Exact finite-size Ising partition functions and energy-degeneracy multiplicities for uniform two-dimensional lattices in a field and three-dimensional lattices

For a finite Ising model the partition function $Z=\sum_{s}\exp(-\beta\mathcal{H}(s))$ can always be written exactly as a sum over the $2^{N}$ spin configurations, and a 2018 preprint by R. Q. Wei reorganized this sum (via a Hubbard–Stratonovich-type Gaussian integral) into an equivalent sum of $2^{N}$ exponentials indexed by subsets of sites, valid in any dimension and any external field. The preprint itself poses two open problems about this expression: (1) how to evaluate the sum of $2^{N}$ exponential terms efficiently for large lattices, especially in a nonzero external field, and (2) how to determine, in closed form or by an efficient algorithm, the multiplicities of the many identical exponent values (the energy-level degeneracies, i.e. the density of states). Exact closed-form partition functions are known in one dimension, and in two dimensions only at zero field (Onsager; finite-size closed forms; polynomial-time Pfaffian evaluation), while the two-dimensional model in a uniform field and the three-dimensional model remain without any closed form or polynomial-time exact algorithm (the best exact method, the transfer matrix, runs in time exponential in the lattice cross-section), and no computational-hardness classification exists for precisely these uniform hypercubic families. The general reading with arbitrary nonuniform couplings and fields is conditionally closed by #P-hardness results (Barahona; Kotek), but the uniform families at the heart of the present problem remain open.

Background

The Ising model is a lattice model of statistical mechanics in which each site $i$ carries a spin $s_{i}=\pm1$ , neighboring spins interact with coupling $J$ , and an external magnetic field $h$ couples to every spin. Its partition function $Z=\sum_{s}\exp(-\beta\mathcal{H}(s))$ , where $\beta$ is inverse temperature and the sum runs over all $2^{N}$ spin configurations of an $N$ -site lattice, encodes the complete equilibrium thermodynamics: the free energy, the specific heat, and the magnetization all follow from derivatives of $\ln Z$ . An exact evaluation of $Z$ is therefore the classical route to exact thermodynamics. Kramers and Wannier (1941) solved the one-dimensional model exactly. Onsager (1944) obtained the exact free energy of the two-dimensional model at zero field, one of the founding results of statistical mechanics; Fisher (1966) showed that the same zero-field two-dimensional partition function on any finite planar lattice can be evaluated exactly in time polynomial in $N$ by reduction to a dimer (perfect-matching) counting problem solved with Pfaffians (determinant-type algebraic objects). Karandashev, Kryzhanovsky, and Malsagov (2017) gave explicit closed-form expressions for the finite-size two-dimensional zero-field partition function on a torus. In contrast, the two-dimensional model in a uniform nonzero field and the three-dimensional model with or without a field have resisted exact solution for eighty years: the transfer-matrix method (building $Z$ row by row) is exact but its cost grows exponentially with the lattice cross-section — subexponential in the total number of sites, yet not polynomial — and no closed form or polynomial-time exact algorithm is known. The problem is widely regarded as a facet of the famous unsolved exact-solution problem for the three-dimensional Ising model. Wei (2018, arXiv:1805.01366, unpublished physics.gen-ph preprint) derived, for a finite $N$ -site Ising lattice in any dimension with uniform couplings and a field, an exact rewriting of $Z$ as a sum of $2^{N}$ exponential terms $\exp(\tfrac12 vK^{-1}v^{T})$ obtained through a Hubbard–Stratonovich transformation (an exact Gaussian-integral identity that linearizes quadratic interaction terms at the cost of introducing auxiliary integration variables); here $v$ is a vector determined by the chosen subset of sites and $K$ is the interaction matrix. The rewriting is an exact identity with one term per spin configuration, so it compresses nothing computationally; the preprint is explicit that for large $N$ the direct summation of $2^{N}$ exponential functions is prohibitive, and that only for a small example ( $2\times6\times2$ , where merely 23 distinct exponent values occur among 16,777,216 terms) could the degeneracies of the repeated exponent values be tabulated empirically. The relevant computational-complexity background: #P is the class of counting problems associated with NP decision problems, and a problem is #P-hard if solving it exactly in polynomial time would collapse complexity classes (FP = #P). Barahona (1982) proved that finding the ground state of an Ising spin glass (couplings of mixed sign) is NP-complete already for a two-dimensional lattice with fields and for a three-dimensional lattice without fields, and Kotek (2012) proved that evaluating the Ising partition function with variable couplings and fields is #P-hard at essentially all rational parameter points, even on bipartite planar lattices. These results conditionally close the door on an efficient exact algorithm in the fully general nonuniform setting, but they do not classify the uniform families (identical couplings, uniform field, hypercubic geometry) that the source paper targets. For approximation, by contrast, Jerrum and Sinclair (1993) gave a polynomial-time randomized approximation scheme (FPRAS) for the ferromagnetic Ising partition function at any temperature and field, so the open question concerns exact evaluation specifically. The rigorous thermodynamic-limit critical behavior of the three-dimensional model is a separate, adjacent open problem on which real progress exists (e.g. Aizenman, Duminil-Copin, and Sidoravicius proved in 2014–2015 that the spontaneous magnetization of the $\mathbb{Z}^{3}$ Ising model vanishes continuously at the critical point); it is context here, not part of the source question.

Problem Statement

For the Ising model with uniform nearest-neighbor couplings on finite uniform lattices — specifically the two-dimensional square lattice in a uniform nonzero external field $h$ , and the three-dimensional simple-cubic lattice with or without a uniform field — provide, with proof, one of the following, for each of the listed families, for arbitrary finite lattice sizes (an $M_{1}\times M_{2}$ square lattice in two dimensions, an $L_{1}\times L_{2}\times L_{3}$ simple-cubic lattice in three dimensions) and for any of the standard boundary conditions (periodic, i.e. on a torus, or free/open): (i) a closed-form expression that can be evaluated in time polynomial in the total number of sites $n$ , or an exact algorithm whose running time is polynomial in the total number of sites $n$ (that is, $n=M_{1}M_{2}$ in two dimensions and $n=L_{1}L_{2}L_{3}$ in three dimensions), for the finite-size partition function $Z(\beta,h)$ ; or (ii) an equivalent closed-form or efficient characterization of the energy-level degeneracies — the multiplicities of the distinct exponent values in the configuration sum, i.e. the joint density of states over bond energy and magnetization, from which $Z$ at all $\beta$ and $h$ follows.

References

Rong Qiang Wei, 'An exact solution to the partition function of the finite-size Ising Model', 2018, arXiv:1805.01366, DOI 10.48550/arXiv.1805.01366, https://arxiv.org/abs/1805.01366
Rong Qiang Wei, 'An empirical partition function for two-dimensional Ising Model in an external magnetic field', 2019, arXiv:1905.04295, DOI 10.48550/arXiv.1905.04295, https://arxiv.org/abs/1905.04295
Rong Qiang Wei, 'An empirical partition function for the simple cubic Ising Model with a zero external magnetic field', 2019, arXiv:1905.06183, DOI 10.48550/arXiv.1905.06183, https://arxiv.org/abs/1905.06183
H. Fakhri and S. B. Seyedein Ardebili, 'Energy levels and their degeneracies for two-ring Ising chains of spins-1/2 with NN and NNN couplings: spin frustration of ferromagnetic and antiferromagnetic orders', The European Physical Journal Plus 136 (2021), DOI 10.1140/epjp/s13360-021-01863-5, https://doi.org/10.1140/epjp/s13360-021-01863-5
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