D-finiteness and exact closed-form representability of the three-dimensional Ising partition function: joint satisfiability of the six necessary conditions

The exact solution of the nearest-neighbor ferromagnetic Ising model on the simple cubic lattice in zero magnetic field has been open since Onsager solved the two-dimensional model in 1944. Viswanathan, Portillo, Raposo and da Luz (2022) consolidated six necessary conditions that any claimed exact expression for the thermodynamic-limit per-site partition function $Z(K_1,K_2,K_3)$ must satisfy, and identified what remains unresolved: whether $Z$ is D-finite (holonomic) or provably non-D-finite, given that the anomalous behavior of partially resummed anisotropic high-temperature series suggests non-D-finiteness while no rigorous proof exists in either direction; whether the six conditions are jointly satisfiable, that is, whether any exact analytic closed form consistent with all six exists at all; the mathematical mechanism by which a nonholonomic exact closed form could satisfy the analyticity and series-coefficient constraints while reproducing the anisotropic resummation anomalies; and the extension of the partial-resummation and D-finiteness diagnostic to other three-dimensional lattices. Subsequent literature and the adjacent rigorous literature leave none of these questions resolved: no proof of D-finiteness or non-D-finiteness of the 3D Ising partition function exists, claimed exact solutions continue to fail the necessary conditions, and the nearest rigorous results concern two-dimensional observables or block only specific three-dimensional solution routes. The problem remains open, with the representability and satisfiability questions as its core.

Background

The Ising model places spins $\sigma_i=\pm 1$ on the sites of a lattice with energy $H=-\sum_{\langle ij\rangle}J_i\sigma_i\sigma_j$ summed over nearest-neighbor pairs, and its partition function $Z=\sum_{\{\sigma\}}e^{-\beta H}$ encodes all its thermodynamics ($\beta$ is the inverse temperature). For the square lattice with isotropic ferromagnetic couplings, Onsager (1944) obtained the exact free energy in zero field, an achievement that fixed what "solving" a lattice model means; for the simple cubic lattice with nearest-neighbor ferromagnetic couplings in zero field, no exact expression is known after eighty years of effort, and the problem is often called the holy grail of statistical mechanics. The object of interest is the thermodynamic-limit per-site partition function $Z(K_1,K_2,K_3)=\lim_{N\to\infty}Z_N^{1/N}$, where $K_i=\beta J_i$ are the dimensionless reduced temperatures (also called couplings); $K_1$, $K_2$, $K_3$ need not be equal, which defines the anisotropic model.

Exact information about the unsolved three-dimensional (3D) model comes from series expansions. In the high-temperature variable $v=\tanh K$ (small $K$ means high temperature), the finite-lattice method, an algorithm that computes series coefficients exactly on finite periodic lattices up to a size limited only by computer memory, yields the expansion of $Z$ to high order; the same holds for the low-temperature variable $u=e^{-4K}$. Hansel, Maillard, Oitmaa and Velgakis (1987) combined these expansions with inversion relations, exact functional identities relating the partition function at transformed couplings to its value at the original ones, and found that partially resummed expansions of the anisotropic Ising partition function and susceptibility on the square and simple cubic lattices develop anomalous singularity structure. (The susceptibility, the response of the magnetization to an infinitesimal field, is the best-studied such series in two dimensions.)

The most natural candidate framework for the exact closed form sought here is the class of D-finite (or holonomic) functions: analytic functions that satisfy a linear differential equation with polynomial coefficients; for a function of several variables, D-finiteness means jointly satisfying a holonomic system of linear partial differential equations with polynomial coefficients, which is stronger than separate holonomy in each variable. This class contains the familiar special functions and is stable under the usual operations, so it is the minimal requirement many physicists would place on a usable exact solution; an exact closed form need not, however, belong to this class. A function possessing a natural boundary, a curve across which analytic continuation is impossible because singularities accumulate densely on it, cannot be D-finite. Guttmann and Enting (1996) introduced numerical diagnostics that indicate whether a statistical-mechanical generating function is "solvable" in the sense of being expressible in terms of D-finite functions, and gave compelling (numerical, not proof-level) evidence that the two-dimensional Ising susceptibility, self-avoiding-walk generating functions, and directed-animal generating functions lie in the unsolvable class. For the 3D Ising partition function itself no analogous result, in either direction, has ever been proved; even the weaker question of whether the two-dimensional Ising susceptibility is differentially algebraic (satisfies a polynomial differential equation, a far larger class than D-finite) remains open (Guttmann, Jensen, Maillard and Pantone, 2016).

