Analytic classification of the invariant sets, positive fixed points, and higher-period orbits of the recursion operator for the Ising model with competing binary and ternary interactions on a third-order Cayley tree

For the Ising model on the semi-infinite Cayley tree of order three with nearest-neighbor, prolonged next-nearest-neighbor, and ternary prolonged next-nearest-neighbor couplings, the partial boundary partition functions reduce to four independent variables whose evolution is governed by an explicitly given cubic polynomial map $F$ on $\mathbb{R}^4_+$ , and every translation-invariant or periodic Gibbs state corresponds to a fixed point or periodic orbit of $F$ . Akın (Condens. Matter Phys. 24, 13001, 2021), working in the slice $c=1$ (i.e. $J_t=0$ , where the ternary coupling is switched off and the model becomes the two-coupling competing model on the order-three tree), proved that fixed points lie on two loci $M_1$ and $M_2$ , solved the fixed-point equations completely only on the symmetric invariant subset (paramagnetic phases), reduced period-2 points there to a degree-6 polynomial, and then explicitly posed three unsolved problems about the operator $F$ : whether invariant sets of $F$ exist beyond the known ones; whether positive fixed points exist outside the identified loci $M_1$ and $M_2$ (the $b=1$ case being singled out as hardest); and whether periodic points of period $p>2$ , governed by cumbersome high-order algebraic equations, admit analytic solution or classification. Two further directions are noted in the same paper as remarks rather than as unsolved problems: phase boundaries located by free-energy minimization rather than orbit iteration, and an analytic (Lyapunov-exponent/attractor) account of the modulated regions.

