The Baxter–Wu model — Ising spins on the triangular lattice coupled by pure three-spin interactions — is exactly solved in zero field, where it undergoes a second-order transition at Tc≈2.2692 J/kBT_c\approx 2.2692\,J/k_B in the four-state Potts universality class. A uniform longitudinal field HH destroys both the solvability and the simple symmetry structure, and the resulting phase diagram for H<0H<0 remains unsettled. Wang–Landau simulations by Velonakis and Martinos (Physica A 392, 2016 (2013)) mapped an apparently continuous critical line Hc(T)H_c(T) running from the zero-field critical point (0,Tc)(0,T_c) to the T=0T=0 point (−6J,0)(-6J,0), where the three ferrimagnetic ground states become degenerate with the fully polarized state, and reported H<0H<0 critical exponents (ν≈1\nu\approx 1, α≈1/2\alpha\approx 1/2, β≈3/4\beta\approx 3/4, γ≈0.4\gamma\approx 0.4) that differ discontinuously from the zero-field values and apparently violate the Rushbrooke, hyperscaling, and Fisher scaling relations. Four interlinked questions about this phase diagram remain open, the fourth of which is settled once the second is: (i) the nature of the T=0T=0, H=−6JH=-6J point; (ii) the robustness and universality status of the H<0H<0 exponents; (iii) whether Hc(T)H_c(T) contains an additional multicritical point or connects continuously to (0,Tc)(0,T_c); and (iv) the origin of the exponent discontinuity between H=0H=0 and H<0H<0.

Background

The Baxter–Wu (BW) model is a spin-1/21/2 lattice model on the triangular lattice whose Hamiltonian is H=−J∑Δσiσjσk−H∑iσi\mathcal{H}=-J\sum_{\Delta}\sigma_i\sigma_j\sigma_k-H\sum_i\sigma_i with σ=±1\sigma=\pm 1 and the first sum over all 2N2N elementary triangles of an NN-site lattice (each site belongs to six triangles). Baxter and Wu solved it exactly in zero field (Phys. Rev. Lett. 31, 1294 (1973)): the model has a second-order transition at kBTc/J≈2.269185k_BT_c/J\approx 2.269185, with four ground states — the all-up state and the three ferrimagnetic states in which one sublattice points up and two point down (net magnetization −1/3-1/3 per site, every triangle carrying an even number of down spins). The zero-field critical exponents coincide with those of the four-state Potts model: α=2/3\alpha=2/3, β=1/12\beta=1/12, γ=7/6\gamma=7/6, ν=2/3\nu=2/3, where α\alpha governs the heat capacity, β\beta the order parameter, γ\gamma the susceptibility, and ν\nu the correlation length. A universality class is the set of models sharing critical exponents, and the scaling relations (e.g. Rushbrooke α+2β+γ=2\alpha+2\beta+\gamma=2, hyperscaling 2−α=dν2-\alpha=d\nu with d=2d=2) link these exponents; a genuine violation of hyperscaling would signal physics outside the standard finite-temperature critical-point framework. Earlier studies of triplet-interaction Ising systems in a field (Wood and Griffiths, J. Phys. C 5, L253 (1972); J. Phys. C 7, 1417 (1974)) used mean-field and series methods on the more general pair-plus-triplet model and left the pure triplet model without an exact solution at nonzero field. In a uniform field the energetics are simple: for H>0H>0 the unique ground state is all-up, so no symmetry remains to break and no transition is expected; for −6J<H<0-6J<H<0 the three ferrimagnetic states are the degenerate ground states, related by permutations of the three sublattices — a residual discrete symmetry the uniform field preserves, since all three states carry the same magnetization −1/3-1/3; at H=−6JH=-6J the fully polarized all-down state becomes degenerate with them (ΔE=4JN+(2/3)HN=0\Delta E = 4JN+(2/3)HN=0), and for H<−6JH<-6J it is the unique ground state. Velonakis and Martinos (Physica A 392, 2016 (2013)) studied the H<0H<0 regime by Wang–Landau entropic sampling (a flat-histogram Monte Carlo method) on lattices up to L=54L=54, benchmarking zero-field results against the exact values before mapping a critical line Hc(T)H_c(T) from (0,Tc)(0,T_c) towards (−6J,0)(-6J,0) (e.g. Tc(−3.5)=1.883±0.007T_c(-3.5)=1.883\pm 0.007, Tc(−5)=0.85±0.02T_c(-5)=0.85\pm 0.02 in units of J/kBJ/k_B). Along this line they estimated ν≈1.00±0.02\nu\approx 1.00\pm 0.02, β≈0.75±0.05\beta\approx 0.75\pm 0.05, γ≈0.40±0.05\gamma\approx 0.40\pm 0.05, α≈0.50±0.01\alpha\approx 0.50\pm 0.01, noting that these values violate the standard exponent equalities (α+2β+γ≈2.4≠2\alpha+2\beta+\gamma\approx 2.4\neq 2; 2−α=1.5≠dν=22-\alpha=1.5\neq d\nu=2; Fisher's relation γ=(2−η)ν\gamma=(2-\eta)\nu — with η\eta the anomalous-dimension exponent governing the critical decay of the spin–spin correlation function — would force η≈1.6\eta\approx 1.6), that their own susceptibility-based diagnostics were inconsistent with specific-heat-based ones at several fields, and that susceptibility peaks were absent altogether for some HH values. The analogy with multicritical points in spin-1 Ising (Blume–Capel-type) models — where a crystal field acting on vacancies generates tricritical or higher multicritical points along the transition line — motivates the question of whether similar multicriticality or a mere crossover occurs on the field axis of the pure spin-1/21/2 model.

