Rigorous changeover threshold and critical behavior of the honeycomb round-the-face Potts model

Whether the continuous-to-first-order changeover integer $q_c$ of the two-dimensional ferromagnetic Potts model depends on the form of the local interaction is an open question linking statistical mechanics, combinatorics, and renormalization-group (RG) theory. For the standard nearest-neighbor model, $q_c = 4$ is exact and rigorous. For the honeycomb-lattice Potts model with round-the-face interaction - each elementary hexagon is rewarded when its six vertex spins are simultaneously equal - Schreiber, Cohen, Haber, Amir, and Barzel (2019) argued that the changeover drops to $q_c \le 3$, with a first-order transition already at $q = 3$. Their claim rests on unproven combinatorial assumptions (dominance of quasi-snake lattice-animal families and a constrained self-avoiding-walk enumeration bound), treats hole-containing configurations only heuristically, and establishes no thermodynamic critical temperature, only pseudo-critical values for lattices up to $L = 49$ and the lower bound $\hat T = 1/\ln(q^2 + 1)$. The objective is to settle this rigorously and physically: prove or refute $q_s < 3$, where $q_s$ is the combinatorial supremum of $\mu(\delta)^{1/\delta}$ over the governing hexagonal lattice-animal families; control the entropy of hole-rich configurations; determine the true critical temperature; and, as source-suggested further open directions, examine whether the $q_c \le 3$ changeover is lattice-independent within the round-the-face class and identify the RG structure (multiple multicritical points or an additional scaling field) that the phenomenon demands.

Background

The ferromagnetic q-state Potts model assigns to each lattice site a spin taking one of q values and rewards equal neighboring spins. In two dimensions its transition depends sharply on q: Baxter solved the square-lattice nearest-neighbor model exactly and found a continuous transition for $q \le 4$ and a discontinuous (first-order) transition with nonzero latent heat for $q > 4$ (Baxter 1973). The changeover integer $q_c = 4$ was later confirmed rigorously through the random-cluster representation - an equivalent percolation formulation of the Potts partition function due to Fortuin and Kasteleyn - which established continuity of the transition for $q \le 4$ (Duminil-Copin, Sidoravicius, Tassion 2016) and discontinuity for $q > 4$ (Duminil-Copin, Gagnebin, Harel, Manolescu, Tassion 2021) on planar lattices. Renormalization-group (RG) theory describes the changeover through three scaling fields (ordering, thermal, and dilution) near a single multicritical point (a point where the critical and tricritical branches of the phase diagram meet and more than the usual two scaling directions are relevant) (Nienhuis, Berker, Riedel, Schick 1979; Cardy, Nauenberg, Scalapino 1980); this picture suggested that $q_c = 4$ should be universal across local ferromagnetic interactions. It is not: in the Potts model with invisible states, non-interacting spin states that contribute entropy but no energy (Tamura, Tanaka, Kawashima 2010), the transition becomes first order even for $q \le 4$ once enough invisible states are added - although that mechanism manipulates entropy through states that never interact.

The source paper studies a different, purely ferromagnetic local modification. In the round-the-face Potts model the interaction is not between spin pairs but around elementary faces: with $K = \beta J > 0$:

$$-\beta H = K \sum_{\text{faces}} \delta_{\sigma_{\text{face}}},$$

where $\delta_{\sigma_{\text{face}}} = 1$ when all $\ell$ spins around a face are simultaneously equal and $0$ otherwise. On the square lattice ($\ell = 4$) this multisite model was analyzed with lattice-animal asymptotics and found to keep the standard changeover $q_c = 4$ (Schreiber, Cohen, Haber 2018). On the honeycomb lattice ($\ell = 6$), the source paper (Schreiber, Cohen, Haber, Amir, Barzel 2019) argues, via the same machinery, that the changeover drops to $q_c \le 3$.