Background

A Cayley tree $\Gamma^k$ of order $k$ is an infinite loopless graph in which every vertex has $k+1$ neighbors; it is the infinite-volume limit of a finitely branching tree and is also called a Bethe lattice. The model below lives on the semi-infinite (rooted) order-three tree, whose generations $W_n$ are the spheres about the root and carry the spin layers between which the competing couplings act. The axial next-nearest-neighbor Ising (ANNNI) model couples each spin with its nearest neighbors along an axis and competitively with spins two steps out; frustration between these couplings produces modulated phases. Vannimenus (1981) studied the ANNNI-type Ising model with two couplings on a Cayley tree numerically by iterating recurrence equations for partial partition functions, finding ferromagnetic, paramagnetic, period-2, period-4 antiphase, and modulated phases meeting at multicritical points; Mariz, Tsallis and Albuquerque (1985) extended this analysis, including the effect of a magnetic field. Ganikhodjaev, Akın, Uğuz and Temir (2011) later studied numerically the model with competing binary and ternary interactions on the order-three tree, the system considered here. A Gibbs measure is the equilibrium probability measure $\mu(\sigma)\propto\exp(-\beta H(\sigma))$ over spin configurations, with $\beta$ the inverse temperature; distinct extremal Gibbs measures are interpreted as distinct thermodynamic phases. On a tree, partial partition functions with fixed boundary spins satisfy exact finite-dimensional recurrences: writing $Z_n$ for the partition function of the depth- $n$ subtree under a fixed boundary pattern of three sibling spins of the last generation together with their parent spin, symmetry reduces the eight patterns $z_1,\dots,z_8$ to four independent ones, and after the change of variables $u_i=\sqrt[3]{z_i}$ ( $i=1,4,5,8$ ) the recurrence closes as a cubic system in $u$ (equation (3.2) of the source); after the further substitution $v_i=\sqrt[3]{u_i}$ ( $i=1,4,5,8$ ) it becomes the explicitly given cubic polynomial map $F:\mathbb{R}^4_+\to\mathbb{R}^4_+$ , $v^{(n+1)}=F(v^{(n)})$ (equation (4.1) of the source, displayed in the problem statement) depending on three positive parameters $a=e^{J/T}$ , $b=e^{J_p/T}$ , $c=e^{J_t/T}$ , with the map's parameter $\alpha=\sqrt[3]{a}$ . The two systems are conjugate under the coordinatewise cubing bijection of $\mathbb{R}^4_+$ , so their fixed points, periodic orbits and invariant sets correspond one-to-one. Fixed points of $F$ correspond to translation-invariant Gibbs measures: a fixed point with $u_1=u_8$ , $u_4=u_5$ (i.e. vanishing order parameters $y=z=0$ ) gives a paramagnetic phase, while $y,z\neq 0$ gives a ferromagnetic phase. A point is periodic with period $p$ if $F^p(u)=u$ with $p$ minimal; period 2 corresponds to an antiferromagnetic phase and period 4 to the antiphase $\langle 2\rangle$ , in which the magnetization alternates in blocks of two generations, $(++--)$ . Non-periodic attractors produce modulated phases, whose commensurate steps are expected to form a devil's staircase (a monotone function constant on infinitely many intervals whose graph resembles a staircase). For such one-dimensional maps of the interval the local expansion rate of nearby orbits is measured by the Lyapunov exponent, whose sign distinguishes periodic from chaotic attractors. Prior rigorous results for related models: Nazarov and Rozikov (2003) determined the invariant sets and the period-2 Gibbs measures for the two-restricted-interaction Ising operator on a Cayley tree; Rozikov, Akın and Uğuz (2014) exactly solved the generalized ANNNI (two-coupling) model on a Cayley tree. For the present model, Akın (2021) fixes $c=1$ (i.e. $J_t=0$ , so the ternary coupling is switched off; the four-dimensional recurrence itself is what is specific to order three) and proves Lemma 4.1: any fixed point of $F$ lies in $M_1=\{u_1=u_8,\ u_4=u_5\}$ or on the locus $M_2$ defined by the equation $(\sqrt[3]{u_4}+\sqrt[3]{u_5})^2-\sqrt[3]{u_4u_5}=\psi\big((\sqrt[3]{u_1}+\sqrt[3]{u_8})^2-\sqrt[3]{u_1u_8}\big)$ , where $\psi(y)=(b^3+\alpha y(1-b^4))/(b^2\alpha(b\alpha y-1))$ is a rational function; he shows the set $A=\{u_1=u_8,\ u_4=u_5,\ c=1\}$ and every set $\{u\in\mathbb{R}^4_+:F^p(u)=u\}$ are invariant; solves the fixed-point equation on the symmetric locus completely (a unique paramagnetic phase for $0<b<1$ ; two or three phases for $b>\sqrt{2}$ across explicit thresholds $\eta_1(b),\eta_2(b)$ ); reduces period-2 points on $M_1$ to the degree-6 polynomial $p_6(x)$ ; and shows that for $b=1$ the symmetric locus is empty (Remark 5.3), so the paramagnetic phase disappears in the studied window. Remark 4.1 of that paper states that the fixed points on $M_2$ (the ferromagnetic phases) are analytically very difficult and were determined only numerically. Lemma 4.1 restricts the fixed points of $F$ to $M_1\cup M_2$ , but it is established only under the assumption $c=1$ , whereas problem (ii) below concerns the operator with general parameters; moreover, in the asymmetric branch ( $v_1\neq v_8$ , $v_4\neq v_5$ ) its reduction divides by the factors $v_1^2+v_1v_8+v_8^2$ and $b\alpha(v_1^2+v_1v_8+v_8^2)-1$ , which can vanish, so those cases are left unverified.

