Exact solution and exact critical temperature of the spin-one Ising model on the honeycomb lattice

The spin-one Ising model on the honeycomb lattice — spins $\sigma_i\in\{-1,0,+1\}$ with pure bilinear nearest-neighbor exchange $H=J\sum_{\langle ij\rangle}(1-\sigma_i\sigma_j)$, equivalent to the Blume–Capel model at zero crystal field — is the simplest two-dimensional lattice spin model with integer spin that has never been solved exactly: neither its free energy nor even its critical temperature is known in closed form, in contrast to the exactly solved spin-one-half honeycomb Ising model. The state of the art is exact enumeration of the integer density of states on $L\times2L$ honeycomb lattices up to $L=14$, from which the ferromagnetic critical point $y_c=\mathrm{e}^{-\beta_cJ}=0.42191(1)$ (i.e. $\beta_cJ\approx0.8630$) is inferred by finite-size extrapolation of partition-function zeros, consistent with the 1973 low-temperature-series estimate $0.4217(10)$. The problem is to obtain an exact solution of the model — a closed-form per-site free energy from which the exact critical point and complex-temperature singularity structure follow — or, at minimum, a rigorous exact determination of the critical temperature. The problem is open: all known exact results for honeycomb spin-one models hold only on special temperature-dependent-coupling subspaces of the generalized (bilinear-plus-biquadratic-plus-crystal-field) Hamiltonian that provably do not contain the pure bilinear model.

Background

The Ising model places a classical spin on each site of a lattice, with nearest-neighbor pairs coupled by an exchange interaction. Its spin-one-half version ($\sigma_i=\pm1$) is the paradigm of an exactly solved phase transition: Onsager (1944) obtained the free energy of the square-lattice model at zero magnetic field, and Houtappel (1950) extended the exact solution to the honeycomb lattice, whose critical coupling is $\beta_cJ=\frac{1}{2}\ln(2+\sqrt{3})$ in the convention $H=-J\sum_{\langle ij\rangle}\sigma_i\sigma_j$ — equivalently $y_c=\sqrt{2-\sqrt{3}}$ in the low-temperature variable $y=\mathrm{e}^{-\beta J}$ used below — with the spontaneous magnetization computed by Naya (1954). These solutions rest on a free-fermion structure — a representation of the partition function as a determinant describing noninteracting ("free") fermions, the Kac–Ward/Onsager algebra — of the $2\times2$ bond-weight matrices, together with planar duality.

The spin-one Ising model upgrades the local spin to $\sigma_i\in\{-1,0,+1\}$ while keeping only bilinear exchange, in the convention $H=J\sum_{\langle ij\rangle}(1-\sigma_i\sigma_j)$ with $J>0$, so bond energies are $0$, $J$, or $2J$. Because the $\sigma=0$ state costs no single-site energy, this is exactly the Blume–Capel model (the spin-one Ising model with a crystal-field term $D\sum_i\sigma_i^2$) at $D=0$: the $\sigma=0$ sites act as annealed vacancies, meaning vacant sites in thermal equilibrium with the occupied ones rather than frozen quenched defects. It is the minimal integer-spin lattice model with annealed dilution, and it orders ferromagnetically at a finite temperature on the honeycomb lattice.

In two dimensions the spin-one Ising model has never been solved exactly on any standard lattice — a standing contrast with the spin-one-half case. The obstruction is structural: the $3\times3$ bond-weight matrix of a $\{-1,0,+1\}$ spin is not reducible to the free-fermion form underlying all spin-one-half solutions, and tracing out one honeycomb sublattice (the honeycomb lattice is bipartite with coordination number 3) produces a spin-one model with three-spin interactions on the triangular lattice rather than a solvable model. No duality that pins the critical point is known either.

