Complete characterization of the disorder-solution manifold of the generalized 2D Ising model with multi-spin interactions: admissible parameter region, branch stratification, and phase transitions in its vicinity

Khrapov (arXiv:2004.02427, 2020) constructed explicit disorder solutions - closed-form exact free energies - for the planar Ising model whose plaquette Hamiltonian contains nearest- and next-nearest-neighbour pair couplings, all four triple interactions, the quadruple interaction, and an external field, on a helical toroidal lattice. The construction yields a seven-parameter family of solutions of the transfer-matrix eigenvector equations, valid only where the algebraic denominators of the closed formulas do not vanish, with one coupling determined by a quadratic equation with two branches, and is illustrated by a single numerical example. The open problems left by the construction - the exact admissible region of parameters yielding real positive solutions, the complete classification of solution branches including degenerate and singular ones, a global numerical characterization of the solution set, and whether genuine thermodynamic phase transitions occur near the disorder manifold - remain open.

Background

The Ising model assigns to each site of a lattice a spin $\sigma\in\{+1,-1\}$ and sums the Boltzmann weights $\exp(-H/(k_BT))$ of an interaction energy $H$ over all spin configurations; the free energy per site in the thermodynamic limit (lattice size tending to infinity) is the central observable, and its non-analyticities are the phase transitions of the model. For the two-dimensional Ising model with nearest-neighbour pair interactions and zero field, Onsager's 1944 solution gives this free energy exactly (Onsager, Phys. Rev. 65, 117 (1944)). Almost every generalization - longer-range pair couplings, three- and four-spin interactions, or a nonzero external field - escapes exact solution. One structural exception is the notion of a disorder solution (also called a disorder variety): a submanifold of the space of coupling constants on which the partition function and free energy become exactly calculable and the model exhibits an effective dimensional reduction, with correlations decaying exponentially at exactly computable rates. Disorder solutions were discovered by Stephenson in the anisotropic triangular-lattice Ising model (J. Math. Phys. 11, 420 (1970)); Jaekel and Maillard gave a local criterion characterizing disorder varieties in any dimension (J. Phys. A 18, 1229 (1985)); Baxter found them for the interactions-round-a-face (IRF, all interactions among the four spins of a square-lattice face) and checkerboard Potts models (J. Phys. A 17, L911 (1984)); and Wu obtained them for the triangular Ising model in a nonzero magnetic field and for the Ising model with crossing and four-spin interactions at the special imaginary field $i(\pi/2)kT$ (J. Stat. Phys. 40, 613 (1985); J. Stat. Phys. 44, 455 (1986)). On the solvable manifold the free energy is analytic; the interest for phase structure lies in the vicinity of the manifold, where Georges, Hansel, Le Doussal and Maillard developed a systematic expansion and exhibited an $\alpha=1/2$ singularity near a disorder variety for pair-interaction models (J. Phys. A 19, 1001 (1986); J. Phys. A 19, L329 (1986)).

The source paper (P. Khrapov, arXiv:2004.02427, 2020, building on the author's earlier generalized Ising and Potts constructions in Mod. Inf. Technol. IT-Educ. 15, 33 (2019) and 15, 312 (2019)) treats the planar Ising model whose Hamiltonian (its Eq. (1.4)) contains, per elementary plaquette, four pair couplings (nearest and next-nearest neighbours), all four triple (three-spin) interactions, the quadruple (four-spin) plaquette interaction, and an external field - ten independent dimensionless couplings $K_1,\dots,K_9$ and $h'$ in total. The lattice is closed by helical (shift-by-one) toroidal boundary conditions: the $L_1\times L_2$ sites are linearly ordered $\tau_0,\tau_1,\dots,\tau_{L-1}$ with the identification $(L_1,t_2)\equiv(0,t_2+1)$. The partition function is written as $Z=\operatorname{Tr}(\Theta^{L})$ of an elementary transfer matrix $\Theta$ (a matrix built from the Boltzmann weights of one plaquette, of size $2^{L_1+1}\times 2^{L_1+1}$, with nonnegative entries), and the free energy per site is $f=-kT\ln\lambda_{\max}$, where $\lambda_{\max}$ is the largest eigenvalue of $\Theta$, unique with a positive eigenvector by the Perron-Frobenius theorem. The author seeks an eigenvector whose components have a highly periodic inductive structure - for $L_1=2$ the eight-component vector $(1,b_2,c,cb_2;cb_3,cb_4,c^2b_3,c^2b_4)$, extended to larger widths by a doubling recursion - and derives the eight equations (its Eq. (2.7)) for the Perron eigenvalue $F$ in terms of $R_i=\exp(K_i)$, $H=\exp(h')$, $c$, $b_2$, $b_3$, $b_4$. Solving this system produces closed-form expressions (its Eqs. (2.12)-(2.19), written in the squared variables $u_i=R_i^2$) in which seven parameters $R_1,R_3,R_4,R_5,R_6,R_7,R_8$ are free, while $R_2$, the squared field factor $h=H^2$, $c$ (through $c^2$), $b_2$, $b_3$, $b_4$ and the eigenvalue $F$ are rational functions of them, with $s_9=R_9^4$ obtained from a quadratic equation (its Eq. (2.8)) admitting two roots; on the solution set the free energy is $-f/(kT)=\ln F$ (its Eq. (2.20)). Physical admissibility requires $R_i>0$, $H>0$, $F>0$, $c>0$, $b_2,b_3,b_4>0$, and the paper states that the formulas hold only on the set of parameter values for which their algebraic denominators do not vanish. A single numerical example (its Eqs. (2.21)-(2.24)), verified by the power method (iterative computation of the dominant eigenvector), shows that nontrivial solutions exist. A remark of the paper notes that the same system and free energy govern a wider class of Hamiltonians in which the second-row couplings are shifted by an arbitrary positive integer.