Problem Statement

Determine the phase structure and critical behavior of the two-dimensional pure Baxter–Wu model (triangular lattice, Hamiltonian H=−J∑Δσiσjσk−H∑iσi\mathcal{H}=-J\sum_{\Delta}\sigma_i\sigma_j\sigma_k-H\sum_i\sigma_i, J>0J>0) in a uniform longitudinal magnetic field H<0H<0. Four interlinked questions jointly constitute this phase-diagram determination: (i) the nature of the T=0T=0 point at H=−6JH=-6J — is it a genuine T=0T=0 critical point with its own scaling and universality properties, or a multicritical or degenerate (first-order) end point? (ii) the true critical behavior along the critical line Hc(T)H_c(T) for −6J<H<0-6J<H<0 — do the reported exponents ν≈1\nu\approx 1, α≈1/2\alpha\approx 1/2, β≈3/4\beta\approx 3/4, γ≈0.4\gamma\approx 0.4 and their apparent violations of the Rushbrooke, hyperscaling, and Fisher scaling relations survive more accurate simulations (Velonakis and Martinos suggest cluster algorithms and larger system sizes), or are they finite-size and algorithmic artifacts — and is the transition continuous at all, rather than weakly first-order or a sharp crossover? (iii) the topology of the critical line — in field–temperature coordinates (H,T)(H,T), does a single continuous line of critical points connect (0,Tc)(0,T_c) to (−6J,0)(-6J,0), or does an additional multicritical point occur along Hc(T)H_c(T) (analogous to the multicritical points on the crystal-field axes of Blume–Capel-type and spin-1 Baxter–Wu models), a question that, as Velonakis and Martinos note, likely requires a renormalization-group transformation tailored to the three-spin model? (iv) the origin and universal status of the discontinuity between the zero-field exponents (four-state-Potts values) and the H<0H<0 exponents — does it reflect distinct fixed points (a new universality class induced by the field) or finite-size and crossover artifacts? Item (iv) is not an independent requirement: it is settled by the resolution of (ii) — if the reported H<0H<0 exponents are finite-size or algorithmic artifacts then no exponent discontinuity remains to be explained, while if they survive the discontinuity reflects a distinct fixed point. A complete determination therefore requires settling (i), (ii), and (iii), with (iv) following from (ii); a partial answer to a proper subset does not resolve the task. The determination concerns the thermodynamic limit across the field range −6J<H<0-6J<H<0, including the behavior as H→0−H\to 0^- and as H→−6J+H\to -6J^+.

References

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