In triangular-lattice Ising antiferromagnets with antiferromagnetic nearest-neighbour coupling J1>0J_1>0 and ferromagnetic second- and third-neighbour couplings J2,J3<0J_2,J_3<0, the low-temperature ferrimagnetic three-sublattice order can melt in two steps through an intermediate power-law ordered (critical) phase. Rakala, Desai, Shivam and Damle (arXiv:2109.03178, 2021) argued that this critical phase pinches off along a one-dimensional multicritical locus in the universality class of the Z6Z_6 parafermion conformal field theory with central charge c=5/4c=5/4 (with a claimed alternate description as the supersymmetric minimal model SM6SM_6), with weakly first-order melting on the far side. That identification rests only on measured correlation-function exponents along two parameter cuts, and the complete characterization remains open: the full multicritical locus in (J1,J2,J3)(J_1,J_2,J_3) space is only approximately located, no direct conformal-data determination (central charge, operator content beyond the measured exponents) exists, the weakly first-order regime is unquantified (no latent-heat or energy-histogram finite-size analysis), and the extended pseudocritical window has only an untested interpretation (a two-step melting region too narrow in temperature to resolve at the accessible system sizes). The problem is to complete this characterization for this classical frustrated magnet: map the whole multicritical locus, determine the multicritical conformal data (thereby testing the Z6Z_6-parafermion and SM6SM_6 identifications), and quantitatively characterize the adjacent weakly first-order regime and pseudocritical window.

Background

The triangular-lattice Ising antiferromagnet places two-state spins σ=±1\sigma=\pm 1 on the sites of the triangular lattice with antiferromagnetic (alignment-opposing) nearest-neighbour coupling J1>0J_1>0; because the three bonds of an elementary triangle cannot be satisfied simultaneously, the system is geometrically frustrated. When ferromagnetic second- and third-neighbour couplings J2,J3<0J_2,J_3<0 are added and are strong enough, the ground state develops three-sublattice ferrimagnetic order: the lattice divides into three interpenetrating sublattices a,b,ca,b,c whose spins form a fixed two-up/one-down pattern, so each triangle carries a net moment. A convenient order parameter is the complex three-sublattice combination ψ(r⃗)=σa(r⃗)+σb(r⃗)e2πi/3+σc(r⃗)e4πi/3\psi(\vec r)=\sigma_a(\vec r)+\sigma_b(\vec r)e^{2\pi i/3}+\sigma_c(\vec r)e^{4\pi i/3}, defined on the up-pointing triangle whose aa-sublattice site is at r⃗\vec r; ψ3\psi^3 is the local magnetization density. Together with the Ising (up/down) choice, ψ\psi effectively takes six values, so its phase θ\theta behaves as a six-state clock variable. The triangular Ising antiferromagnet with competing further-neighbour couplings has been studied since the early 1980s (Landau 1983).

The statistical mechanics of planar spins with qq-fold anisotropy is classical: José, Kadanoff, Kirkpatrick and Nelson (1977) showed that for q≥5q\ge 5 there is an intermediate critical (quasi-long-range-ordered) phase in which correlations decay as power laws with a temperature-dependent exponent, bounded by two Berezinskii-Kosterlitz-Thouless (BKT) transitions driven by vortex binding and unbinding; for q=6q=6 the exponent varies between 1/91/9 and 1/41/4. Damle (2015) predicted that easy-axis triangular- and kagome-lattice antiferromagnets melt their three-sublattice order in exactly this two-step manner, via an intermediate power-law phase with a divergent uniform susceptibility as a simple experimental signature; Heidarian and Damle (2018) confirmed the scenario for S=1S=1 easy-axis models, and Biswas and Damle (2018) studied the transverse-field quantum sibling. On the conformal-field-theory (CFT) side, Zamolodchikov and Fateev (1985) constructed self-dual critical points of ZNZ_N-symmetric statistical systems described by the ZNZ_N parafermion CFTs, where parafermions are nonlocal currents generalizing fermions; the central charge (the CFT's measure of effective degrees of freedom) is c=2(N−1)/(N+2)c=2(N-1)/(N+2), giving c=5/4c=5/4 for N=6N=6. The self-dual point of the six-state clock model, where ordered, disordered and critical phases meet, is expected to be described by this Z6Z_6 parafermion CFT; the phase diagrams of discrete Z6Z_6 spin models were studied numerically by Dorey, Provero, Tateo and Vinti (1999).

