Duality criterion for the three-state Potts antiferromagnet on plane quadrangulations: rigorous proof or classification of exceptions

Whether the two-dimensional three-state Potts antiferromagnet (AF) is critical or ordered at zero temperature depends on the lattice. Lv, Deng, Jacobsen, Salas and Sokal (Phys. Rev. E 97, 040104 (2018)) proposed a criterion based on planar graph duality: for a periodic plane quadrangulation $\Gamma$ arising from a dual pair of plane graphs $(G_0,G_1)$, the $q=3$ AF has a zero-temperature critical point with central charge $c=1$ if $G_0$ is isomorphic to its own dual (self-dual type), and a finite-temperature transition in the three-state Potts ferromagnet universality class otherwise (non-self-dual type). Monte Carlo, transfer-matrix and critical-polynomial computations on eleven quadrangulations agree with the criterion without exception, but the supporting height-model and entropy arguments are heuristic, and the authors leave open whether exceptions exist — quadrangulations of self-dual type whose stiffness exceeds the locking threshold $K_c=\pi/2$ and order at $T=0$, or non-self-dual quadrangulations with a first-order transition — as well as an analytic derivation of the stiffness from lattice geometry and the mechanism selecting the ordered sublattice in equal-density cases. The criterion is used as a working rule but remains unproven, and no counterexample is known.

Background

The $q$-state Potts model assigns one of $q$ spin states ('colors') to each vertex of a graph and energetically favors equal colors on adjacent vertices in the ferromagnetic (FM) case and different colors in the antiferromagnetic (AF) case. In two dimensions the FM Potts models are governed by universality — their critical behavior depends only on $q$ — whereas AF Potts models depend strongly on the microscopic lattice. At zero temperature the $q$-state AF reduces to the uniform measure on proper $q$-colorings of the lattice graph, and one expects for each periodic lattice $\mathcal{L}$ a threshold $q_c(\mathcal{L})$ such that the model is critical at $T=0$ exactly at $q=q_c(\mathcal{L})$, while for $q>q_c(\mathcal{L})$ correlations decay exponentially at all temperatures including $T=0$. Determining $q_c(\mathcal{L})$ — equivalently, whether the zero-temperature coloring ensemble is critical (a 'spin liquid') or ordered — must be done lattice by lattice, and is directly relevant to frustrated-magnet materials and to the mathematics of the chromatic polynomial.

A sharp setting is the class of plane quadrangulations: periodic planar lattices all of whose faces are quadrilaterals. Such a lattice is bipartite, its vertex set splits into even and odd sublattices $V_0,V_1$, and drawing a diagonal across each face inside each sublattice produces a pair of planar graphs $(G_0,G_1)$ that are planar duals of one another (the dual $G^*$ of a plane graph $G$ has a vertex in each face of $G$ and an edge across each edge of $G$). This construction is a bijection between plane quadrangulations and dual pairs. Following the source authors, a quadrangulation is said to be of self-dual type if $G_0$ is isomorphic to its own dual $G_0^*$, and of non-self-dual type otherwise; the square lattice is of self-dual type, while the diced lattice (dual of the kagome lattice) is of non-self-dual type.

For the $q=3$ AF on any plane quadrangulation, the zero-temperature model admits a height representation: a many-to-one map from proper 3-colorings to integer-valued height functions on the faces, whose ensemble must be either 'rough' (massless Gaussian) — in which case the spin model is critical, with central charge $c=1$ (the label of the conformal field theory describing the critical fluctuations) and exponents determined by a single stiffness constant $K$ through Coulomb-gas relations such as $(\gamma/\nu)_{\mathrm{stagg}}=2-\pi/(18K)$ — or 'smooth' (locked), in which case the spin model is ordered. The border sits at a critical stiffness $K_c$ where the locking potential becomes marginal: $K_c=\pi/2$ for quadrangulations of self-dual type, and $K_c=\pi/8$ for quadrangulations of non-self-dual type. Only two anchor cases are settled: the square lattice is exactly solved ($K=\pi/6$, so the model is critical at $T=0$ with $c=1$ and exponents $(\gamma/\nu)_{\mathrm{stagg}}=5/3$, $(\gamma/\nu)_{\mathrm{u}}=2/3$), and the diced lattice is rigorously proven, by a Peierls-type contour argument, to have an ordered low-temperature phase. Quadrangulations with $q_c$ arbitrarily large are also known, so no naive criterion can work.

