Thermodynamic-limit phase and critical structure of the triangular-lattice $Q$-state Potts antiferromagnet and its $A_{p-1}$ restricted-solid-on-solid counterpart

Jacobsen, Salas and Scullard (J. Phys. A 50, 345002 (2017)) mapped the phase diagram of the triangular-lattice $Q$-state Potts antiferromagnet in the real $(Q,v)$ plane - locating the antiferromagnetic critical curve and its T-point bifurcation for $Q\le 4$ - and studied the related $A_{p-1}$ restricted-solid-on-solid (RSOS) height models at the Beraha values $Q=B_p=4\cos^2(\pi/p)$ on toroidal triangular strips. Their finite-width transfer-matrix and critical-polynomial analysis uncovered structures whose thermodynamic-limit status and field-theory identification remain unsettled: for $Q>4$, intricate small phases and regions dominated by complex-conjugate pairs of dominant transfer-matrix eigenvalues; for the RSOS models, narrow central-charge peaks just below the antiferromagnetic critical curve ($c=1.198(2)$, conjectured exactly $6/5$, at $p=5$; $c\approx1.53$ at $p=7$; $c\approx1.64$ at $p=8$; $c=1.265(15)$ at $p=6$), plateaux and renormalisation-group flows inside the Berker-Kadanoff interval, intervals with large unphysical fitted central charges, and the failure of the expected Fortuin-Kasteleyn (FK) eigenvalue cancellations on the non-planar torus at non-integer $Q$. The problem is to determine which of these structures are genuine thermodynamic-limit physics, to identify the nature and conformal field theories of the RSOS critical points, to map the renormalisation-group flows among them, and to establish the criterion for RSOS-FK equivalence on toroidal geometry.

Background

The $Q$-state Potts model assigns to each vertex of a lattice a spin $\sigma_i\in\{1,\ldots,Q\}$ and a Boltzmann weight $e^{J\delta_{\sigma_i,\sigma_j}}$ on each edge. Writing $v=e^J-1$, the antiferromagnetic (AF) regime is $v<0$, and zero temperature is $v=-1$, where the partition function becomes the chromatic polynomial counting proper $Q$-colourings (the chromatic line). In the Fortuin-Kasteleyn (FK) random-cluster representation, $Z=\sum_{A\subseteq E}v^{|A|}Q^{k(A)}$ with $k(A)$ the number of connected components of the spanning subgraph $(V,A)$, both $Q$ and $v$ can be continued off their physical values, so the phase diagram is naturally studied in the real $(Q,v)$ plane. For the square lattice this diagram is exactly known: Baxter's solution gives the self-dual curves $v=\pm\sqrt{Q}$ and the mutually dual AF curves $v_\pm=-2\pm\sqrt{4-Q}$; the ferromagnetic (FM) critical curve is $v=+\sqrt{Q}$, while the lower self-dual branch acts as the attractor of a finite Berker-Kadanoff (BK) critical phase - an extended critical region, bounded by $v_\pm$, in which the temperature perturbation is irrelevant and some magnetic exponents are negative (hence unphysical). Crossing the AF curves $v_\pm$, the square-lattice transition is simultaneously first and second order (Jacobsen-Saleur 2006), and its continuum limit has central charge $c_{\rm PF}=2-6/p$ (an $sl(2,\mathbb{R})/u(1)$ Euclidean black-hole sigma model, i.e. a sigma model of one compact and one non-compact coupled boson), where $p$ is defined by $Q=B_p=4\cos^2(\pi/p)$; the numbers $B_p$ are the Beraha numbers, at which planar Potts partition functions exhibit massive eigenvalue cancellations.

