The spin-1 (dilute) Baxter-Wu model on the triangular lattice adds annealed vacancies and a crystal field to the exactly solved three-spin-interaction Baxter-Wu model, producing a phase diagram with a continuous transition line that crosses over to a first-order line. Whether this crossover is mediated by an isolated pentacritical point, and whether the continuous segment retains the 4-state Potts universality of the pure model, have been debated since 1981. The most recent combined transfer-matrix and Monte Carlo study finds no evidence for an isolated multicritical point, but instead an extended crossover region in which the effective scaling dimensions drift monotonically away from the 4-state Potts values while the central charge stays near c=1c=1; it cannot, however, distinguish genuinely continuously varying critical exponents from a slow renormalization-group crossover. The open problem is to settle this discrimination - by asymptotically controlled simulations or by a field-theoretical/renormalization-group construction - and thereby fix the multicritical or crossover structure of the model's transition line.

Background

The Baxter-Wu model is a triangular-lattice spin system in which the basic interaction is a product of three Ising spins, one on each corner of every elementary triangle. It was exactly solved by Baxter and Wu in 1973; its ground state is fourfold degenerate (one ferromagnetic state plus three ferrimagnetic states distinguished by which sublattice carries the minority spins), and its critical behavior is that of the four-state Potts model: it is described by a conformal field theory (the continuum description of a two-dimensional critical point, characterized by a central charge cc and scaling dimensions xαx_\alpha that determine the critical exponents) with c=1c=1 and correlation-length exponent ν=2/3\nu=2/3, but - unlike the four-state Potts model itself - without logarithmic corrections to scaling (subleading factors of ln⁡L\ln L that contaminate finite-size data).

The dilute, or spin-1, Baxter-Wu model introduces a crystal field (a single-ion anisotropy) Δ\Delta and allows each spin to take the values σi∈{−1,0,1}\sigma_i\in\{-1,0,1\}, an occupied spin (σi2=1\sigma_i^2=1) costing crystal-field energy Δ\Delta, so that a vacancy (σi=0\sigma_i=0) has zero crystal-field energy:

H=−J∑⟨ijk⟩σiσjσk+Δ∑iσi2.\mathcal{H}=-J\sum_{\langle ijk\rangle}\sigma_i\sigma_j\sigma_k+\Delta\sum_i\sigma_i^2 .

This is the Baxter-Wu analogue of the Blume-Capel model, the spin-1 Ising model with a crystal field whose second-order (Ising-universality) line meets a first-order line at a tricritical point. Unlike its pure limit, the spin-1 Baxter-Wu model has no exact solution. Its (Δ,T)(\Delta,T) phase diagram contains ordered phases separated from the paramagnetic phase by a transition line with a second-order segment at small Δ\Delta and a first-order segment at low temperature; along the first-order segment three ferrimagnetic phases and one ferromagnetic phase coexist with the paramagnetic phase (a quintuple line), long believed to terminate at a pentacritical point where the five coexisting phases merge, and at T=0T=0 all five phases coexist at Δ/J=2\Delta/J=2.

The literature on this crossover is conflicting. An early transfer-matrix finite-size-scaling analysis (the transfer matrix relates the eigenvalues of an infinite strip of width MM with periodic boundary conditions to the thermodynamics and conformal data of the infinite system) concluded that a continuous transition occurs only in the limit Δ→−∞\Delta\to-\infty. Later work established a multicritical point at finite Δ\Delta but disagreed sharply on its location - conformal-invariance transfer-matrix analysis gave (Δpp,Tpp)≈(0.8902,1.4)(\Delta_{\rm pp},T_{\rm pp})\approx(0.8902,1.4), whereas entropic (Wang-Landau-type) sampling gave (1.68288(62),0.98030(10))(1.68288(62),0.98030(10)) - and on the universality along the second-order segment, with claims of continuously varying critical exponents, of 4-state Potts criticality, and even of a mixed-order transition (combining first-order and second-order features). Combined Wang-Landau and multicanonical simulations (in which a flat histogram in the crystal-field energy is sampled to overcome free-energy barriers), together with a study of the zeros of the energy probability distribution (a finite-size proxy for Fisher's partition-function zeros, from which critical temperatures and 1/ν1/\nu can be extracted), supported 4-state Potts universality at least for Δ<0\Delta<0, attributing residual deviations to strong finite-size effects, and formulated the open agenda: determine the pentacritical point accurately and perform a comprehensive finite-size-scaling study around it using a hybrid cluster/heat-bath update (a combination of cluster moves that flip correlated domains of spins and local heat-bath single-spin updates), including possible multiplicative or additive logarithmic corrections and field mixing (the separation of order-parameter-like from energy-like scaling fields near a multicritical point).