The unresolved question thus arises directly from the construction: the exact solvable set is given only parametrically, with its real-positive admissible region, its branch and singular structure, and its global numerical topology left undetermined, and the author observes that the high symmetry and inductive construction of the eigenvector components disappear beyond the solvable set, which is stated in the abstract and introduction as an occasion to search for phase transitions in the vicinity of this set of disorder solutions. The question is the natural multi-spin, nonzero-field counterpart of the classical vicinity-of-disorder-variety program, which was carried out only for pair-interaction models.

Problem Statement

For the generalized 2D Ising model with multi-spin interactions of Khrapov (arXiv:2004.02427) - the plaquette Hamiltonian (1.4)/(1.5) with couplings $K_1,\dots,K_9$ and field $h'$, helical toroidal boundary conditions, and elementary transfer matrix $\Theta$ with $Z=\operatorname{Tr}(\Theta^{L})$ - complete the characterization of the disorder-solution manifold produced by the periodic eigenvector ansatz and the closed-form solution (2.12)-(2.19). The four requested results jointly support one objective, the full geometric and physical account of this exactly solvable set:

(1) Admissible region. Determine the set of free parameters $(R_1,R_3,R_4,R_5,R_6,R_7,R_8)\in\mathbb{R}_+^7$ for which the closed-form expressions (2.12)-(2.19) produce, on each branch of the quadratic (2.8), real values with $h>0$ (the paper's $h=H^2$), $c>0$, $R_2>0$, $b_2>0$, $b_3>0$, $b_4>0$, $F>0$, $s_9>0$ and all denominators nonzero; give its defining inequalities explicitly and characterize its topology and measure, including how it is cut by the loci where the denominators of (2.12)-(2.19) vanish.

(2) Branch and stratification structure. Classify all solution branches of the eigenvector system (2.7), equivalently its polynomial form (3.2) in the variables $u_i=R_i^2$ with the solved expressions (3.3)-(3.7): the two branches of the quadratic (2.8) for $s_9$, the degenerate locus where its discriminant vanishes, and any further degenerate or singular branches arising where denominators vanish; identify which branches carry physically admissible (real positive) solutions and how the branches connect or terminate at the singular loci.

(3) Global numerical structure. Provide a robust numerical characterization of the solution set across the admissible parameter space - existence, multiplicity, and stability of solutions of the system as functions of the seven free parameters, where 'stability' means the conditioning of each branch under small perturbations of the free parameters - going beyond the single worked example (2.21)-(2.24), which demonstrates only one admissible point on one branch. As the branch-by-branch numerical counterpart to the exact characterizations (1)-(2), it must supply the explicit solution values, multiplicities, and conditioning at parameter points across the admissible region, and state the coverage of the admissible region and the accuracy to which the solutions are determined.

(4) Phase transitions in the vicinity. Determine whether genuine thermodynamic phase transitions - non-analyticities of the free energy per site $f=-kT\ln\lambda_{\max}$ in the thermodynamic limit - occur at or near the disorder-solution manifold: locate the phase-transition surface of the Hamiltonian (1.4) in the ten-dimensional classical coupling space relative to the seven-dimensional disorder manifold, and establish whether the manifold is a phase boundary, adjacent to one, or lies in a region of analyticity.