In this setting Lv, Deng, Jacobsen, Salas and Sokal proposed the duality criterion (their Conjecture 1, restated as Conjecture 1.1 in the companion article): (a) if the quadrangulation $\Gamma$ is of self-dual type, the 3-state Potts AF has a zero-temperature critical point with $c=1$ (so $q_c=3$), and is disordered at every $T>0$; (b) if $\Gamma$ is of non-self-dual type, the model has long-range order at low temperature and a finite-temperature second-order transition in the universality class of the 3-state Potts ferromagnet ($c=4/5$). They tested it by Monte Carlo simulations using the Wang–Swendsen–Kotecký (WSK) cluster algorithm (a Markov chain that interchanges two of the three colors on connected single-color clusters), by transfer matrices, and by the critical-polynomial method (systematically improvable graph-polynomial approximations to critical manifolds), on four quadrangulations of self-dual type (Q(hextri), Q(house), Q(martini-B), Q(cmm-pmm)) and seven of non-self-dual type (Q(diced), Q(martini), Q(ruby), Q(asanoha), Q(cross), G''_2 = Q(union-jack) and G''_3 = Q(decorated four-eight)) — the Rapid Communication treating four of each type and the companion article extending the non-self-dual set — beside the two settled anchors: the exactly solved square lattice (self-dual type) and the rigorously proven diced lattice (non-self-dual type). Agreement was found in every case: central charges extrapolating to $c\approx1$ in case (a), and exponents matching the 3-state FM Potts values $(\gamma/\nu)_{\mathrm{stagg}}=26/15$, $\nu=5/6$ in case (b). The supporting reasoning is nonetheless heuristic, and the authors state explicit failure modes: the measured stiffness grows with the maximum vertex degree and could exceed $K_c$ for large degree; a non-self-dual quadrangulation could conceivably undergo a first-order transition (as the triangular-lattice 3-state AF does); and when the two sublattices have equal vertex densities (Q(diced), Q(ruby)) the entropy argument does not even predict which sublattice orders — empirically it is the one containing the sub-sublattice of largest vertex degree, with only partial ferromagnetic order.

Problem Statement

Resolve the status of the duality criterion (Conjecture 1 of the source paper, restated as Conjecture 1.1 in the companion article) for the three-state Potts antiferromagnet on periodic plane quadrangulations — prove it, or refute it by exhibiting counterexamples.

For the $q=3$ Potts antiferromagnet on an arbitrary periodic plane quadrangulation $\Gamma$ arising from a dual pair $(G_0,G_1)$:

(a) Self-dual type. If $G_0\cong G_0^*$, prove that the zero-temperature model is critical with central charge $c=1$ — equivalently, that its height representation is rough with stiffness $K(\Gamma)<K_c=\pi/2$, so that $q_c(\Gamma)=3$ and the model is disordered at every positive temperature — or else exhibit quadrangulations of self-dual type for which this fails (for example, $K(\Gamma)\ge\pi/2$ with long-range order at $T=0$).

(b) Non-self-dual type. If $G_0\not\cong G_0^*$, prove that the model has long-range order at low temperature and a finite-temperature transition in the universality class of the three-state Potts ferromagnet — or else exhibit quadrangulations of non-self-dual type deviating from this prediction, in particular ones whose transition is first order.

The sources also discuss two predictive gaps inside the criterion, without making their closure a condition for resolving it: in case (a), the stiffness $K(\Gamma)$ — and hence the continuously varying critical exponents — is currently known only through numerical estimates (e.g. $K=0.6177(6)$, $0.5865(6)$, $0.5778(8)$, $0.5609(6)$ for Q(cmm-pmm), Q(hextri), Q(house), Q(martini-B), alongside the exact square-lattice value $K=\pi/6$) rather than by an analytic derivation from the lattice geometry; in case (b), the entropy argument does not predict which sublattice is the ordered one in the equal-density cases Q(diced) and Q(ruby), for which the currently available entropy-maximization account is heuristic. These are companion open issues discussed by the sources, not part of the objective above.

The same source also conjectures a dynamic property of the WSK cluster algorithm (absence of critical slowing-down at the $T=0$ critical point of every self-dual-type quadrangulation) and raises a methodological question about parity ($L\bmod4$) effects in transfer-matrix extrapolations; these are companion open issues with separate completion boundaries and are not part of the objective above.

References

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