Disorder varieties of finite-support Ising-like models: characterizing restricted Perron eigenvectors of the transfer matrix and the exhaustiveness of the Khrapov parametrization

Khrapov (arXiv:2103.08425, 2021) constructed parametric families of disorder solutions — exact closed-form free energies in the thermodynamic limit — for Ising-like lattice models of any dimension with arbitrary multispin interactions on a finite Hamiltonian support and boundary conditions with a shift. The construction assumes that the Perron (leading) eigenvector of the elementary transfer matrix depends only on the spins in a distinguished subset $\Phi(\Omega)$ of the support; this reduces the eigenvalue problem to an explicitly invertible linear system in which half of the multispin couplings and the eigenvector coordinates are free parameters and the remaining couplings follow from closed formulas. The source leaves both the soundness and the completeness of this construction open: it neither gives necessary-and-sufficient conditions on the couplings and closures under which the leading transfer-matrix eigenvector actually admits the restricted dependence (nor the exact conditions on the free parameters under which the constructed vector is a genuine Perron eigenvector of the full matrix, so that the free-energy formula is valid), nor does it address whether the parametrized submanifolds exhaust all disorder solutions of a finite-support model. The problem posed here asks for exactly that: a complete characterization of the restricted-eigenvector disorder varieties of finite-support Ising-like models, and a determination — proof of exhaustion or explicit unreachable disorder points with a characterization of the complement — of whether the parametrization captures all of them.

Background

Ising-like models and their transfer matrices. An Ising-like model assigns a spin $\sigma_t\in\{+1,-1\}$ to every site $t$ of a $\nu$-dimensional lattice and takes a translation-invariant Hamiltonian that is an arbitrary linear combination of products of spins on translates of one fixed finite set $\Omega\subset\mathbb{Z}^{\nu}$, the support of the Hamiltonian:

$$H \;=\; -\sum_{i}\; \sum_{\emptyset\neq S\subseteq \Omega+\tau^{i}} J_{S}\prod_{\omega\in S}\sigma_{\omega+\tau^{i}},$$

where the inner sum runs over all non-empty subsets $S$ of the support. A multispin interaction is any term involving a product of two or more spins (single-site terms are fields, pair terms are ordinary two-body couplings), and the support may be any finite point set — a simplex, a cube, the plaquette-plus-diagonals cluster of the axial next-nearest-neighbour Ising (ANNNI) model, etc. Khrapov (2021) studies these models on lattices of size $L_1\times\cdots\times L_\nu$ with cyclic boundary conditions with a shift (helical-type closure), under which the sites are linearly ordered $\tau^{0}<\tau^{1}<\cdots<\tau^{L-1}$ by traversing the first coordinate fastest with wrap-around identifications. Writing $\omega^{0}$ and $\omega^{r}$ for the first and last points of $\Omega$ in this natural order, and $i_{\max}$ for the natural-order index of $\omega^{r}$, the partition function is the trace of a power of the elementary transfer matrix $\Theta$: a $2^{i_{\max}}\times 2^{i_{\max}}$ nonnegative matrix whose nonzero entries are Boltzmann weights $\exp(\sum_{S\subseteq\Omega}K_{S}\prod_{\omega\in S}\sigma_{\omega})$, with $K_{S}=J_{S}/k_{B}T$, connecting consecutive spin tuples of length $i_{\max}$. By the Perron–Frobenius theorem (a nonnegative matrix has a nonnegative eigenvector belonging to its spectral radius, its eigenvalue of largest modulus, and this eigenvector is unique and strictly positive when the matrix is primitive, i.e. when some power of it has all entries strictly positive), the free energy per site in the thermodynamic limit is $f=-k_{B}T\ln\lambda_{\max}$ with $\lambda_{\max}$ the largest eigenvalue of $\Theta$ — provided one can actually control the leading eigenvector. Disorder solutions and varieties. A disorder solution (the locus in coupling space where it holds is a disorder variety) is a classical notion for models with competing interactions: a submanifold, lying in the disordered phase, on which the model undergoes an exact dimensional reduction — the free energy and correlation functions reduce to those of an effectively lower-dimensional model, and correlations along lattice directions become powers of nearest-neighbour correlations, exactly as in a one-dimensional chain. Stephenson (1970) found the first example, the disorder point of the anisotropic antiferromagnetic Ising model on the triangular lattice, given by $\tanh K_{3}+\tanh K_{1}\tanh K_{2}=0$. Many further examples were then produced by heterogeneous techniques: crystal-growth mappings for the interaction-round-a-face model (IRF, i.e. all interactions inside one face of the square lattice), kinetic-model representations, and vertex-model analyses. Baxter (1984) showed that the disorder points of the IRF model correspond to the leading transfer-matrix eigenvector taking a simple product form — a restricted-dependence condition on the Perron eigenvector. Jaekel and Maillard (1985) gave a local criterion valid in any dimension: the Boltzmann weight of an elementary cell, summed over a suitably chosen set of spins, is independent of the remaining spins; this decoupling explains the effective dimensional reduction and was used to obtain disorder solutions and correlation functions for Ising and Potts models in two and three dimensions (Georges, Hansel, Le Doussal and Maillard, 1987). Batchelor and van Leeuwen (1989) review the classical theory. In the language of probabilistic graphical models, a disorder variety is a locus where the lattice Gibbs measure acquires extra conditional-independence structure; Pelizzola (2000, 2005) showed that on disorder varieties the cluster variation method (CVM, a hierarchy of cluster-based variational approximations to the free energy) becomes exact, which gives these loci a practical role as exact benchmarks. Khrapov's construction. Khrapov (2021), continuing his earlier two-dimensional work (arXiv:2004.02427, arXiv:2011.10259), observed that these case-by-case results can be approached uniformly through the leading eigenvector of the elementary transfer matrix. He introduces the subset