The decisive follow-up combined transfer matrices up to strip width M=21M=21, Monte Carlo simulations up to L=288L=288 at Δ=0.5\Delta=0.5, and multicanonical simulations for the first-order regime. It finds no numerical evidence for an isolated pentacritical point, but rather an extended crossover region with onset of first-order behavior near Δ≈0.85\Delta\approx0.85 (T≈1.42T\approx1.42); the central charge remains close to c=1c=1 while the effective scaling dimensions drift monotonically away from the 4-state Potts values (νeff\nu_{\rm eff} decreasing from ≈0.667\approx0.667 at Δ=−10\Delta=-10 to ≈0.60\approx0.60 at Δ=1\Delta=1; Monte Carlo at Δ=0.5\Delta=0.5: η=0.226(4)\eta=0.226(4), ν=0.627(3)\nu=0.627(3)); and the first-order regime is firmly established through phase coexistence, finite interfacial tension, and volume scaling of the maxima of the specific heat and susceptibility. This pattern resembles the Ashkin-Teller and eight-vertex models (two other exactly solved two-dimensional models whose c=1c=1 critical lines carry continuously varying scaling dimensions), but for the dilute Baxter-Wu model no field-theoretical description is available, and the data cannot exclude a slow renormalization-group crossover whose ultimate asymptotic behavior is 4-state Potts criticality.

Problem Statement

For the spin-1 (dilute) Baxter-Wu model on the triangular lattice with Hamiltonian

H=−J∑⟨ijk⟩σiσjσk+Δ∑iσi2,σi∈{−1,0,1},\mathcal{H}=-J\sum_{\langle ijk\rangle}\sigma_i\sigma_j\sigma_k+\Delta\sum_i\sigma_i^2,\qquad\sigma_i\in\{-1,0,1\},

determine the asymptotic renormalization-group structure of the crossover between the continuous and first-order segments of the transition line in the (Δ,T)(\Delta,T) plane. The single overarching objective is to decide which of two competing scenarios is realized and to characterize it: (a) an isolated pentacritical (multicritical) point terminates the quintuple coexistence line, in which case its location, scaling behavior, and possible logarithmic corrections must be determined; or (b) there is no isolated multicritical point but an extended crossover region, in which case the asymptotic criticality along the continuous line must be determined - either genuinely continuously varying critical exponents (universality breaking relative to the 4-state Potts class of the pure model) or a slow crossover whose ultimate asymptotic behavior is 4-state Potts criticality. A complete answer must be asymptotically controlled: either (i) numerical evidence - simulations and/or transfer-matrix analyses at system sizes and strip widths, with a controlled correction-to-scaling structure, sufficient to discriminate the scenarios beyond the systematic uncertainties of the extrapolations and to fix the effective exponents and their Δ\Delta-dependence together with the location and nature (or absence) of any multicritical point; or (ii) an analytical construction - an exact solution, a Coulomb-gas/conformal-field-theory treatment (mapping the critical lattice model onto a free Gaussian field whose coupling parametrizes the critical line, the mechanism underlying the varying exponents of the Ashkin-Teller model), or a renormalization-group analysis - that identifies the scaling fields, reproduces the observed phase-boundary topology and exponent flow, and settles the same discrimination.

References

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