Griffiths–Kelly–Sherman-type correlation inequalities for the eight-vertex model with non-uniform vertex weights

Björnberg and Lees (arXiv:2201.00176; Rev. Math. Phys. 35 (2023) 2350029) proved that the uniform eight-vertex model on the square-lattice torus — equivalently, diagonal observables of Kitaev's toric code — satisfies Griffiths–Kelly–Sherman (GKS)-type positive-correlation inequalities: events forcing the vertices of a set $C$ to carry arrow configuration $\eta$ or its complete reversal are positively correlated across two sets $C_1, C_2$ (under a topological side condition). The proof represents the model as the strong-coupling limit of a ferromagnetic four-body Ising model and applies the classical GKS inequalities. Remark 3.2 of the same paper observes that for general vertex weights $(a,b,c,d)$ the generalized quantum Hamiltonians realizing the measure $\mu_{a,b,c,d}$ require sign-indefinite operators, so the non-negativity underlying the proof is lost, and explicitly suggests that the inequalities may fail for non-uniform weights. The open problem is to determine whether these correlation inequalities hold for eight-vertex measures with non-uniform weights: prove them in the weight regimes where they are valid, exhibit explicit finite-volume counterexamples in the regimes where they fail, and thereby characterize the subset of the weight simplex on which the inequalities hold. No published proof or refutation is known to date; the question remains open.

Background

The eight-vertex model lives on a finite square lattice with periodic boundary conditions (a torus). One places an arrow on each edge, and an allowed configuration must satisfy the ice rule: an even number of arrows point into every vertex. There are then eight possible local vertex configurations, denoted I, ..., VIII — the six types of the six-vertex model (two arrows in, two out) plus the all-in sink (VII) and all-out source (VIII). With general vertex weights $(a,b,c,d) \ge 0$ the model is governed by the Gibbs measure $\mu_{a,b,c,d}(\omega) \propto a^{\#I+\#II} b^{\#III+\#IV} c^{\#V+\#VI} d^{\#VII+\#VIII}$, counting how many vertices of each type occur; the uniform model has $a=b=c=d$, and the six-vertex model is the subcase $d=0$. Baxter computed the free energy of the general model exactly, and its phase diagram splits into ferroelectric and antiferroelectric regimes (long-range order in aligned and in staggered vertex patterns, respectively) separated by a disordered (critical, no long-range order) regime; but exact solvability says little about monotonicity or correlation inequalities between events.

For ferromagnetic classical spin systems such inequalities are classical: the Griffiths–Kelly–Sherman (GKS) inequalities state that in an Ising model with non-negative couplings, the expectation of any product of spins is non-negative, and any two such spin monomials are positively correlated. GKS-type inequalities are the standard tool behind monotonicity in couplings and comparison arguments, and they are what Björnberg and Lees transported to the vertex-model setting.

The quantum side of the source is Kitaev's toric code: qubits on the edges, commuting star operators $Z_s$ (product of the diagonal Pauli matrices $\sigma^{(3)}$ on the four edges meeting a vertex $s$) and plaquette operators $X_p$ (product of $\sigma^{(1)}$ around a face), and the Hamiltonian $H = -\sum_p J^x_p X_p - \sum_s J^z_s Z_s$ with $J^x, J^z \ge 0$, whose Gibbs and ground states are the equilibrium states of interest. Björnberg and Lees showed that the expectation of any observable diagonal in the $\sigma^{(3)}$ product basis equals its expectation under a classical Ising model with four-spin (four-body) interaction $P_J(\sigma) \propto \exp(\sum_s J_s \prod_{e \sim s} \sigma_e)$ with $J_s \ge 0$ — a ferromagnetic model, so GKS applies — and that the uniform eight-vertex measure is recovered as all $J_s \to \infty$. Consequently (their Theorem 1.1): for $\eta \in \{I,\dots,VIII\}$, letting $A^\eta_C$ be the event that every vertex in $C$ carries configuration $\eta$ or its complete arrow reversal, and $A^{\eta,\nu}_C$ ($\nu \ne \eta$) the event that every vertex in $C$ carries $\eta$, $\nu$, or their reversals, one has $\mu(A^\eta_{C_1} \cap A^\eta_{C_2}) \ge \mu(A^\eta_{C_1})\mu(A^\eta_{C_2})$ and the analogous inequality for $A^{\eta,\nu}$, provided $C_1 \cup C_2$ is contractible on the torus or all its non-contractible loops have even length; the same holds for the toric-code observables built from the projectors $Q^\epsilon_C$ and $Q^{\epsilon,\delta}_C$.