$$\Phi(\Omega)\;=\;\{\omega\in\Omega:\;\omega+(1,0,\ldots,0)\in\Omega\}$$

of support points whose translate by one step along the first coordinate stays inside the support, and makes the ansatz that the Perron eigenvector of $\Theta$ depends on the spin tuple only through the spins at $\Phi(\Omega)$ (eq. (2.21) of the source). Under this ansatz the $2^{i_{\max}}$ eigenvector equations are asserted to collapse to $2^{|\Omega|-1}$ reduced equations, which after eliminating $\lambda$ form a linear system in the couplings of the subsets not containing the maximal point $\omega^{r}$; the system matrix, built from $\pm 1$-valued spin products and of Walsh–Hadamard type, has an explicit closed-form inverse (eq. (2.35)), so that any choice of the free parameters — the couplings of the subsets containing $\omega^{r}$, plus the coordinates $W$ of the restricted eigenvector (normalized in the source by $W_{0}=1$) — determines the remaining couplings by closed formulas (eqs. (2.36), (2.43)), and the free energy follows from $f=-k_{B}T\ln\lambda_{\max}$. For Hamiltonians invariant under global sign reversal the formulas simplify (odd multispin couplings vanish and the eigenvector is centrally symmetric). The paper works out the simplex, cube and ANNNI supports in arbitrary dimension and checks agreement with the earlier exact results of Wu (1985) for the triangular Ising model in a field on numerical examples. What is missing. The construction is a sufficient parametrization: it produces disorder solutions on explicit finite-dimensional submanifolds of coupling space. It does not prove in general that the reduced system implies the full eigenvector equation for every finite support and closure (the collapse of equations is asserted in one sentence, not proven); it does not state the conditions — strict positivity of the restricted eigenvector in particular — under which the vector built from the reduced equations is a genuine Perron eigenvector of the full transfer matrix rather than a formal solution of the reduced equations only; and it says nothing about whether these submanifolds exhaust all disorder solutions of a finite-support model, or how to describe the complement. These gaps are precisely the content of the present problem.

Problem Statement

Consider Ising spins $\sigma_{t}\in\{+1,-1\}$ on the $\nu$-dimensional lattice with cyclic boundary conditions with a shift (helical-type closure), whose sites are linearly ordered $\tau^{0}<\tau^{1}<\cdots$, and a translation-invariant Hamiltonian with finite support $\Omega=\{\omega^{0},\ldots,\omega^{r}\}$,

$$H \;=\; -\sum_{i}\;\sum_{\emptyset\neq S\subseteq \Omega+\tau^{i}} J_{S}\prod_{\omega\in S}\sigma_{\omega+\tau^{i}},$$

with arbitrary multispin couplings $J_{S}$ and dimensionless couplings $K_{S}=J_{S}/k_{B}T$. Let $\Theta$ be the associated elementary transfer matrix (a $2^{i_{\max}}\times 2^{i_{\max}}$ nonnegative matrix, $i_{\max}$ being the natural-order index of the maximal point $\omega^{r}$), and let $\Phi(\Omega)=\{\omega\in\Omega:\omega+(1,0,\ldots,0)\in\Omega\}$. (1) Characterization. For an arbitrary but fixed finite support $\Omega$ and dimension $\nu$, determine necessary and sufficient conditions on the couplings $\{K_{S}\}$ and on the closure data (the side lengths $L_1,\dots,L_\nu$, the shift vector, and the induced wrap-around identification) under which the Perron (leading) eigenvector of $\Theta$ depends on the spin configuration only through the spins on $\Phi(\Omega)$ — the ansatz underlying the reduction in the source. In particular: (i) prove, or refute by an explicit support and closure, the assertion that every eigenvector equation of the full transfer matrix coincides with one of the $2^{|\Omega|-1}$ reduced equations whenever the vector has this restricted dependence; and (ii) characterize exactly the choices of free parameters — the couplings of the subsets containing $\omega^{r}$ together with the coordinates $W$ of the restricted vector — for which the vector produced by the explicit formulas of the source is a genuine Perron eigenvector of the full transfer matrix (strictly positive, with $\lambda$ equal to the spectral radius), so that $f=-k_{B}T\ln\lambda$ is the true free energy per site. A single explicit counterexample discharges sub-item (i); the necessary-and-sufficient characterization demanded by part (1) is still required. (2) Exhaustiveness. The notion of disorder solution is fixed as the classical dimensional-reduction notion defined in the Background: a locus in coupling space on which the model's free energy, and equivalently its correlations, reduce exactly to those of a lower-dimensional model — the loci characterized in any dimension by the Jaekel-Maillard local criterion that the Boltzmann weight of an elementary cell, summed over suitably chosen spins, is independent of the remaining spins, which the source itself invokes as characterizing disorder varieties for any dimensionality. The restricted-eigenvector property of part (1) must not itself be adopted as the definition of disorder solution; part (2) asks whether the parametric varieties capture all classically defined disorder loci, not whether the parametrization is surjective onto its own defining condition. For an arbitrary but fixed finite support $\Omega$ and dimension $\nu$, under the helical closure and natural transfer order fixed above, determine whether the parametric varieties produced in (1) exhaust all disorder solutions of the model, across all choices of the free parameters for which the constructed vector is a genuine Perron eigenvector in the sense of part (1)(ii). If they do not, exhibit disorder points unreachable by the parametrization and characterize the complement — that is, give defining conditions for, or prove the emptiness of, the disorder points outside the parametrized submanifolds, not merely name them. The two parts form one objective: a complete, rigorous characterization of the exactly reducible (disorder) loci of finite-support Ising-like models, establishing at once the soundness and the completeness of the restricted-eigenvector approach.

References

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