Rakala, Desai, Shivam and Damle (2021; unpublished arXiv preprint) studied the classical triangular-lattice Ising antiferromagnet with J1>0J_1>0 and J2,J3<0J_2,J_3<0 by large-scale Monte Carlo using dual-worm cluster algorithms (cluster Monte Carlo methods that build worm paths on the dual lattice and thereby tame the critical slowing down that afflicts frustrated Ising models). Working with the ratios X=J1/J3X=J_1/J_3, Y=J2/J3Y=J_2/J_3 (J3<0J_3<0), they found that the finite-temperature critical phase associated with two-step melting pinches off along a one-dimensional multicritical locus as (X,Y)→(−1,1)(X,Y)\to(-1,1), the corner of the ground-state three-sublattice region X<−1X<-1, Y>1Y>1; crossing this locus, melting becomes a single weakly first-order transition. Along their two detailed cuts (the line R≡−(J2+J3)/J1=2R\equiv-(J_2+J_3)/J_1=2, parameterized by κ≡−(J2−J3)/J1\kappa\equiv-(J_2-J_3)/J_1, and the line Y=1.05Y=1.05, parameterized by XX) they located the multicritical point approximately at κ≈0.1425\kappa\approx 0.1425 and X≈−1.365X\approx -1.365, respectively. Their evidence for the universality class is that the exponents ηψ\eta_\psi, ηϕ\eta_\phi, ησ\eta_\sigma of the correlation functions of ψ\psi, ψ2\psi^2, ψ3\psi^3 approach the Zamolodchikov-Fateev self-dual values at the multicritical point. On this basis they argued that the multicritical locus is in the Z6Z_6 parafermion CFT universality class with c=5/4c=5/4, which they state also admits an alternate description as the supersymmetric minimal CFT SM6SM_6 of the Friedan-Qiu-Shenker superconformal series, making the multicriticality a rare candidate for thermally accessible emergent supersymmetry, detectable in easy-axis triangular antiferromagnets at temperatures of order the exchange couplings. Between the two-step regime and the resolved weakly first-order regime they observed an extended pseudocritical window that looks continuous at accessible system sizes, and which they ascribed to a two-step melting region too narrow in temperature to resolve in their numerics.

What the source did not establish, and what no later work has supplied: the full multicritical locus was mapped only approximately along two representative cuts; the conformal data of the multicritical point beyond the three measured exponents (such as the central charge from finite-size scaling of the free energy, and the scaling-dimension spectrum including parafermionic operators) were never directly determined, so the identification rests on three measured exponents and on a Binder-cumulant analogy with the self-dual locus of the generalized six-state clock model; the weakly first-order transitions were characterized only through dips of Binder cumulants (dimensionless ratios of moments of an order-parameter or energy distribution, whose drift with system size signals the phase coexistence of a first-order transition; no latent-heat, energy-histogram, or finite-size-scaling quantification); and the interpretation of the pseudocritical window as an unresolvably narrow two-step melting region, though motivated by Landau-theory arguments, was not directly tested.

Problem Statement

For the classical triangular-lattice Ising antiferromagnet with antiferromagnetic nearest-neighbour coupling J1>0J_1>0 and ferromagnetic second- and third-neighbour couplings J2,J3<0J_2,J_3<0 (ferrimagnetic three-sublattice ground-state order for X=J1/J3<−1X=J_1/J_3<-1, Y=J2/J3>1Y=J_2/J_3>1), provide a complete characterization of the multicritical threshold at which the finite-temperature critical (power-law ordered) phase of two-step melting pinches off, specifically: (a) map the entire one-dimensional multicritical locus in the (X,Y)(X,Y) plane along which the critical phase terminates as (X,Y)→(−1,1)(X,Y)\to(-1,1), with quantified precision of its coordinates beyond the two representative cuts already studied (the R=2R=2 line, multicritical point at κ≈0.1425\kappa\approx 0.1425, and the Y=1.05Y=1.05 line, multicritical point at X≈−1.365X\approx -1.365); (b) determine the conformal data of the multicritical point directly - the central charge (testing the prediction c=5/4c=5/4) and the scaling-dimension spectrum, including parafermionic operator content, going beyond the previously measured exponents ηψ\eta_\psi, ηϕ\eta_\phi, ησ\eta_\sigma - and thereby test the claimed identification with the self-dual Z6Z_6 parafermion conformal field theory, including whether it also admits the supersymmetric SM6SM_6 description; and (c) quantitatively characterize the melting on the first-order side of the locus - latent heat, double-peak structure of energy histograms and its finite-size scaling, and the dependence of the strength of the first-order character on parameters - and test the source's interpretation of the intervening pseudocritical regime as a two-step melting region too narrow in temperature to resolve at the accessible system sizes. The source's outlook additionally invites identifying candidate experimental materials; that is an application of these results and lies outside this problem.

