Invisible-state transmutation thresholds for the q = 3 and q = 4 Potts models beyond mean field: planar random graphs and two-dimensional lattices

In the ferromagnetic Potts model with invisible states, each spin can take q interacting 'visible' states plus r non-interacting 'invisible' states that add entropy but no energy; increasing r past a critical count r_c converts the continuous transition into a first-order one. For q = 2 the threshold is known in many settings: about 3.6 in mean field, exactly $r_c(z)=\frac{4z}{3(z-1)}\left(\frac{z-1}{z-2}\right)^z$ on Bethe lattices and z-regular random graphs (16 for z = 3, 9 for z = 4), roughly 29 on the square lattice (2025), and about 223 on 4-regular planar random graphs via matrix models — r_c is strongly non-universal. For q = 3 and q = 4, however, no determination exists in any genuinely non-mean-field geometry: mean-field-like settings already have a first-order transition for q ≥ 3, so r_c is meaningful only where the pure transition is continuous, namely two-dimensional lattices and planar random graphs coupled to two-dimensional quantum gravity. This problem asks for r_c(q = 3) and r_c(q = 4) on planar random graphs via the external-field matrix-model computation that Ananikian et al. (2013) stated was possible in principle but never carried out, and/or on two-dimensional lattices where r_c is currently only bracketed; the resulting determinations would also illuminate how the non-universal threshold depends on q and on geometry.

Background

The model. The ferromagnetic q-state Potts model places a spin with q possible values ('colours') at each site and favours equal neighbouring spins; in two dimensions its transition is continuous for $q\le 4$ and first order for $q>4$. Tamura, Tanaka and Kawashima (2010) introduced a generalization in which each spin may additionally take any of $r$ 'invisible' states: a spin in an invisible state does not interact with its neighbours, but the invisible states are summed over in the partition function, so they add entropy without changing the internal energy, the symmetry, or the number of ground states. The motivation was experimental: some quasi-two-dimensional systems whose order parameter breaks a $Z_q$ symmetry (the cyclic group of order $q$, the symmetry of a $q$-state clock-like variable) show transition orders different from the Potts prediction. Because entropy favours the invisible states, increasing $r$ lowers the transition temperature and, past a critical count $r_c$, converts the continuous transition into a first-order one. This entropy-driven transmutation of transition order has also been proven rigorously to occur for $q>1$ at sufficiently large $q+r$ (van Enter, Iacobelli and Taati), although the proofs give no values of $r_c$. A useful reformulation rewrites the invisible states as a single non-interacting 'vacancy' value $\sigma=0$ carrying a temperature-dependent external field $-T\ln r$; the partition function is unchanged, so the $(q,r)$-state Potts model is equivalently a q-state Potts model diluted by annealed vacancies of fugacity $r$ ('annealed' meaning that the vacancies are thermal degrees of freedom summed over in the partition function, not frozen quenched disorder). For $q=2$ this maps exactly onto the spin-1 Blume-Emery-Griffiths (BEG) model (spins $t_i=\pm 1,0$ with a quadrupolar and a biquadratic coupling of equal strength and a crystal field) with crystal field $\mu=T\ln r$, and $r_c$ is fixed by the tricritical point of the BEG model — the point where the BEG transition line changes from second to first order.

Known values of $r_c$ for $q=2$. A Bragg-Williams mean-field treatment gives $r_c\simeq 3.6$ (first order for $r\ge 4$), and a complete-graph (mean-field-exact) analysis gives $r_c=3.65(5)$ (Krasnytska et al. 2016). On a Bethe lattice — the infinite regular tree, the standard interpolation between mean field and finite dimensions — with $z$ neighbours, Ananikian, Izmailyan, Johnston, Kenna and Ranasinghe (2013) derived