What exact results do exist concern the generalized honeycomb spin-one Hamiltonian that adds a biquadratic exchange term $K\sum_{\langle ij\rangle}\sigma_i^2\sigma_j^2$ (coupling the squared spins of neighboring sites) and a single-ion crystal-field term $\Delta\sum_i\sigma_i^2$ (distinguishing the energy of the $\sigma=0$ state from that of the $\sigma=\pm1$ states) — the Blume–Emery–Griffiths form. Horiguchi (1986) discovered a solvable case, clarified by F. Y. Wu (1986) and worked out by X. N. Wu and F. Y. Wu (1988): under the constraint $K=-\ln\cosh J$ on the dimensionless couplings, the model maps exactly onto the spin-one-half Ising ferromagnet, yielding its free energy and spontaneous magnetization. Urumov (1987) obtained the exact magnetization along a line in the coupling–temperature plane, and Kolesík and Šamaj (1992) systematized such transformations, recovering the known and finding new solvable subspaces of the general model. Crucially, these solvable loci are temperature-dependent coupling relations: at the pure bilinear point $K=0$ the constraint $K=-\ln\cosh J$ collapses to the trivial $\beta J=0$, so none of them solves the model at hand. Rosengren and Häggkvist (1989) claimed a rigorous solution of a two-dimensional Blume–Emery–Griffiths model, which F. Y. Wu (1990) disputed; no accepted exact solution of the pure model resulted.

The numerical and rigorous state of the art for the pure model is as follows. Fox and Guttmann (1973) derived low-temperature series for the $S=1$ Ising model on several two- and three-dimensional lattices, estimating the honeycomb critical point $y_c=0.4217(10)$ in the low-temperature variable $y=\mathrm{e}^{-\beta J}$ and finding critical exponents consistent with the spin-one-half values. Warnaar, Nienhuis and Seaton (1992) showed that the solvable restricted solid-on-solid dilute-$A_3$ model (an exactly solvable lattice height model of the $A_3$ Dynkin-diagram family) is related to the two-dimensional critical spin-one Ising model in a magnetic field, fixing the critical-isotherm exponent $\delta=15$; the exponents are thus settled by integrable and conformal-field-theory relatives, but the critical temperature is not. Braga, Ferreira and Sá Barreto (1994) derived rigorous upper bounds on the critical temperature of the two-dimensional spin-one Blume–Emery–Griffiths model from correlation inequalities. Most recently, Kim (2025) enumerated the exact integer density of states $\Omega(E)$ of the pure model on $L\times2L$ honeycomb lattices ($N_s=2L^2$ sites, $N_b=3L^2-2L$ bonds) up to $L=14$, constructed the exact partition-function polynomials $Z(y)=\sum_E\Omega(E)y^E$, and studied the Fisher zeros — the zeros of $Z$ in the complex temperature plane, whose accumulation in the thermodynamic limit locates the singularities of thermodynamic functions. Bulirsch–Stoer extrapolation (a rational-function finite-size extrapolation) of the zero closest to the positive real axis gives $y_c=0.42191(1)$, i.e. $\beta_cJ\approx0.8630$, with thermal scaling exponent $y_t=1$ as expected for the two-dimensional Ising universality class, and locates inferred complex-temperature singularities at $y_1=-0.398(6)$, $y_2\approx-0.5+0.2i$, $y_3\approx0.563+0.826i$. All of these are inferences from exact finite-lattice data; nothing about the infinite honeycomb lattice is established exactly.

Problem Statement

Determine, exactly, the thermodynamics of the spin-one Ising model on the honeycomb lattice: the ferromagnet with Hamiltonian $H=J\sum_{\langle i,j\rangle}(1-\sigma_i\sigma_j)$, where each spin takes values $\sigma_i\in\{-1,0,+1\}$ and $J>0$, on the honeycomb lattice in the thermodynamic limit at zero magnetic field. The complete objective is an exact solution — a closed-form expression for the per-site free energy (equivalently the partition function per site) valid at all temperatures, from which the exact ferromagnetic critical point $y_c=\mathrm{e}^{-\beta_cJ}$, the exact critical behavior, and the exact complex-temperature (Fisher-zero) singularity structure all follow; the antiferromagnetic critical point, in the same variable $y=\mathrm{e}^{-\beta J}$ with antiferromagnetic coupling $J<0$ (so $y>1$), is $1/y_c$ by the bipartite symmetry of the honeycomb lattice. The minimal standalone objective is the exact critical temperature, currently bracketed numerically at $y_c=0.42191(1)$ (low-temperature series: $0.4217(10)$) but not known in closed form. These two deliverables are nested — the critical point is a component of the full solution — and jointly serve one objective: replacing finite-size inference by exact knowledge of this model. The objective is an exact, closed-form determination by rigorous derivation: a high-precision numerical enclosure, however tight, is not a solution, and neither are exact results for the generalized model on special temperature-dependent coupling loci (such as $K=-\ln\cosh J$), which are adjacent but different problems.

References

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