p)$, and critical behavior of the site-diluted q = 4 and q = 5 Potts models on the hexagonal lattice as a function of dilution.

Background

The $q$-state Potts model places on every site of a lattice a variable $\sigma_i$ taking one of $q$ values, with a ferromagnetic Hamiltonian $H=-J\sum_{\langle i,j\rangle}\delta(\sigma_i,\sigma_j)$ that rewards equal values on neighboring sites ($J>0$; $\delta$ is the Kronecker symbol). In two dimensions the phase structure is exactly known on the square lattice and is lattice-independent: the order-disorder transition is continuous for $q\le 4$ (the case $q=4$ is marginal, with logarithmic corrections) and first order for $q\ge 5$ — that is, accompanied by a latent heat, a finite discontinuity in the energy density — already at $q=5$, where the first-order character is weak. These models are not purely academic: the honeycomb lattice is the basal plane of graphite, and Potts models on it are standard descriptions of commensurate adsorbed monolayers and of intercalation in graphite-like substrates.

The hexagonal (honeycomb) lattice, in which every site has three nearest neighbors, is dual to the triangular lattice (the model on a lattice is exactly equivalent to the model on the dual lattice with a transformed coupling), and a duality/star-triangle argument fixes the critical coupling of the clean $q$-state Potts model on it; for $q=4$ it gives $J/(k_BT_c)=\ln 5$, that is $k_BT_c/J=1/\ln 5\approx 0.6213$. Monte Carlo work by Murtazaev and Babaev (2022), using the Wolff single-cluster algorithm (a Monte Carlo method that updates correlated spin clusters to overcome critical slowing down) for linear sizes $L=21$–$180$, confirmed the expectations on this lattice: a second-order transition at $q=4$ with $T_c=0.620\,J/k_B$, and a first-order transition at $q=5$. Independent simulations agree on the $q=4$ honeycomb result: a Wang–Landau density-of-states study (Fadeeva and Shchur 2022; a Monte Carlo method that estimates the energy density of states directly) found second-order behavior with the standard $q=4$ exponents, and very large simulations (Peng and Jiang) found honeycomb energy histograms single-peaked up to $L=2048$ while pseudo-first-order signatures (bimodal energy histograms whose valley deepens rather than fills in as the system grows) persist on the square lattice — a reminder that classifying the order of these transitions at finite size is delicate.

Here disorder means quenched, non-magnetic (site) dilution: an independently chosen random fraction $1-p$ of the sites is occupied by non-magnetic impurities that cannot carry a spin; the impurity configuration is drawn once and frozen while the remaining spins are thermally averaged, and observables are averaged over impurity configurations ($p$ is the spin concentration). Two general results frame what dilution can do. First, the Harris criterion states that quenched disorder is a relevant perturbation of a continuous transition when the clean specific-heat exponent satisfies $\alpha>0$; for the two-dimensional $q=4$ Potts model $\alpha=2/3$, so weak dilution must modify the critical behavior. At a first-order transition in two dimensions, the Aizenman–Wehr theorem proves (for continuously distributed quenched disorder) that randomness rounds the transition: the latent heat is destroyed and the transition becomes continuous. Second, extensive numerical work on random-bond Potts models — in which the coupling $J$ of each bond is itself quenched random, rather than sites being removed — e.g. the randomness-induced second-order transition in the $q=8$ square-lattice model (Chen, Ferrenberg and Landau 1992) — together with the renormalization-group and transfer-matrix (exact finite-strip partition-function) analyses of Cardy and Jacobsen shows that disordered two-dimensional Potts models possess their own critical points whose exponents vary continuously with disorder strength; whether all of them are instead governed by one superuniversal set of exponents, the same for every $q$ and every disorder strength, is still debated (Delfino and Tartaglia 2017). This literature is reviewed by Berche and Chatelain (2004).

For site dilution specifically, the pattern has been established directly on the square and triangular lattices: slight non-magnetic dilution turns the first-order transition of the $q=5$ model into a second-order one on both the square lattice (Murtazaev, Babaev and Ataeva 2020) and the triangular lattice (Ataeva, Babaev and Murtazaev 2024), while for $q=4$ the transition stays second order. On the hexagonal lattice — the lattice of the 2022 source study — the same group subsequently answered the weak-dilution $q=4$ part: at spin concentrations $p=0.90$ and $p=0.80$ the transition remains second order, with critical exponents defining a universality class (the set of critical exponents governing scaling near the transition) different from that of the pure model (Babaev and Murtazaev 2023; A. K. Murtazaev, A. B. Babaev and M. A. Babaev 2026). What has not been determined anywhere is the site-diluted $q=5$ model on the hexagonal lattice — precisely the case whose clean limit is first order, so that dilution can change the order of the transition and not merely its exponents — and the strong-dilution regime $p<0.80$ of both models, down to the site-percolation threshold of the honeycomb lattice, $p_c\approx 0.6970$ (the occupied-site concentration below which the lattice loses long-range connectivity and no magnetic transition can survive).

Problem Statement

Determine the nature of the thermal phase transition in the two-dimensional ferromagnetic $q$-state Potts model on the hexagonal (honeycomb) lattice with quenched non-magnetic disorder — a random fraction $1-p$ of the sites occupied by frozen non-magnetic impurities — as a function of the spin concentration $p$, for $q=4$ and $q=5$.

In the clean limit $p=1$ the transition is second order for $q=4$ (with $k_BT_c/J=1/\ln 5\approx 0.6213$) and first order for $q=5$. The open scientific objective, posed explicitly by the source study of Murtazaev and Babaev (2022) and only partially settled since, is to complete the disorder-versus-transition-order map for these two models on this lattice:

1. For $q=5$, whose clean transition is first order: does the introduction of non-magnetic impurities round the transition into a second-order one (as the Aizenman–Wehr theorem and the square- and triangular-lattice analogues suggest), or does it remain first order, become sharp in the order parameter while the latent heat is destroyed, or smeared (no sharp thermodynamic singularity), or disappear; and, if continuous, what are the critical line $T_c(p)$ and the universality class as a function of $p$?

2. For both $q=4$ and $q=5$, the strong-dilution regime down toward the site-percolation threshold of the honeycomb lattice, $p_c\approx 0.6970$: does the transition persist as a second-order one for all $p>p_c$, and how does its critical behavior cross over toward percolation-controlled criticality as $p\to p_c$?

The weak-dilution regime of the $q=4$ model ($p\ge 0.80$) has already been determined — the transition remains second order with a disorder-dependent universality class — so an answer that only re-derives that regime does not resolve the question. The common objective uniting the parts above is the transition-order classification and the critical behavior of the two diluted hexagonal-lattice Potts models across the dilution range; each part supplies one piece of that single map.

References

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