$r_c(z)=\frac{4z}{3(z-1)}\left(\frac{z-1}{z-2}\right)^{z},$

e.g. $r_c(3)=16$ (so 17 invisible states are needed) and $r_c(4)=9$, with the $z\to\infty$ limit $4e/3$ recovering mean field. The same values hold on $z$-regular random graphs ('thin graphs'), because the saddle-point equations of the graph enumeration coincide with the Bethe-lattice fixed points (Johnston and Ranasinghe 2013, where the 3-regular value 17 was first obtained). Other geometries give strikingly different thresholds. On random planar graphs — the Potts model coupled to two-dimensional quantum gravity, treatable exactly by matrix models, i.e. hermitean matrix integrals whose large-$N$ perturbative expansion generates the ensemble of decorated planar maps — the coupling to fluctuating geometry 'softens' the transition (the Ising transition there is third order), and Johnston and Ranasinghe (2013), transcribing the numerical triple point of the equal-coupling BEG model solved on 4-regular (quartic) planar graphs by the three-matrix model of Fukazawa, Hamada and Sato (1990), obtained $r_c\approx 223$. On the fixed square lattice, where first order has been known since 2010 at $r=30$, Wang-Landau Monte Carlo (a method that samples the energy density of states directly) combined with partition-function zeros (zeros of $Z$ in the complex temperature or field plane, whose behaviour signals the transition order) recently gave $r_c\approx 29$ (Lee, Park and Kim 2025). In one dimension an exact transfer-matrix solution shows no finite-temperature transition for $r\ge 0$, and on annealed scale-free networks (networks whose node-degree distribution follows a power law) the degree exponent, not $r$, controls criticality. The picture is that $r_c$ is strongly non-universal — it spans two orders of magnitude across geometries — and its geometry-dependence is known only in fragments.

Why $q=3,4$ is the open case. In all mean-field-like settings (Bragg-Williams, complete graph, Bethe lattices, z-regular random graphs, annealed networks) the ordinary $q\ge 3$ Potts transition is already first order, so no transmutation threshold exists there. The question 'how many invisible states are needed?' for $q=3$ and $q=4$ is meaningful precisely in the geometries where the pure model's transition is continuous: two-dimensional lattices ($q\le 4$) and planar random graphs, where the pure $q=2,3,4$ Potts transitions are continuous (with gravity-modified exponents). On two-dimensional lattices, simulations establish first order only at large $r$ (latent heat for the $(3,25)$, $(3,27)$, $(4,20)$, $(4,22)$ models), and the only threshold estimate is a rough $r_c(q{=}4)\approx 20$ from a Wang-Landau density-of-states conference study (Surungan, Komura and Okabe). On planar random graphs no determination at all exists: Ananikian et al. close their paper by remarking that the matrix models for the $q=3,4$ state Potts model, augmented with the external-field coupling to an invisible state described above, would 'in principle' allow a determination of $r_c$ for $q=3,4$ in a non-mean-field context — a calculation that has still not been carried out. A 2023 review of the whole invisible-states Potts literature (Krasnytska, Sarkanych, Berche, Holovatch, Kenna) records no $q\ge 3$ non-mean-field threshold, and the most recent mean-field analyses of tricriticality in this family (Kim, Lee and Kahng 2024) do not touch non-mean-field geometries.

Problem Statement

Determine the critical number of invisible states $r_c$ at which the phase transition of the ferromagnetic $(q,r)$-state Potts model changes from continuous to first order for $q=3$ and $q=4$ in genuinely non-mean-field geometries, where the pure $q$-state Potts transition is continuous and an entropy-driven transmutation threshold therefore exists. The named targets, following Ananikian et al. (2013):

(1) Planar random graphs (the Potts model coupled to two-dimensional quantum gravity, e.g. the ensemble of quartic/4-regular planar maps): construct the matrix model for the $(q,r)$-state Potts model with the external-field vertex weight representing the invisible states (the vacancy formulation with temperature-dependent field $T\ln r$), solve its large-$N$ saddle-point equations, locate the tricritical point separating the continuous (gravity-modified) transition from the first-order one, and extract $r_c$ for $q=3$ and $q=4$ — the calculation that Ananikian et al. stated would in principle allow such a determination but never carried out. In this formulation the $r=0$ limit reduces to the pure $q=3,4$ Potts model on planar random graphs, whose critical points are known, and the $q=2$ case reduces to the existing equal-coupling BEG transcription giving $r_c\approx 223$ on 4-regular planar random graphs.

(2) Two-dimensional lattices (e.g. the square lattice), where first order is currently established only at large $r$ (e.g. $(3,25)$, $(4,20)$) and the only threshold estimate is a rough $r_c(q{=}4)\approx 20$: determine $r_c(3)$ and $r_c(4)$ with controlled uncertainties, using indicators such as latent-heat scaling, partition-function zeros, energy-histogram bimodality, and order-parameter discontinuities, in the spirit of the recent $q=2$ determination $r_c\approx 29$.

A complete answer must deliver located tricritical/changeover thresholds for both $q=3$ and $q=4$ in at least one genuinely non-mean-field setting (the planar-random-graph matrix-model determination and the two-dimensional-lattice determination are the canonical instances named in the source lineage). Determinations restricted to mean-field-like settings (Bragg-Williams, complete graph, Bethe lattices, $z$-regular random graphs, annealed networks) do not answer the question, since in those settings the $q\ge 3$ transition is already first order and no transmutation threshold exists.

References

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