For the triangular lattice the analogous exactly known object is Baxter's cubic $v^3+3v^2=Q$ (Baxter-Temperley-Ashley), whose three branches play the roles of the FM critical curve $v_{\rm FM}$, the BK critical curve $v_{\rm BK}$, and the lower boundary $v_-$. The chromatic line $v=-1$ is also integrable (Baxter): the exact solution exhibits the BK phase (Regime I) for $0<Q<Q_0=3.81967\ldots$ and a distinct critical phase (Regime IV), governed by the coset conformal field theory $SU(2)_4\times SU(2)_{p-6}/SU(2)_{p-2}$ (a quotient of $SU(2)$ current-algebra theories whose surviving fields furnish the critical operator content), for $Q_0<Q\le 4$ (Vernier-Jacobsen-Salas 2016). The missing ingredient was the AF critical curve $v_{\rm AF}(Q)$, the upper boundary of the BK phase. Jacobsen, Salas and Scullard (2017) computed it by combining transfer matrices (TM) for triangular strips of width $L$ with periodic (toroidal) transverse boundary conditions in the FK representation with the critical-polynomial method (a finite-size approximation to critical curves obtained from graph polynomials). They found that $v_{\rm AF}$ bifurcates at a T-point (a triple point where three phase boundaries meet) $T=(Q_T,v_T)\approx(3.77,-0.95)$, one branch ending at $(Q,v)=(4,-1)$ and the other passing through $(Q_0,-1)$ and continuing to $(4,v_1)$ with $v_1=-1.286\pm0.004$, and concluded that the $(Q,v)$ phase diagram is complete for $0\le Q\le 4$.

At Beraha values the Potts model admits an equivalent restricted-solid-on-solid (RSOS) description: a height model whose integer heights $h\in\{1,\ldots,p-1\}$ live on the triangular lattice (with complementary parity on the dual hexagonal lattice, giving even and odd sectors) and must change by exactly one unit between nearest neighbours (the $A_{p-1}$ graph adjacency, hence the name $A_{p-1}$ RSOS model). On planar graphs the RSOS and Potts/FK partition functions are then locked together by the Beraha eigenvalue cancellations. The 2017 study built the toroidal TM of this RSOS model for $p=4,\ldots,8$ and widths $L=3,6,9,12,15$ ($L\le 12$ for $p=7,8$, chosen as multiples of 3 to avoid mod-3 parity effects), fitted the free energy to the conformal finite-size-scaling (FSS) ansatz to extract an effective central charge $c$ (the parameter characterising the conformal field theory, obtained from the $1/L$ correction to the free energy at a critical point), and compared with the FK and spin TMs. Two exact values anchor the analysis: $c_{\rm FM}=1-6/[p(p-1)]$ (unitary minimal models, the ferromagnetic Potts values) and $c_{\rm PF}=2-6/p$. Beyond the expected peaks carrying these two values, the data showed, for each $p$, a rich structure whose interpretation could not be settled at the accessible widths: narrow peaks just below $v_{\rm AF}$ with fitted $c\gtrsim1$ ($c=1.198(2)$ at $x=v/\sqrt{Q}\approx-0.4883$ for $p=5$, conjectured exactly $c=6/5$; $c\approx1.53$ at $p=7$; $c\approx1.64$ at $p=8$; $c=1.265(15)$ at $p=6$, coinciding with the weak first-order transition of the three-state model at $x=-0.460106(12)$ found by Adler-Brandt-Janke-Shmulyian), narrow plateaux inside the BK interval alternating between values compatible with $c_{\rm PF}$ and $c_{\rm FM}$, intervals (around $x\approx-1$) where the fits give large unphysical central charges, disagreement of the RSOS and FK free energies across the whole BK interval with no eigenvalue cancellations at non-integer $B_p$ on the torus, and, on the Potts side for $Q>4$, small phases and regions dominated by complex-conjugate pairs of dominant TM eigenvalues. Deciding what, if anything, in this catalogue survives the thermodynamic limit $L\to\infty$ is the open problem.

