Extremality of the Disordered Phase of the Ising Model with Competing Interactions on a Cayley Tree

For the Ising model with competing interactions — nearest-neighbor coupling $J$ and next-nearest-neighbor (distance-two) coupling $J_1$ — on a Cayley tree (the Vannimenus model), the disordered phase is the symmetric translation-invariant Gibbs measure $\mu_0$. For the purely nearest-neighbor Ising model on a Cayley tree it is a classical result (Bleher; Ioffe) that $\mu_0$ is extreme (a pure phase) exactly above the spin-glass critical temperature, i.e. when $k\theta^2\le 1$ with $\theta=\tanh(\beta J)$. For the competing-interaction model, the source paper (Mukhamedov–Barhoumi–Souissi, 2016) proved only an algebraic separation — non-quasi-equivalence of the quantum Markov chain state associated with $\mu_0$ from the states of the other translation-invariant Gibbs measures — and explicitly left the extremality of $\mu_0$ open. Later rigorous work established the translation-invariant Gibbs measures, the phase transition, and free energies for this model, and solved the analogous disordered-phase extremality problem for neighboring models (the $\lambda$-model and the mixed spin-$(1,1/2)$ Ising model), but not for the spin-$(1/2)$ competing-interaction model itself. The problem is to determine whether the disordered phase of the Ising model with competing interactions on the Cayley tree is an extreme Gibbs measure, and to identify the regions of the parameter space $(J,J_1,\beta)$ in which it is extreme and in which it is not.

Background

The Cayley tree $\Gamma^k$ of order $k$ (also called the Bethe lattice) is the infinite connected graph with no cycles in which every vertex has exactly $k+1$ neighbors; it is the standard arena for exactly solvable lattice models, since Boltzmann–Gibbs distributions on it satisfy recursive equations along the generations of the tree. Ising spins $\sigma(x)\in\{-1,+1\}$ live on the vertices, and the Ising model with competing interactions (introduced by Vannimenus in 1981 as the tree analogue of the ANNNI model, the axial next-nearest-neighbor Ising model) is defined by the Hamiltonian

$$H(\sigma)=-J\sum_{\langle x,y\rangle}\sigma(x)\sigma(y)-J_1\sum_{\langle\langle x,y\rangle\rangle}\sigma(x)\sigma(y),$$

where the first sum runs over nearest-neighbor pairs $\langle x,y\rangle$ and the second over prolonged next-nearest-neighbor pairs $\langle\langle x,y\rangle\rangle$ — the branchwise ancestor–descendant pairs at graph distance two (grandparent–grandchild pairs along a branch), which exclude the sibling or “one-level” pairs at distance two — with real couplings $J,J_1$ and inverse temperature $\beta$. A Gibbs measure for this Hamiltonian is a probability measure on infinite spin configurations whose finite-volume conditional distributions are given by the Boltzmann–Gibbs rule; the set of all Gibbs measures is a nonempty convex compact set, and a phase transition means that this set contains more than one element. Its extreme points (those not expressible as a nontrivial convex combination of other Gibbs measures) are called pure phases; equivalently, a Gibbs measure is extreme if and only if it is trivial on the tail sigma-algebra, i.e. on events determined by the spins arbitrarily far from any fixed finite region. Extremality of a phase therefore expresses exactly that it is a single macroscopic state rather than a mixture, and on trees it is equivalent to a question about information flow from the boundary to the root (the reconstruction problem of information theory).

For the classical Ising model with nearest-neighbor interactions only, the Gibbs measures on the Cayley tree are well understood: the translation-invariant (independent of the generation) splitting Gibbs measures — measures determined by boundary fields satisfying a recursive functional equation — consist of the symmetric disordered phase $\mu_0$ (free boundary conditions, zero effective field), existing at all temperatures, plus a pair of ordered ferromagnetic phases $\mu_\pm$ at low temperature; and the extremality of $\mu_0$ is completely characterized: writing $\theta=\tanh(\beta J)$, $\mu_0$ is extreme if and only if $k\theta^2\le 1$ (Bleher 1990; Ioffe 1996), the threshold $k\theta^2=1$ being the Kesten–Stigum bound — a second-eigenvalue threshold from the theory of branching processes that also governs non-reconstruction for the associated broadcast channel, the information-theoretic model in which a symbol sent from the root is copied noisily down the tree. The Vannimenus model has a numerically established rich phase diagram (including modulated phases), and its rigorous Gibbs-measure theory was developed by Mukhamedov, Akın and Khakimov (2017): because the next-nearest-neighbor coupling links every other generation, the natural recursion is two-step (the conditional law of one generation depends on the two preceding generations), the splitting Gibbs measures are determined by a genuinely two-dimensional fixed-point system, phase transition was proved rigorously, and free energies were computed — but extremality was not addressed. For the order-two tree, the source paper (Mukhamedov–Barhoumi–Souissi, 2016) showed that the translation-invariant Gibbs measures correspond to diagonal quantum Markov chains (finitely correlated states on the quasi-local $C^*$-algebra $\bigotimes_{x\in V}M_2(\mathbb{C})$, i.e. a quantum analogue of a Markov random field on the tree), and used this correspondence to prove that the state of the disordered phase is not quasi-equivalent (not related by an isomorphism of the associated von Neumann algebras) to the states of the other translation-invariant Gibbs measures — an algebraic separation result, but one that says nothing about the classical probabilistic question of extremality, which the authors explicitly leave open. That extremality question is the present problem: it asks for the competing-interaction model the analogue of what Bleher and Ioffe proved for the nearest-neighbor model, and its difficulty is tied to the two-step memory structure of the model, which prevents a direct application of the one-step tree-indexed Markov chain and Kesten–Stigum eigenvalue criteria used in all neighboring solved cases.

Problem Statement

For the Ising model with competing interactions on the Cayley tree — Hamiltonian $H(\sigma)=-J\sum_{\langle x,y\rangle}\sigma(x)\sigma(y)-J_1\sum_{\langle\langle x,y\rangle\rangle}\sigma(x)\sigma(y)$ with spins $\sigma(x)\in\{-1,+1\}$ and inverse temperature $\beta$, on the Cayley tree of order $k$ (the source paper treats $k=2$; the question is stated for the model in general) — determine whether the disordered phase $\mu_0$ (the symmetric translation-invariant Gibbs measure corresponding to free boundary conditions) is an extreme point of the convex set of all Gibbs measures. More precisely, determine the region of the parameter space $(J,J_1,\beta)$ in which $\mu_0$ is extreme (equivalently, trivial on the tail sigma-algebra, i.e. a pure phase) and the region in which it is not extreme, as a function of the tree order $k$ (the binary question above being decided as its special case); this is the analogue, for the two-step recursion governing this model, of the Bleher–Ioffe characterization ($k\theta^2\le 1$) known for the nearest-neighbor Ising model.

References

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