Kocharovsky and Kocharovsky (Phys. Lett. A 379, 2520 (2015)) reduced the nearest-neighbor 3D Ising model, written as constrained Holstein-Primakoff spin bosons with nonpolynomial occupation cutoffs handled by an exact recurrence-equation method for partial contractions, to a finite closed system of 2(1+p)2(1+p) self-consistency equations for the nearest-neighbor basis correlations, and gave exact formulas expressing the true (constraint-respecting) occupation distributions, correlation functions, and spontaneous magnetization through the inverse correlation matrix and circulant-matrix permanents. The explicit solution of that system, the evaluation of the constrained observables, and the extraction of the critical temperature and critical exponents were explicitly deferred by the authors. The explicit solution of that system and the constrained observables have remained open in the decade of follow-up work. The problem is to carry the program through: solve the closed self-consistency system at least for the isotropic simple-cubic ferromagnet at zero field, evaluate the true constrained observables, extract the critical behavior (critical temperature, η\eta, and other indices), and validate against exact 2D Ising results and established 3D benchmarks — or refute the formalism by demonstrating inconsistency with them.

Background

The Ising model — spins ±1/2\pm 1/2 on a lattice with nearest-neighbor exchange coupling — is the prototypical model of a continuous phase transition. Its one-dimensional version was solved in 1925, and Onsager's 1944 solution of the two-dimensional zero-field model remains the archetype of an exact critical-point analysis; the three-dimensional (3D) model, however, has resisted exact solution for a century. Its universal content for the simple cubic lattice (critical coupling Kc=J/(kBTc)≈0.22165K_c=J/(k_BT_c)\approx 0.22165, anomalous dimension η≈0.0363\eta\approx 0.0363, and the other critical indices) is known only numerically, from Monte Carlo, high-temperature-series, renormalization-group, and conformal-bootstrap studies (Pelissetto and Vicari 2002; Hasenbusch 2010). Kocharovsky and Kocharovsky proposed a new exact attack, announced in the source paper (Phys. Lett. A 379, 2520 (2015)) and detailed in the companion preprint arXiv:1510.07327. Each spin s=1/2s=1/2 is written in the Holstein-Primakoff representation — the standard mapping of a spin onto two bosonic modes per site whose total occupation is fixed to 2s2s — upgraded to 'constrained spin bosons': the physical restriction is enforced by nonpolynomial cutoff operators θ^=∏rθ(2s−n^r)\hat\theta=\prod_{\mathbf r}\theta(2s-\hat n_{\mathbf r}), where θ\theta is the Heaviside step function and n^r\hat n_{\mathbf r} is the occupation operator. Because such nonpolynomial factors defeat standard diagrammatic perturbation theory, the authors had earlier introduced (Phys. Lett. A 379, 466 (2015); Physica Scripta 90, 108002 (2015)) a method of recurrence equations for partial contractions: contractions of two operators with functions of the occupation numbers reproduce themselves, and the resulting recurrence equations admit an exact solution in the form of an inverse 2N×2N2N\times 2N matrix. Applied to the 3D Ising model on a cubic lattice of NN sites with coordination number pp (p=6p=6 for the simple cubic lattice), this yields three families of exact results. First, the total irreducible Matsubara self-energy Σ\Sigma — the irreducible part of the two-point Green's function in the Dyson equation — is expressed exactly through only 2(1+p)2(1+p) basis nearest-neighbor correlations g1j(l)g_{1j}(l), l=0,…,pl=0,\dots,p, j=1,2j=1,2, where jj is the Nambu-like creation/annihilation index (g11g_{11} are normal and g12g_{12} anomalous correlations). Second, the exact characteristic functions of the (unconstrained) occupation numbers at one, two, or all sites are obtained in closed form. Third, and crucially, the basis correlations are themselves fixed by an exact closed system of 2(1+p)2(1+p) self-consistency equations (Eq. (31) of the source paper; Sec. VI of the companion) — a finite-dimensional nonlinear fixed-point system involving only elementary functions and a Brillouin-zone Fourier sum, which reduces to two equations in the isotropic case. The authors liken its status to that of a mean-field self-consistency equation, but exact and valid through the entire critical region. Finally, the physically meaningful, constraint-respecting ('true') quantities — the cutoff occupation distributions ρ′\rho', the true correlation functions g′g', and the order parameter (spontaneous magnetization) — are given by exact formulas involving the inverse correlation matrix g−1g^{-1} (Eq. (33) of the source paper), in principle evaluable by the Toeplitz/circulant-matrix techniques that underlie Onsager's 2D solution, and, on the branch with vanishing anomalous correlations, by permanents (determinants without permutation signs) of circulant matrices (matrices constant along diagonals, diagonalized by the discrete Fourier transform) built from gg. What the source paper explicitly leaves open — and what ten years of follow-up have not delivered — is the passage from this formal exact solution to explicit results: solving the self-consistency system, evaluating the true constrained observables, and extracting the long-distance asymptotics ∼r2−d−η\sim r^{2-d-\eta} and the critical exponents. The authors themselves state that this analysis 'could take an effort of many researchers for many years, as it was with analysis of Onsager's exact solution for 2D Ising model'.