Problem Statement

Determine the thermodynamic-limit phase and critical structure of the triangular-lattice $Q$-state Potts antiferromagnet and its $A_{p-1}$ restricted-solid-on-solid (RSOS) counterpart, as left open by Jacobsen-Salas-Scullard (2017). The common objective is to decide, for each finite-width structure found in that work, whether it is genuine $L\to\infty$ physics or a finite-width artefact, and to identify the universality classes and renormalisation-group (RG) flows of whatever critical behaviour is genuine. Concretely:

(1) $Q>4$ Potts structures. Do the intricate finite-width structures found for $Q>4$ - the small phases and the regions dominated by complex-conjugate pairs of dominant transfer-matrix eigenvalues (the phases $\ell^*$, $0\le\ell\le3$, which also appear near the lower branch $v_-$ for $0\le Q\le1$ and as new bands in the critical-polynomial approach) - survive as $L\to\infty$ as bona fide thermodynamic phases or continuous accumulation regions of partition-function zeros, or do they shrink and vanish as finite-width artefacts?

(2) Nature of the RSOS peaks. For the $A_{p-1}$ RSOS models on the triangular lattice, what is the nature of the narrow central-charge peaks just below the antiferromagnetic critical curve (parametrised by the reduced variable $x=v/\sqrt{Q}$) - at $x\approx-0.4883$ with $c=1.198(2)$ for $p=5$ (conjectured exactly $c=6/5$, with the suggested decomposition $6/5=1/2+7/10$ as a ferromagnetic-model-plus-Ising-fermion theory), $c\approx1.53$ for $p=7$, $c\approx1.64$ for $p=8$, and $c=1.265(15)$ for $p=6$ coinciding with the known weak first-order point $x=-0.460106(12)$ - first-order, critical, or mixed first- and second-order as on the square-lattice AF curve; and if (some are) critical, which conformal field theories describe them?

(3) RG flows and unphysical central charges. Map the RG flows among the RSOS plateaux and peaks inside the Berker-Kadanoff interval (the extended critical phase of the Potts model) - the sequences $c_{\rm PF}\to c_{\rm FM}\to$ other fixed points, where $c_{\rm PF}=2-6/p$ and $c_{\rm FM}=1-6/[p(p-1)]$ are the central charges of the antiferromagnetic and ferromagnetic Potts fixed points - identify the relevant and irrelevant operators controlling the crossovers and the universality classes of the putative fixed points, and clarify the intervals around $x\approx-1$ where finite-size-scaling fits give large unphysical central charges: non-unitary continuum limits, breakdown of the scaling ansatz, or finite-size contamination.

(4) RSOS-FK equivalence on toroidal geometry. Establish the criterion for the $A_{p-1}$ RSOS model to reproduce the Potts/FK thermodynamics on toroidal (non-planar) graphs when $Q=B_p=4\cos^2(\pi/p)$ (the Beraha numbers) is not an integer - in particular the conjectured restriction of eigenvalue cancellations to positive integer $Q$ on non-planar graphs (the generalised Beraha conjecture), and the role of the periodicity sectors $a$ in the thermodynamic limit.

Deliverables (1)-(4) are jointly needed: (1) completes the Potts phase diagram beyond $Q=4$; (2) and (3) identify the new critical points and flows observed in the RSOS surrogate through which the BK regime is probed; and (4) determines when that surrogate's data can be read as Potts physics. Two further items recorded in the source - sharper estimates of the T-point $(Q_T,v_T)$ and of $v_1$ as $Q\to4$ with controlled error bars, and an understanding of which bases or twists make the critical-polynomial method cover the AF branches it currently misses (the interval $2<Q<3$ and the branch from the T-point down to $(4,v_1)$) - are precision and methodological refinements of the same phase diagram and are recorded as auxiliary goals; an answer need not deliver them for the objective above to count as met.