The obstruction beyond the uniform point is explicit in the source. Ardonne, Fendley and Fradkin constructed generalized quantum Hamiltonians whose ground states realize general eight-vertex measures; Björnberg and Lees note in Remark 3.2 that the operators which must be added to the toric-code Hamiltonian to reproduce the weights $(a,b,c,d)$ do not satisfy the non-negativity properties used in the proofs, so the GKS route breaks down, and that this suggests the inequalities of Theorem 1.1 may not hold for non-uniform weights.

The question matters beyond the uniform model: the Ashkin–Teller model (a pair of interacting Ising copies whose spin products form an eight-vertex configuration) has regimes, for instance with a strongly antiferromagnetic four-spin coupling, in which no correlation inequalities are available — Dober (2025) had to circumvent exactly this absence when analyzing antiferromagnetic Ashkin–Teller regimes — and recent work on classical representations of toric-code Gibbs states (Páez-Velasco et al., preprint 2025, published 2026, via a poly-depth duality — conjugation by a polynomial-depth quantum circuit — to decoupled classical Ising chains) delivers sampling but no ferromagnetic representation of eight-vertex observables and no GKS-type inequality.

Problem Statement

Consider the eight-vertex model on the square-lattice torus $\Lambda$ with general vertex weights $(a,b,c,d) \ge 0$ (not all equal, and including the six-vertex subcase $d=0$), i.e. the measure

$$\mu_{a,b,c,d}(\omega) \propto a^{\#I+\#II}\, b^{\#III+\#IV}\, c^{\#V+\#VI}\, d^{\#VII+\#VIII}$$

on arrow configurations with an even number of arrows into every vertex. For $\eta \in \{I,\dots,VIII\}$ let $A^\eta_C$ denote the event that every vertex in $C \subseteq V$ carries the local arrow configuration $\eta$ or its complete reversal, and for $\nu \ne \eta$ let $A^{\eta,\nu}_C$ denote the event that every vertex in $C$ carries $\eta$, $\nu$, or their complete reversals, exactly as in Theorem 1.1 of Björnberg–Lees. Determine whether, under the same side conditions as in that theorem (namely that $C_1 \cup C_2$ is contractible on the torus, or that every non-contractible loop in $C_1 \cup C_2$ has even length), the inequalities

$$\mu_{a,b,c,d}(A^\eta_{C_1} \cap A^\eta_{C_2}) \ge \mu_{a,b,c,d}(A^\eta_{C_1})\,\mu_{a,b,c,d}(A^\eta_{C_2})$$

and

$$\mu_{a,b,c,d}(A^{\eta,\nu}_{C_1} \cap A^{\eta,\nu}_{C_2}) \ge \mu_{a,b,c,d}(A^{\eta,\nu}_{C_1})\,\mu_{a,b,c,d}(A^{\eta,\nu}_{C_2})$$

hold for non-uniform weights. A complete answer settles the question raised in Remark 3.2 of the source by (i) proving the inequalities in the weight regimes where they are valid — that is, for every finite torus and every admissible choice of $\eta$, $\nu$ and $C_1, C_2$ satisfying the side condition — and (ii) exhibiting explicit finite-volume counterexamples — weights, torus, and sets $C_1, C_2$ — in the regimes where they fail, thereby characterizing the subset of the weight simplex $(a,b,c,d) \ge 0$ on which the inequalities hold; equivalently for the quantum formulation, establishing the corresponding positive-correlation statements (or counterexamples) for diagonal observables of the generalized, Ardonne–Fendley–Fradkin-type quantum Hamiltonians whose ground states realize $\mu_{a,b,c,d}$. Constructing a classical non-negative (ferromagnetic) representation for such Hamiltonians is the natural proof route blocked in the source, and a solution may either construct one or bypass it; proving the inequalities in the valid regimes and exhibiting counterexamples in the failing regimes are the two complementary halves of this single objective — deciding the reach of GKS-type positive correlation beyond the uniform eight-vertex point — and neither alone settles it.

References

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