Easy-axis antiferromagnets on triangular and kagome lattices develop three-sublattice order at low temperature, and this order melts either in two steps, through an intermediate phase with power-law (quasi-long-range) three-sublattice order, or through a pair of transitions in the three-state Potts and Ising universality classes. A 2015 Physical Review Letters study by Damle showed that a coarse-grained theory keeping the uniform easy-axis magnetization mode coupled to the clock-like phase of the three-sublattice order parameter generically connects these two melting routes at a single multicritical point M, with the c-theorem bounding its central charge to 1<cM<3/21<c_M<3/2 and Monte Carlo estimates of the anomalous dimensions at one point in parameter space. The universal conformal field theory of M has not been identified: later work identified a different multicritical point of the same family, where two-step melting gives way to weakly first-order melting, as the Z6\mathbb{Z}_6 parafermion CFT with c=5/4c=5/4, but its exponents differ sharply from those measured at M. The open problem is to determine the universal critical theory of M itself: its exact central charge, its operator content, and the scaling dimensions of the operators representing eipθe^{ip\theta} (p=1,2,3p=1,2,3) and τ\tau that give a first-principles account of - or a quantified correction of - the published Monte Carlo exponent estimates.

Background

In an easy-axis antiferromagnet, the magnetic moments (spins) pay an energy cost for pointing anywhere except along one crystalline axis. On the triangular and kagome lattices, the three sites of each elementary triangle cannot all satisfy their antiferromagnetic bonds, so weak further-neighbour couplings or quantum fluctuations select, at low temperature, 'three-sublattice order': the spins distinguish the three sublattices of the underlying triangular Bravais lattice, either in a ferrimagnetic pattern with a net easy-axis moment (the pattern of nearly half-filled monolayers of adsorbed gases on triangular substrates and of artificial 'kagome ice' networks of magnetic wires) or in an antiferromagnetic pattern with no net moment (as in some S=1S=1 triangular-lattice magnets).

The thermal melting of this order has an unusual feature. The complex three-sublattice order parameter ψ=∣ψ∣eiθ\psi=|\psi|e^{i\theta} has, by symmetry, a six-fold anisotropy favouring six values of the phase θ\theta, so the Landau theory of the ordering transition is a generalized six-state clock model: a planar spin whose phase sees a six-well anisotropy. Two-dimensional clock models with five or more states do not melt at a single transition. They possess an intermediate 'floating' phase with quasi-long-range order, in which correlations decay as powers, ⟨eiθ(r⃗)e−iθ(0)⟩∼r−η\langle e^{i\theta(\vec r)}e^{-i\theta(0)}\rangle\sim r^{-\eta}, with a temperature-dependent exponent; for the six-state case η(T)∈(1/9,1/4)\eta(T)\in(1/9,1/4), the two ends marking where the six-fold anisotropy and where vortices become relevant (the latter a Kosterlitz-Thouless, i.e. vortex-unbinding, transition) [Jose-Kadanoff-Kirkpatrick-Nelson 1977; Dorey et al. 1999]. The same symmetries alternatively allow the order to melt in two consecutive continuous transitions: a three-state Potts transition (the Potts model generalizes the Ising model to spins taking three values; its critical point forms a standard universality class) losing the three-sublattice modulation, followed by an Ising transition losing the residual easy-axis magnetization.

In the standard Landau treatment the uniform easy-axis magnetization is a slave of the clock phase. Damle (2015) observed that this is not correct in general: the direction of the local uniform magnetization is correlated with, but not tied to, the phase of ψ\psi. Keeping it as an independent Ising variable τr⃗=±1\tau_{\vec r}=\pm 1 gives the coarse-grained theory

Heff=−Jxy∑⟨r⃗r⃗′⟩cos⁡(θr⃗−θr⃗′)−h6∑r⃗cos⁡(6θr⃗)−JIsing∑⟨r⃗r⃗′⟩τr⃗τr⃗′−Jθτ∑r⃗τr⃗cos⁡(3θr⃗)H_{\mathrm{eff}} = -J_{xy}\sum_{\langle \vec r\vec r'\rangle}\cos(\theta_{\vec r}-\theta_{\vec r'}) - h_6\sum_{\vec r}\cos(6\theta_{\vec r}) - J_{\mathrm{Ising}}\sum_{\langle \vec r\vec r'\rangle}\tau_{\vec r}\tau_{\vec r'} - J_{\theta\tau}\sum_{\vec r}\tau_{\vec r}\cos(3\theta_{\vec r})

on a coarse-grained triangular lattice. Its zero-field phase diagram in the temperature-JIsingJ_{\mathrm{Ising}} plane connects the two melting routes: at small JIsingJ_{\mathrm{Ising}}, clock-like two-step melting through the floating phase; at large JIsingJ_{\mathrm{Ising}}, the Ising-plus-Potts pair; in between, a single multicritical point M at which the two boundaries of the floating phase meet the Ising and the three-state Potts critical lines. The generalized clock model by itself cannot produce this four-line meeting - this is precisely why M is a new critical phenomenon.