References

1. Jesper Lykke Jacobsen, Jesus Salas, Christian R. Scullard, Phase diagram of the triangular-lattice Potts antiferromagnet, J. Phys. A: Math. Theor. 50 (2017) 345002. DOI: 10.1088/1751-8121/aa778f; arXiv:1702.02006. https://doi.org/10.1088/1751-8121/aa778f
2. Eric Vernier, Jesper Lykke Jacobsen, Jesus Salas, Q-colourings of the triangular lattice: exact exponents and conformal field theory, J. Phys. A: Math. Theor. 49 (2016) 174004. DOI: 10.1088/1751-8113/49/17/174004; arXiv:1509.02804. https://doi.org/10.1088/1751-8113/49/17/174004
3. Jesper Lykke Jacobsen, Hubert Saleur, The antiferromagnetic transition for the square-lattice Potts model, Nucl. Phys. B 743 (2006) 207-248. DOI: 10.1016/j.nuclphysb.2006.02.041; arXiv:cond-mat/0512058. https://doi.org/10.1016/j.nuclphysb.2006.02.041
4. Jesper Lykke Jacobsen, Jesus Salas, A generalized Beraha conjecture for non-planar graphs, Nucl. Phys. B 875 (2013) 678-718. DOI: 10.1016/j.nuclphysb.2013.07.012; arXiv:1303.5210. https://doi.org/10.1016/j.nuclphysb.2013.07.012
5. J. Adler, A. Brandt, W. Janke, S. Shmulyian, Three-state Potts antiferromagnet on the triangular lattice, J. Phys. A: Math. Gen. 28 (1995) 5117-5129. DOI: 10.1088/0305-4470/28/18/005. https://doi.org/10.1088/0305-4470/28/18/005
6. R. J. Baxter, H. N. V. Temperley, S. E. Ashley, Triangular Potts model at its transition temperature, and related models, Proc. R. Soc. Lond. A 358 (1978) 535-559. DOI: 10.1098/rspa.1978.0026. https://doi.org/10.1098/rspa.1978.0026
7. R. J. Baxter, q colourings of the triangular lattice, J. Phys. A: Math. Gen. 19 (1986) 2821-2839. DOI: 10.1088/0305-4470/19/14/019. https://doi.org/10.1088/0305-4470/19/14/019
8. R. J. Baxter, Chromatic polynomials of large triangular lattices, J. Phys. A: Math. Gen. 20 (1987) 5241-5261. DOI: 10.1088/0305-4470/20/15/037. https://doi.org/10.1088/0305-4470/20/15/037
9. Jesper Lykke Jacobsen, Kay Joerg Wiese, Making complex CFTs real: The two-dimensional Potts model for Q>4 and complex Q, arXiv:2606.18125 (2026). https://arxiv.org/abs/2606.18125
10. Jesus Salas, Phase diagram for the bisected-hexagonal-lattice five-state Potts antiferromagnet, Phys. Rev. E 102 (2020) 032124. DOI: 10.1103/PhysRevE.102.032124; arXiv:2006.04866. https://doi.org/10.1103/PhysRevE.102.032124
11. Augustin Lafay, Azat M. Gainutdinov, Jesper Lykke Jacobsen, U_q(sl_n) web models and Z_n spin interfaces, J. Stat. Mech. (2021) 053104. DOI: 10.1088/1742-5468/abfcb3; arXiv:2101.00282. https://doi.org/10.1088/1742-5468/abfcb3
12. Mireille Bousquet-Mélou, Hadrien Notarantonio, The 3-state Potts model on planar triangulations: explicit algebraic solution, arXiv:2510.08414 (2025). https://arxiv.org/abs/2510.08414
13. G. M. Sommers, B. Placke, R. Moessner, S. L. Sondhi, From hard spheres to hard-core spins, Phys. Rev. B 103 (2021) 104407. DOI: 10.1103/PhysRevB.103.104407; arXiv:2010.04714. https://doi.org/10.1103/PhysRevB.103.104407