Problem Statement

Within the exact formulation of the 3D Ising model of Kocharovsky and Kocharovsky (Phys. Lett. A 379, 2520 (2015), detailed in arXiv:1510.07327) — spins s=1/2s=1/2 on a cubic lattice with coordination number pp, represented as constrained spin bosons with cutoff θ^=∏rθ(2s−n^r)\hat\theta=\prod_{\mathbf r}\theta(2s-\hat n_{\mathbf r}), whose exact irreducible self-energy, occupation characteristic functions, and true-observable formulas are obtained by the recurrence-equation method for partial contractions — carry the exact formal solution through to explicit physical results. At minimum for the homogeneous isotropic ferromagnetic nearest-neighbor model on the simple cubic lattice (p=6p=6) at zero external field, where the exact closed system of 2(1+p)2(1+p) self-consistency equations (Eq. (31) of the source paper; two equations in the isotropic reduction) fixes the basis correlations g1j(l)g_{1j}(l), l=0,…,pl=0,\dots,p, j=1,2j=1,2, the required deliverables are jointly: (1) solve this closed self-consistency system, analytically or numerically (including the finite-lattice Brillouin-zone sum and its thermodynamic limit), throughout the disordered and ordered phases, locating the critical point at which the solution changes branch; (2) evaluate from that solution the true constrained observables — the θ^\hat\theta-cutoff occupation distributions ρ{nr}′\rho'_{\{n_{\mathbf r}\}} and their one- and two-site marginals, the order parameter (spontaneous magnetization) as a function of temperature, and the true equal-time correlation functions — implementing the g−1g^{-1} inversion (Eq. (33) of the source paper) or the equivalent circulant-permanent evaluation; (3) extract the critical behavior of the model: the critical temperature, the long-distance decay of the true correlation function and the exponent η\eta, and any further critical indices and critical functions accessible from the solution; and (4) validate the outcome — reproduce, within the same formalism, the exactly known two-dimensional Ising results (Onsager's solution and its correlation-function refinements) in the d=2d=2 reduction, and compare the 3D critical temperature and exponents with the established numerical benchmarks (e.g., Kc≈0.22165K_c\approx 0.22165, η≈0.0363\eta\approx 0.0363). An answer that instead demonstrates that this exact formalism, or the self-consistency system it yields, is internally inconsistent or incompatible with the exactly known 2D results or the established 3D benchmarks also resolves the problem, by refuting the program.

References

  1. Vitaly V. Kocharovsky and Vladimir V. Kocharovsky, 'Towards an exact solution for the three-dimensional Ising model: A method of the recurrence equations for partial contractions', Physics Letters A 379, 2520-2523 (2015). DOI: 10.1016/j.physleta.2015.07.026, https://doi.org/10.1016/j.physleta.2015.07.026
  2. Vitaly V. Kocharovsky and Vladimir V. Kocharovsky, 'Exact general solution to the three-dimensional Ising model and a self-consistency equation for the nearest-neighbors' correlations', arXiv:1510.07327 (v3, March 2016). https://arxiv.org/abs/1510.07327
  3. Vitaly V. Kocharovsky and Vladimir V. Kocharovsky, 'Microscopic theory of a phase transition in a critical region: Bose-Einstein condensation in an interacting gas', Physics Letters A 379, 466-470 (2015). DOI: 10.1016/j.physleta.2014.10.052, https://doi.org/10.1016/j.physleta.2014.10.052
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