At a two-dimensional critical point the long-distance physics is described by a conformal field theory (CFT), characterized by a central charge cc that counts the effective critical degrees of freedom; Zamolodchikov's c-theorem states that cc cannot increase along renormalization-group flows. Since M is reached from the decoupled combination of an Ising CFT (c=1/2c=1/2) and a floating free-compact-boson theory (c=1c=1) at its vortex end, and must itself flow to the c=1c=1 floating theory, the c-theorem bounds 1<cM<3/21<c_M<3/2. Monte Carlo simulations at one parameter set (Jxy=h6=1.0J_{xy}=h_6=1.0, Jθτ=0.25J_{\theta\tau}=0.25) located M and estimated the anomalous dimensions ηθ≈0.258(5)\eta_\theta\approx 0.258(5), ηθ2≈0.353(6)\eta_{\theta^2}\approx 0.353(6), and ητ=ηθ3≈0.201(20)\eta_\tau=\eta_{\theta^3}\approx 0.201(20) (uncertainties in the last quoted digit; the last equality follows from symmetry). These match no nearby known critical behaviour: in the floating phase ηpθ=p2η\eta_{p\theta}=p^2\eta, on the Potts line ηθ=ηθ2=4/15\eta_\theta=\eta_{\theta^2}=4/15, and on the Ising line ητ=1/4\eta_\tau=1/4. The same theory predicts a divergent uniform easy-axis susceptibility χ(B)∼∣B∣−(4−18η)/(4−9η)\chi(B)\sim|B|^{-(4-18\eta)/(4-9\eta)} through much of the floating phase, since confirmed numerically in microscopic models.

Two adjacent multicritical points of the same family have been identified since, and both are different from M. First, in the generalized six-state clock model - the limit in which the magnetization is slaved to the clock phase - the floating phase pinches off against a direct first-order transition at a self-dual multicritical point controlled by the Z6\mathbb{Z}_6 parafermion CFT constructed by Zamolodchikov and Fateev (a CFT with an extended algebra of fractional-spin 'parafermion' currents, generalizing the three-state Potts CFT; in the terminology of the 2021 follow-up it also admits a description as the supersymmetric minimal model SM6_6), with c=5/4c=5/4 and anomalous dimensions 5/245/24, 1/31/3, 3/83/8 [Zamolodchikov-Fateev 1985; Dorey et al. 1999; Rakala et al. 2021; Dagnino-Szabo 2026]. Second, in a generalized six-state clock chain interpolating towards the Potts model, an exactly decoupled Ising ⊗\otimes three-state-Potts multicritical point (c=13/10c=13/10) appears [Dagnino-Szabo 2026]. Neither matches the exponents measured at M - the Z2\mathbb{Z}_2-odd sector would have η=3/8\eta=3/8 in the parafermion theory and η=1/4\eta=1/4 in the decoupled theory, versus 0.201(20)0.201(20) at M - and neither involves the independent magnetization mode whose fluctuations define M.

Problem Statement

Determine the universal critical theory FM\mathcal{F}_M of the multicritical point M of the coarse-grained theory

Heff=−Jxy∑⟨r⃗r⃗′⟩cos⁡(θr⃗−θr⃗′)−h6∑r⃗cos⁡(6θr⃗)−JIsing∑⟨r⃗r⃗′⟩τr⃗τr⃗′−Jθτ∑r⃗τr⃗cos⁡(3θr⃗)H_{\mathrm{eff}} = -J_{xy}\sum_{\langle \vec r\vec r'\rangle}\cos(\theta_{\vec r}-\theta_{\vec r'}) - h_6\sum_{\vec r}\cos(6\theta_{\vec r}) - J_{\mathrm{Ising}}\sum_{\langle \vec r\vec r'\rangle}\tau_{\vec r}\tau_{\vec r'} - J_{\theta\tau}\sum_{\vec r}\tau_{\vec r}\cos(3\theta_{\vec r})

(on a coarse-grained triangular lattice, with τ=±1\tau=\pm 1 the uniform easy-axis magnetization mode and θ\theta the phase of the three-sublattice order parameter), which describes the thermal melting of three-sublattice order in easy-axis antiferromagnets on triangular and kagome lattices. Specifically, at M - the point in the zero-field phase diagram at which the two boundaries of the intermediate power-law (floating) phase meet the Ising critical line (loss of uniform easy-axis magnetization) and the three-state Potts critical line (loss of three-sublattice order) - an answer must establish:

(1) the exact central charge cMc_M (currently only bounded, 1<cM<3/21<c_M<3/2, by the c-theorem) and the operator content of FM\mathcal{F}_M, with the transformation properties of its scaling operators under the S3×Z2S_3\times\mathbb{Z}_2 symmetry of HeffH_{\mathrm{eff}};

(2) the scaling dimensions of the operators representing eipθe^{ip\theta} (p=1,2,3p=1,2,3) and τ\tau, so as to give a first-principles account of - or a quantified correction of - the published estimates ηθ≈0.258(5)\eta_\theta\approx 0.258(5), ηθ2≈0.353(6)\eta_{\theta^2}\approx 0.353(6), ητ=ηθ3≈0.201(20)\eta_\tau=\eta_{\theta^3}\approx 0.201(20).

The task is thus to identify FM\mathcal{F}_M - a specific conformal field theory, or a controlled construction of the new interacting fixed point - together with the universal data of items (1) and (2). Experimental realization of M in artificial kagome ice, S=1S=1 triangular magnets, or adsorbed-gas monolayers is a recorded follow-on application of the answer, not part of the completion boundary.

References

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