The combinatorial framework is the following. A lattice animal is a connected set of $k$ faces with $m$ sites and boundary size $b$; a beast is a collection of animals sharing joint sites; a beast with holes encloses clusters colored differently from the surrounding bulk. The number of animals of size $k$ grows like $c\lambda^k k^{-1}$ with a growth constant $\lambda$ that is not known exactly for any two-dimensional lattice - a classical open enumeration problem dating to Klarner (1967), for which Madras's pattern theorem (1999) supplies rigorous bounds. Writing $m/k = \rho + \delta$ with the minimal asymptotic number of sites per face $\rho = 2$ on the honeycomb lattice, making a family of quasi-fractal beasts growing at rate $\mu(\delta)$ monochromatic is entropically disadvantageous at the transition only if $\mu(\delta)/q^{\delta} \le 1$; the changeover is therefore governed by the combinatorial supremum $q_s = \sup_\delta \mu(\delta)^{1/\delta}$, with a first-order transition for $q \ge q_s$. Bounding animal boundaries by self-avoiding walks (SAWs, lattice paths that never revisit a site) and using the exactly known honeycomb connective constant $\mu_c = \sqrt{2 + \sqrt{2}}$ - the exponential growth rate of the number of $n$-step SAWs (Duminil-Copin, Smirnov 2012) - gives $q_s \le \mu_c^2 \approx 3.4$, hence a first-order transition for $q \ge 4$. Pushing below $q = 3$ requires the assumptions that the supremum is governed by quasi-snakes - snake-like animals whose single-side boundary walk overlaps at most $\varepsilon \le 3$ edges per face - yielding the conditional bound $q_s \le \mu_c^{3/2} \approx 2.51 < 3$ and hence a first-order transition at $q = 3$. Wang-Landau entropic sampling - a flat-histogram Monte Carlo method that estimates the energy density of states directly (lattices up to $L = 49$, supplemented by Metropolis runs up to $L = 100$) - supports this picture: double-peaked energy distributions and specific-heat maxima scaling as $L^2$ at $q = 3, 4$, and Ising-like continuous behavior at $q = 2$. A lower bound $\hat T = 1/\ln(q^2 + 1)$ on the first-order critical temperature is derived in an appendix, but the derivation excludes beasts with holes by assumption and in fact shows that a hole-generated entropy term must be present at $\hat T$; the thermodynamic critical temperature itself remains undetermined.

Problem Statement

The task is the rigorous and physical completion of the changeover claim for the honeycomb round-the-face Potts model, whose source (Schreiber et al., Phys. Rev. E 100, 052119 (2019)) presents its own $q > 3$ and, conditionally, $q = 3$ first-order results as resting on unproven combinatorial assumptions and incomplete numerics. The common objective is to turn those conditional and finite-size results into a complete characterization of the changeover phenomenon in the honeycomb round-the-face Potts model. An answer must settle the following linked core components.

(1) Transition order at $q = 3$ (combinatorial core). Prove, or refute, without the unproven assumptions of quasi-snake dominance and the constrained-self-avoiding-walk enumeration bound with parameter $\varepsilon \le 3$, whether the combinatorial supremum $q_s = \sup_\delta \mu(\delta)^{1/\delta}$ over the hexagonal lattice-animal families governing the model satisfies $q_s < 3$ - which by the paper's free-energy criterion implies a first-order transition already at $q = 3$ - or whether $q_s \ge 3$. This entails the exact asymptotic enumeration, or sharp rigorous two-sided bounds on the growth constants, of the relevant hexagonal animal families (snakes, quasi-snakes, quasi-fractal beasts), together with a rigorous treatment of the entropy of beasts with holes, which is presently only bounded heuristically and which underlies both the supremum criterion and the lower bound $\hat T = 1/\ln(q^2 + 1)$ on the first-order critical temperature.

(2) Critical temperature. Determine the thermodynamic-limit critical temperature $T_c$ in the first-order regime (the source reports only pseudo-critical temperatures (finite-size estimates: $T_L$, the location of the specific-heat maximum, and $T_{LW}$, where the ordered- and disordered-phase energy weights balance) for $5 \le L \le 49$ and the bound $\hat T$), and confirm whether the finite-size indications - double-peaked energy distributions and specific-heat maxima growing as $L^2$ at $q = 3$ and $q = 4$ - persist in the infinite-volume limit.

Two further questions suggested by the source are open in their own right, but are not mandatory completion requirements for the core problem:

(3) Lattice independence. Determine whether the changeover at $q_c \le 3$ is a general property of round-the-face Potts models on two-dimensional lattices: the source's counting arguments extend only the boundary bound, not the value of $q_c$, to the triangular and square lattices, and the square-lattice four-site round-the-face model was earlier found to have $q_c = 4$, so the lattice-(in)dependence of the changeover within this model class is unsettled.

(4) Renormalization-group structure. Determine which RG description the changeover requires: whether the single-multicritical-point scaling theory in the ordering, thermal, and dilution fields must be replaced by multiple multicritical points or augmented by a new scaling field allowing $q_c$ to vary. Components (1) and (2) are the load-bearing core of the source's open problem; components (3) and (4) are the source-suggested generalization and interpretation.

References

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