Viswanathan, Portillo, Raposo and da Luz (2022) systematized the constraints on any future claim. Their six necessary (explicitly not sufficient) conditions for a claimed exact $Z(K_1,K_2,K_3)$ are: (1) in the thermodynamic limit the per-site partition function is independent of boundary conditions; (2) it must reduce to Onsager's two-dimensional solution when any one coupling $K_i$ is set to zero; (3) in the isotropic case $K_1=K_2=K_3=K$ it must be analytic for $0\le K<K_c$ and for $K_c<K<\infty$, with the numerically established critical coupling $K_c=0.221654626\ldots$; (4) its high- and low-temperature series must match the established series to all known orders; (5) it must be invariant under permutations of $(K_1,K_2,K_3)$ and convex in those variables; and (6) it must explain the anomalous behavior of the partially resummed anisotropic high-temperature series. They stress that, on the strength of that anomaly, it is believed the model might "not be solvable" in the sense that $Z$ is not D-finite, while emphasizing that mathematical proof is lacking and that an exact solution built from nonholonomic functions is not thereby excluded. They also note the logical possibility that no exact analytic $Z$ satisfying all six conditions exists at all, conjecturing, without proof, that the conditions are mutually consistent. Every claimed exact solution examined in their paper, spanning seven decades of attempts, violates at least one of the conditions.

Problem Statement

Determine the closed-form solvability class of the three-dimensional nearest-neighbor ferromagnetic Ising partition function, and the coherence of the necessary-condition framework around it. As left open by Viswanathan et al. (2022) and unresolved by subsequent literature, the problem has a mandatory core of two questions (items 1 and 2), a mechanism question contingent on the branch established in the core (item 3), and one flagged, optional generalization direction (item 4), all serving the single objective of establishing in which functional class, if any, the exact $Z(K_1,K_2,K_3)$ is representable. Throughout, an exact analytic closed form means an explicitly exhibited analytic expression in $K_1,K_2,K_3$ (not necessarily D-finite), as opposed to the finite-lattice polynomials or the unspecified limit function itself. A complete resolution of either item 1 or item 2 counts as a resolution of the task; item 3 is required only if its antecedent holds; item 4 is optional and does not affect completion:

1. Representability. Prove, with full rigor, either that the thermodynamic-limit per-site partition function $Z(K_1,K_2,K_3)$ of the simple-cubic nearest-neighbor ferromagnetic Ising model in zero field is D-finite (holonomic) in its temperature variables, or that it is not D-finite. The anomalous behavior of the partially resummed anisotropic high-temperature series is evidence for non-D-finiteness; a rigorous proof of either alternative is required.

2. Joint satisfiability. Determine whether the six necessary conditions are jointly satisfiable by an explicitly exhibited expression: either exhibit an exact analytic closed form consistent with all six conditions (since the conditions are necessary but not sufficient, this establishes joint satisfiability rather than, by itself, a completed solution), or prove that no such function exists (which would settle negatively the possibility of an analytic exact solution of the 3D Ising model). A resolution must turn on such an exhibited expression, with each of the six conditions, condition 6 included, established for it; merely asserting the existence of some function satisfying the conditions (for instance the thermodynamic-limit $Z$) is not an admissible resolution.

3. Mechanism, contingent on the branch established in 1 and 2. If a nonholonomic exact closed form exists, identify the mathematical mechanism by which it can satisfy the analyticity and series-coefficient constraints (conditions 3-4) while reproducing the anisotropic partial-resummation anomalies (condition 6).

4. Generalization direction flagged by the same source. Extend the partial-resummation and D-finiteness diagnostic (condition 6) to three-dimensional lattices beyond the simple cubic.

Note on the object: on any finite lattice $Z_N$ is a polynomial in the $v_i=\tanh K_i$ and hence trivially D-finite; the question concerns the infinite-volume limit $Z=\lim_N Z_N^{1/N}$, taken along any sequence of finite boxes whose boundary becomes negligible, as a function of the anisotropic reduced temperatures.

References

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