Solving the exact closed self-consistency system for the nearest-neighbour correlations of the three-dimensional Ising model

A detailed 2015 preprint by V. V. Kocharovsky and Vl. V. Kocharovsky (arXiv:1510.07327, never published in a journal) reduces the Ising model on a $d$-dimensional lattice to an exact closed system of $2(1+p)$ nonlinear self-consistency equations for a finite set of unconstrained nearest-neighbour basis correlations $g_{1j}(l)$ (their Eqs. (112)-(118); $p$ is the coordination number of nearest neighbours, so $2(1+p)=14$ correlation quantities for the simple cubic lattice, and two equations in the isotropic case with vanishing anomalous correlations). All physical observables — the constrained partition function, spontaneous magnetization, correlation functions, and joint occupation statistics — are given by explicit algebraic formulas, built from permanents of large circulant matrices, in these same numbers, and the critical exponents would follow from the singular behaviour of the exact self-energies near ${\bf k}=0$. The authors themselves stress that this is only the first step toward a full solution of the 3D Ising model: actually solving the system (analytically or numerically), extracting the critical temperature, magnetization, susceptibility, correlation length, and exponents of the 3D Ising universality class, and validating against 2D exact results and 3D Monte Carlo, series, and conformal-bootstrap data all remain undone.

Background

The Ising model — spins taking values $\pm 1$ on a lattice, coupled between nearest neighbours — is the archetypal statistical-mechanical model of a continuous phase transition. The one- and two-dimensional models were solved exactly (Onsager's 1944 free energy for the zero-field 2D case), but after roughly a century of effort no exact solution of the three-dimensional (3D) model exists. Its critical properties are known only numerically, to high but finite precision, from Monte Carlo simulations and high-temperature series (reviewed by Pelissetto and Vicari, 2002) and from the conformal bootstrap (a numerical method that carves out allowed regions of scaling dimensions from crossing symmetry of the 4-point function; Kos et al., 2016), e.g. $\eta\approx 0.036$ and $\nu\approx 0.630$.

In a long 2015 preprint, V. V. Kocharovsky and Vl. V. Kocharovsky proposed a new reduction of the 3D Ising model, building on their earlier 'constrained spin boson' formulation. In the Holstein–Primakoff representation (which rewrites spin operators in terms of bosonic creation and annihilation operators), each spin-1/2 becomes a pair of bosonic modes whose total occupation is constrained to equal one per site; the Ising partition function is thereby a constrained many-boson problem. Their method of recurrence equations for 'partial contractions' (operator contractions evaluated on the constrained Hilbert space) yields the exact irreducible self-energy of the auxiliary, unconstrained bosonic theory — the self-energy being the object that dresses the bare propagator in the Dyson equation of many-body theory. The paper's central result is an exact closed self-consistency system: a finite set of $2(1+p)$ coupled nonlinear equations (Eqs. (112)-(118) of the preprint, restated verbatim as Eq. (74) of the authors' 2020 Entropy review) for the nearest-neighbour basis correlations $g_{1j}(l)$, $j=1,2$, $l=0,\dots,p$, where $p$ is the coordination number of nearest neighbours ($p=6$ for the simple cubic lattice, hence $2(1+p)=14$ correlation quantities; in the isotropic case with vanishing 'anomalous' correlations — off-diagonal, mode-1/mode-2 correlators whose self-energy block is $\Sigma_{12}$ — the authors note the system collapses to two equations). Once the $g_{1j}(l)$ are found, the physical observables are explicit functions of them: for example, the spontaneous magnetization is a ratio of permanents of submatrices of $(2+g^{-1})$, where the permanent is the determinant with all signs omitted and $g$ is the (circulant, i.e. cyclic-shift-invariant) correlation matrix of the auxiliary problem. The critical exponents, such as $\eta$ governing the long-range decay $\sim r^{2-d-\eta}$ of the correlation function, would follow from the singular behaviour of the exact self-energies and eigen-energies near ${\bf k}=0$ in Fourier space.

The authors are explicit that this is a formulation rather than a solution: 'the presented results constitute only the first, basis steps towards a full solution of the 3D Ising model. A lot of further work is required for a computation of various statistical and thermodynamic quantities, critical functions and exponents.' They flag two further obstacles: the permanent-based observables require evaluating permanents of large matrices, a #P-hard problem in general (#P-hardness is the counting-problem analogue of NP-hardness), for which only asymptotic regimes — not a controlled approximation hierarchy — are given; and the key theorem on the Taylor expansion of diagonal elements of an inverse matrix (their Eqs. (64)-(65)) was announced with a proof 'to be published elsewhere', which has never appeared. The preprint was never published in a refereed journal and has never been independently verified, and no resolution of the self-consistency system is known.

Problem Statement

Turn the claimed exact reduction of the three-dimensional Ising model into a solved, validated computation. Within the framework of Kocharovsky and Kocharovsky (arXiv:1510.07327), solve the exact closed self-consistency system — the $2(1+p)$ nonlinear equations of their Eqs. (112)-(118), restated as Eq. (74) of Entropy 22, 322 (2020) — for the unconstrained basis nearest-neighbour correlations $g_{1j}(l)$, $j=1,2$, $l=0,\dots,p$, of the Ising model on the simple cubic lattice, as functions of temperature, across the relevant phases and including regimes with nonzero anomalous correlations $\Sigma_{12}\neq 0$ (the isotropic $\Sigma_{12}=0$ reduction to two equations is the source's own identified minimal milestone of the same objective, not a separate problem). From the solution, compute the quantities the framework expresses through these numbers: the critical temperature $T_c$, the spontaneous magnetization (for which the source's permanent formula, their Eq. (120), is stated only in the particular case of vanishing anomalous correlations and non-zero normal correlations $g^{1{\bf r}'}_{1{\bf r}}=g^{2{\bf r}'}_{2{\bf r}}$; on $\Sigma_{12}\neq 0$ branches the source's general observable formulas apply), the susceptibility, the correlation length, and the critical exponents ($\eta$, $\nu$, $\beta$) and critical functions, from the Fourier-space singular behaviour of the exact self-energy and eigen-energies near ${\bf k}=0$ — which requires evaluating the involved large circulant-matrix permanents with controlled error. Finally, carry out the validation comparisons the source itself identifies as the required further work: reproduce exact results available within the same formalism in lower dimensions (e.g. the 2D Ising benchmark against Onsager's solution) and benchmark the 3D values against established Monte Carlo, high-temperature-series, and conformal-bootstrap data. These three components form one mutually dependent scientific objective: the observables are explicit functions of the solution of the system, and the benchmarks are the acceptance test of both. An answer of the opposite sign — a demonstration, by explicit computation within the framework's own definitions, that the self-consistency system is not equivalent to the Ising partition function (for instance, a controlled failure of the 2D benchmark) — also resolves the problem, provided it is supported by explicit computation or proof rather than by an uncontrolled approximation.

References

1. V. V. Kocharovsky and Vl. V. Kocharovsky, 'Exact general solution to the three-dimensional Ising model and a self-consistency equation for the nearest-neighbors' correlations', arXiv:1510.07327 [cond-mat.stat-mech] (2015, v3 2016), DOI: 10.48550/arXiv.1510.07327, https://arxiv.org/abs/1510.07327
2. V. V. Kocharovsky and Vl. 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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