Deliverables (1)-(3) jointly complete the geometric description of the exactly solvable manifold, and (4) is its physical payoff in the spirit of the classical vicinity-of-disorder-variety analysis. All four deliverables are required and jointly constitute the resolution of the task: (1)-(3) form its geometric part and (4) its physical part.

References

1. Pavel Khrapov, 'Disorder solutions for generalized 2D Ising Model with multi-spin interaction', arXiv:2004.02427 (2020), https://arxiv.org/abs/2004.02427
2. Pavel Khrapov, 'Disorder solutions for the partition functions of the two-dimensional Ising-like models', arXiv:2011.10259 (2020), https://arxiv.org/abs/2011.10259
3. Pavel V. Khrapov, 'Disorder solutions for the free energy of the Ising-like models', arXiv:2103.08425 (2021), https://arxiv.org/abs/2103.08425
4. Pavel Khrapov and Anastasia Loginova, 'The exact solution of the Axial Next-Nearest-Neighbor Ising model on width 2 spin chains', arXiv:2202.01834 (2022), https://arxiv.org/abs/2202.01834
5. Pavel Khrapov and Veronika Nikishina, 'Exact solution Ising-like $2\times 2\times\infty$ models with multispin interactions', arXiv:2205.15640 (2022), https://arxiv.org/abs/2205.15640
6. Pavel Khrapov and Stepan Shchurenkov, 'Exact Solution and Correlation Functions of Generalized Double Ising Chains', arXiv:2511.00451 (2025), https://arxiv.org/abs/2511.00451
7. Pavel Khrapov and Stepan Shchurenkov, 'Exact solution of generalized gauge-invariant Ising chains with multi-spin interactions', arXiv:2605.23228 (2026), https://arxiv.org/abs/2605.23228
8. Lars Onsager, 'Crystal Statistics. I. A Two-Dimensional Model with an Order-Disorder Transition', Physical Review 65, 117-149 (1944), DOI: 10.1103/PhysRev.65.117, https://doi.org/10.1103/PhysRev.65.117
9. John Stephenson, 'Ising-Model Spin Correlations on the Triangular Lattice. IV. Anisotropic Ferromagnetic and Antiferromagnetic Lattices', Journal of Mathematical Physics 11, 420-431 (1970), DOI: 10.1063/1.1665155, https://doi.org/10.1063/1.1665155
10. M. T. Jaekel and J.-M. Maillard, 'A criterion for disorder solutions of spin models', Journal of Physics A: Mathematical and General 18, 1229-1238 (1985), DOI: 10.1088/0305-4470/18/8/023, https://doi.org/10.1088/0305-4470/18/8/023
11. R. J. Baxter, 'Disorder Points of the IRF and Checkerboard Potts Models', Journal of Physics A: Mathematical and General 17, L911-L917 (1984), DOI: 10.1088/0305-4470/17/17/001, https://doi.org/10.1088/0305-4470/17/17/001
12. F. Y. Wu, 'Exact solution of a triangular Ising model in a nonzero magnetic field', Journal of Statistical Physics 40, 613-620 (1985), DOI: 10.1007/BF01009892, https://doi.org/10.1007/BF01009892
13. F. Y. Wu, 'Two-dimensional Ising model with crossing and four-spin interactions and a magnetic field $i(\pi/2)kT$', Journal of Statistical Physics 44, 455-463 (1986), DOI: 10.1007/BF01011305, https://doi.org/10.1007/BF01011305
14. A. Georges, D. Hansel, P. Le Doussal and J.-M. Maillard, 'The vicinity of the disorder varieties: a systematic expansion', Journal of Physics A: Mathematical and General 19, 1001-1005 (1986), DOI: 10.1088/0305-4470/19/6/027, https://doi.org/10.1088/0305-4470/19/6/027
15. A. Georges, D. Hansel, P. Le Doussal and J.-M. Maillard, 'An $\alpha=1/2$ singularity in the vicinity of a disorder variety and its random walk interpretation', Journal of Physics A: Mathematical and General 19, L329-L333 (1986), DOI: 10.1088/0305-4470/19/6/006, https://doi.org/10.1088/0305-4470/19/6/006
16. P. V. Khrapov, 'Disorder Solutions for Generalized Ising and Potts Models with Multispin Interaction', Modern Information Technologies and IT-Education 15(1), 33-44 (2019), DOI: 10.25559/SITITO.15.201901.33-44, https://doi.org/10.25559/SITITO.15.201901.33-44
17. P. V. Khrapov, 'Disorder Solutions for Generalized Ising Model with Multispin Interaction', Modern Information Technologies and IT-Education 15(2), 312-319 (2019), DOI: 10.25559/SITITO.15.201902.312-319, https://doi.org/10.25559/SITITO.15.201902.312-319