The nearest-neighbor Ising model on the square lattice has an exact thermodynamic free energy only at zero magnetic field (Onsager, 1944); at any nonzero real field the problem has remained open for eighty years. Jha (2025) conjectured a closed-form expression for the free energy per site f(β,h)f(\beta,h) at real field hh — a logarithmic term plus a 4F3{}_4F_3 hypergeometric series in an auxiliary variable ξ\xi, reducing exactly to the known hypergeometric form of Onsager's result at h=0h=0 — and verified it against tensor-renormalization-group numerics to within about 2% across the explored parameters, while stating explicitly that it is neither exact nor proved. The open problem is to replace this purely empirical status with mathematics: settle the exactness of the conjectured formula on its domain, or obtain a controlled derivation or correction of it, in either case with explicit, provable error bounds on the discrepancy from the true f(β,h)f(\beta,h); the broader parent challenge is an exact closed form for f(β,h)f(\beta,h) at arbitrary real hh. The conjecture has no rigorous treatment with quantified error bounds, and the finite-field lattice problem remains unsolved.

Background

The Ising model is the archetypal lattice model of a ferromagnet: spins σi=±1\sigma_i=\pm 1 sit on the sites of a lattice, nearest-neighbor pairs interact ferromagnetically, and a uniform magnetic field couples to every spin. On the square lattice with coupling J=1J=1 the partition function is

Z=∑{σ}exp⁡(β∑⟨ij⟩σiσj+βh∑iσi),Z=\sum_{\{\sigma\}}\exp\Big(\beta\sum_{\langle ij\rangle}\sigma_i\sigma_j+\beta h\sum_i\sigma_i\Big),

where β\beta is the inverse temperature, hh the reduced field, and the central object of equilibrium thermodynamics is the free energy per site in the thermodynamic limit (infinite-volume limit), f(β,h)=−1βlim⁡N→∞1Nln⁡Zf(\beta,h)=-\frac{1}{\beta}\lim_{N\to\infty}\frac{1}{N}\ln Z, from which all bulk observables follow by differentiation. Onsager (1944) solved the zero-field case exactly — the first exact solution of a model exhibiting a phase transition — with a critical point at sinh⁡(2βc)=1\sinh(2\beta_c)=1 (βc≈0.4407\beta_c\approx 0.4407). His solution, and the equivalent combinatorial derivation of Kac and Ward (1952), exploit structure that exists only at h=0h=0: the row-to-row transfer matrix (the operator that appends one lattice row to the partition sum) is then reducible, via a Jordan–Wigner transformation mapping spins to free fermions, to a determinant. A uniform field destroys this free-fermion structure, and no exact closed form for f(β,h)f(\beta,h) at real h≠0h\neq0 is known on any regular two-dimensional lattice despite eighty years of effort; this is one of the celebrated open problems of equilibrium statistical mechanics.

What is known at nonzero field circumscribes the problem. Yang (1952) computed the spontaneous magnetization m(β)=lim⁡h→0+−∂f/∂hm(\beta)=\lim_{h\to0^+}-\partial f/\partial h, strictly positive below TcT_c, which makes h=0h=0 a line of first-order transitions for T<TcT<T_c: f(β,h)−f(β,0)∼−m(β)∣h∣f(\beta,h)-f(\beta,0)\sim-m(\beta)|h|. Lee and Yang (1952) proved that all zeros of ZZ in the variable z=e−2βhz=e^{-2\beta h} lie on the unit circle ∣z∣=1|z|=1 (the Lee–Yang circle theorem), so for real h≠0h\neq0 the free energy is analytic in hh — established rigorously by Lebowitz and Penrose (1968) — while the complex-hh plane carries the Yang–Lee edge singularities — the non-analytic endpoints of the zero distribution — at its boundary. Two special points are exactly solvable in closed form: z=1z=1 (h=0h=0, Onsager) and z=−1z=-1, the imaginary field h=iπ/(2β)h=i\pi/(2\beta), anticipated by Lee and Yang and derived by Merlini (1974). On the quantitative side, high-order series expansions exist — notably the zero-field susceptibility χ(β)=−∂2f/∂h2∣h=0\chi(\beta)=-\partial^2f/\partial h^2\big|_{h=0}, analyzed to hundreds of terms by Orrick, Nickel, Guttmann and Perk (2001) — and the complex-temperature and complex-field singular structure at nonzero field has been mapped by Matveev and Shrock (1996). In the scaling limit (T→TcT\to T_c, h→0h\to0) the continuum theory is exactly solvable: Zamolodchikov's (1989) integrable Ising field theory gives the mass spectrum and form-factor expansions of the free energy (systematic series built from the theory's exact multi-particle matrix elements), developed by Delfino and Mussardo (1995) and by Fonseca and Zamolodchikov (2001), who analyzed the analytic properties of the scaling free energy across the Yang–Lee and Langer branch cuts. These continuum results do not, however, give the lattice free energy at finite β\beta and hh.