Problem Statement

For the Ising model on the semi-infinite third-order Cayley tree with Hamiltonian $H(\sigma)=-J\sum_{\langle x,y\rangle}\sigma(x)\sigma(y)-J_p\sum_{\rangle x,z\langle}\sigma(x)\sigma(z)-J_t\sum_{\rangle x,y,z\langle}\sigma(x)\sigma(y)\sigma(z)$ , where $\langle x,y\rangle$ runs over nearest-neighbor pairs, $\rangle x,z\langle$ over prolonged next-nearest-neighbor pairs (a vertex and its grandchild, two generations apart on the same branch), and $\rangle x,y,z\langle$ over ternary prolonged next-nearest-neighbor triples (three consecutive generations along a branch), complete the analytic theory of the orbit structure of the induced recursion operator $F$ . Writing $T$ for temperature, set $a=e^{J/T}$ , $b=e^{J_p/T}$ , $c=e^{J_t/T}$ and $\alpha=\sqrt[3]{a}$ ; after the standard reductions of the partial boundary partition functions ( $u_i=\sqrt[3]{z_i}$ , then $v_i=\sqrt[3]{u_i}$ for $i=1,4,5,8$ ), $F$ is the cubic polynomial map $v^{(n+1)}=F(v^{(n)})$ on $\mathbb{R}^4_+$ given by the source's equations (3.2) and (4.1): $v_1'=\frac{\alpha}{bc}\left((bc)^2v_1^3+v_4^3\right),\qquad v_4'=\frac{1}{\alpha bc}\left(b^2v_5^3+c^2v_8^3\right),$ $v_5'=\frac{1}{\alpha bc}\left(v_1^3+(bc)^2v_4^3\right),\qquad v_8'=\frac{\alpha}{bc}\left(c^2v_5^3+b^2v_8^3\right),$ whose fixed points solve the polynomial system (4.2). The source states this objective through three linked unsolved problems, which together carry one scientific goal — turning the numerically plotted phase diagram of this model into a rigorous classification: (i) determine whether $F$ has invariant subsets of $\mathbb{R}^4_+$ other than the trivial ones ( $\mathbb{R}^4_+$ itself and the empty set) and the known ones (the set $A=\{u_1=u_8,u_4=u_5,c=1\}$ , which the source proves invariant in the slice $c=1$ , and the periodic-point sets $\{u:F^p(u)=u\}$ ), and describe all invariant loci analytically, an invariant locus here being a nontrivial algebraic subset of $\mathbb{R}^4_+$ (a locus cut out by equations in $v_1,v_4,v_5,v_8$ , distinct from $\mathbb{R}^4_+$ and the empty set, and mapped into itself by $F$ ), with a complete description classifying the maximal invariant loci and the containment relations among them; the loci $M_1=\{u_1=u_8,u_4=u_5\}$ and $M_2$ are the sets on which the fixed points are known to lie (Lemma 4.1), not invariant subsets of $\mathbb{R}^4_+$ for general parameters; (ii) determine whether positive fixed points of $F$ exist outside the identified loci $M_1$ and $M_2$ , and find them analytically — in particular settle the degenerate case $b=1$ , where the symmetric locus is empty and the source states that finding fixed points analytically is much more difficult — and, as part of the same objective, complete the description of the ferromagnetic fixed points on $M_2$ , which the source could only determine numerically; (iii) determine whether periodic points of $F$ with period $p>2$ , which satisfy high-order algebraic systems (period 2 on $M_1$ already reduces to a degree-6 univariate polynomial), admit an analytic solution (a closed-form or parametric description of the period- $p$ orbits) or a structural classification (explicit criteria or a finite decision procedure that, for each $p>2$ , decides the existence and number of period- $p$ orbits and their dependence on the parameters), thereby characterizing the periodic Gibbs phases beyond period 2, where establishing either branch resolves (iii). The source establishes its completed results (Lemma 4.1, the paramagnetic trichotomy, the period-2 reduction, the $b=1$ discussion) in the slice $c=1$ ( $J_t=0$ ); the three sub-problems (i)–(iii) are each part of the single objective and are posed for the operator $F$ with general positive parameters $(a,b,c)$ ; the $c=1$ slice analyzed in the source is the minimal case. Two further directions are noted in the same paper as remarks rather than as unsolved problems: Remark 5.1 observes that phase boundaries obtained by iterating the recurrences are only approximate and that computing the free energy (e.g. by the Gujrati scheme) 'will not be considered here', and Remark 5.2 observes that the modulated regions call for a detailed analysis via Lyapunov exponents and attractors whose details 'we do not give'; these remarks mark the paper's own non-coverage and are not part of the completion boundary of this problem. The extension of the same analysis to Cayley trees of order $k>3$ and to Cayley-like (chandelier) lattices is explicitly announced by the source as a separate future program and is not part of the completion boundary of this problem.

References

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