References

  1. G. Rakala, N. Desai, S. Shivam, K. Damle, "Melting of three-sublattice order in triangular lattice Ising antiferromagnets: Power-law order, Z6Z_6 parafermionic multicriticality, and weakly first order transitions", arXiv:2109.03178 (2021), DOI 10.48550/arXiv.2109.03178, https://arxiv.org/abs/2109.03178 (unpublished preprint)
  2. K. Damle, "Melting of three-sublattice order in easy-axis antiferromagnets on triangular and Kagome lattices", Phys. Rev. Lett. 115, 127204 (2015), DOI 10.1103/PhysRevLett.115.127204, https://arxiv.org/abs/1507.08393
  3. D. Heidarian, K. Damle, "Two-step melting of three-sublattice order in S=1S=1 easy-axis triangular lattice antiferromagnets", Eur. Phys. J. B 91, 202 (2018), DOI 10.1140/epjb/e2018-90353-8, https://arxiv.org/abs/1512.01346
  4. A. B. Zamolodchikov, V. A. Fateev, "Parafermionic currents in the two-dimensional conformal quantum field theory and self-dual critical points in Z(n) invariant statistical systems", Sov. Phys. JETP 62, 215-225 (1985) [original Zh. Eksp. Teor. Fiz. 89, 380-399 (1985)], https://inspirehep.net/literature/219875
  5. J. V. José, L. P. Kadanoff, S. Kirkpatrick, D. R. Nelson, "Renormalization, vortices, and symmetry-breaking perturbations in the two-dimensional planar model", Phys. Rev. B 16, 1217 (1977), DOI 10.1103/PhysRevB.16.1217, https://doi.org/10.1103/PhysRevB.16.1217
  6. D. P. Landau, "Critical and multicritical behavior in a triangular-lattice-gas Ising model: Repulsive nearest-neighbor and attractive next-nearest-neighbor coupling", Phys. Rev. B 27, 5604 (1983), DOI 10.1103/PhysRevB.27.5604, https://doi.org/10.1103/PhysRevB.27.5604
  7. P. Dorey, P. Provero, R. Tateo, S. Vinti, "On the phase diagram of the discrete Z6Z_6 spin models", J. Phys. A: Math. Gen. 32, L151-L158 (1999), DOI 10.1088/0305-4470/32/13/002, https://doi.org/10.1088/0305-4470/32/13/002
  8. D. Friedan, Z. Qiu, S. Shenker, "Superconformal invariance in two dimensions and the tricritical Ising model", Phys. Lett. B 151, 37 (1985), DOI 10.1016/0370-2693(85)90819-6, https://doi.org/10.1016/0370-2693(85)90819-6
  9. S. Biswas, K. Damle, "Singular ferromagnetic susceptibility of the transverse-field Ising antiferromagnet on the triangular lattice", Phys. Rev. B 97, 085114 (2018), DOI 10.1103/PhysRevB.97.085114, https://arxiv.org/abs/1603.06473
  10. G. Rakala, K. Damle, "Cluster algorithms for frustrated two-dimensional Ising antiferromagnets via dual worm constructions", Phys. Rev. E 96, 023304 (2017), DOI 10.1103/PhysRevE.96.023304, https://doi.org/10.1103/PhysRevE.96.023304
  11. A. Kouta Dagnino, A. Szabó, "Parafermionic and decoupled multicritical points in a frustrated Z6Z_6 clock chain", arXiv:2605.03001 (2026), https://arxiv.org/abs/2605.03001
  12. Y. Okabe, H. Otsuka, "Berezinskii-Kosterlitz-Thouless phase transitions of the antiferromagnetic Ising model with ferromagnetic next-nearest-neighbor interactions on the kagome lattice", Phys. Rev. B 113, 014441 (2026), DOI 10.1103/599c-896h, https://arxiv.org/abs/2602.07401
  13. M. Gen, H. Suwa, "Nematicity and fractional magnetization plateaus induced by spin-lattice coupling in the classical kagome-lattice Heisenberg antiferromagnet", Phys. Rev. B 105, 174424 (2022), DOI 10.1103/PhysRevB.105.174424, https://arxiv.org/abs/2202.09077
  14. S. Pujari, H. Nigam, "'Anticommuting' Z2Z_2 quantum spin liquids", Phys. Rev. B 113, 064402 (2026), DOI 10.1103/dhh4-8wkj, arXiv:2506.03866, https://arxiv.org/abs/2506.03866