The conjecture builds on a recent integral-free rewrite of the zero-field solution: f(β,0)=−1β(ln⁡(2cosh⁡2β)−κ2 4F3(1,1,32,32;2,2,2;16κ2))f(\beta,0)=-\frac{1}{\beta}\big(\ln(2\cosh2\beta)-\kappa^2\,{}_4F_3(1,1,\tfrac{3}{2},\tfrac{3}{2};2,2,2;16\kappa^2)\big) with κ=sinh⁡(2β)/2cosh⁡2(2β)\kappa=\sinh(2\beta)/2\cosh^2(2\beta) (Hucht, Grüneberg and Schmidt (2011); Viswanathan (2014)), where pFq{}_pF_q is the generalized hypergeometric series pFq(a;b;z)=∑n≥0(a1)n⋯(ap)n/[(b1)n⋯(bq)n n!] zn{}_pF_q(\mathbf a;\mathbf b;z)=\sum_{n\ge0}(a_1)_n\cdots(a_p)_n/[(b_1)_n\cdots(b_q)_n\,n!]\,z^n with Pochhammer symbols (a)n=a(a+1)⋯(a+n−1)(a)_n=a(a+1)\cdots(a+n-1) ((a)0=1(a)_0=1). Jha (2025) proposed, for real hh, the ansatz

f(β,h)=−1βln⁡(cosh⁡(βh)e2β+e−4β+sinh⁡2(βh)e4β)+ξ2β 4F3(1,1,32,32;2,2,2;16ξ2)f(\beta,h)=-\frac{1}{\beta}\ln\Big(\cosh(\beta h)e^{2\beta}+\sqrt{e^{-4\beta}+\sinh^2(\beta h)e^{4\beta}}\Big)+\frac{\xi^2}{\beta}\,{}_4F_3\Big(1,1,\tfrac{3}{2},\tfrac{3}{2};2,2,2;16\xi^2\Big)

with

ξ=cosh⁡(βh)e2β−e−4β+sinh⁡2(βh)e4β(cosh⁡(βh)e2β+e−4β+sinh⁡2(βh)e4β)2.\xi=\frac{\cosh(\beta h)e^{2\beta}-\sqrt{e^{-4\beta}+\sinh^2(\beta h)e^{4\beta}}}{\big(\cosh(\beta h)e^{2\beta}+\sqrt{e^{-4\beta}+\sinh^2(\beta h)e^{4\beta}}\big)^2}.

At h=0h=0 one has ξ=κ\xi=\kappa and the expression reduces exactly to the hypergeometric Onsager form; it has no singularity for real hh (consistent with the analyticity results above); and it was checked numerically by the tensor renormalization group (TRG), a deterministic coarse-graining algorithm acting on the tensor-network representation of ZZ whose cost grows only logarithmically with system size (Levin and Nave (2007)), with a maximum deviation of about 2% over the explored range of β\beta and real hh. The author states explicitly that the result is 'neither exact nor has a proof' and frames it as a step toward the long-standing problem.

Problem Statement

For the square-lattice nearest-neighbor Ising model with partition function Z=∑{σ}exp⁡(β∑⟨ij⟩σiσj+βh∑iσi)Z=\sum_{\{\sigma\}}\exp(\beta\sum_{\langle ij\rangle}\sigma_i\sigma_j+\beta h\sum_i\sigma_i), settle the mathematical status of the conjectured closed form for the thermodynamic free energy per site f(β,h)f(\beta,h) at real magnetic field — Eq. (10) of Jha (2025), reproduced in the Background — on its proposed domain 0≤z<10\le z<1 with z=e−2βhz=e^{-2\beta h}:

  1. Exactness. Settle rigorously whether the conjectured expression equals f(β,h)f(\beta,h) for all β≥0\beta\ge0 and real hh (or on the stated domain), with quantitative control of the verdict: an assertion of equality must be supported by the controlled derivation of branch 2, and an assertion of inexactness must be supported by explicit error bounds — explicit parameter values together with explicit bounds showing that the true f(β∗,h∗)f(\beta^*,h^*) disagrees with the conjectured value beyond those bounds. A qualitative refutation that does not supply quantified error bounds does not complete this branch.

  2. Control. Obtain a controlled derivation: a systematic, provably convergent approximation scheme, developed from the partition function, that yields the conjectured expression (or an explicitly corrected version of it) and quantifies its error by rigorous bounds.

The common objective is a mathematically controlled closed-form description of the finite-field free energy anchored on this conjecture: completion is achieved by a proof of exactness supported by a controlled derivation (branch 1), a refutation of exactness supported by explicit error bounds (branch 1), or a controlled derivation with provable error bounds (branch 2) — any of these replaces the present purely empirical ∼2%\sim2\% agreement with proved mathematics. The motivating parent challenge posed alongside them — an exact closed form for f(β,h)f(\beta,h) at arbitrary real nonzero hh on the square lattice — would supersede both branches and remains an open theoretical challenge.

References

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