Long-range order in discrete spin systems

TL;DR

This work proves long-range order for a broad class of discrete spin systems on $\mathbb{Z}^d$ with nearest-neighbor interactions under a symmetry assumption, valid in sufficiently high dimensions and in certain lattice-augmented settings. The authors develop an entropy-contour framework that identifies ordered regions via dominant patterns and analyzes breakups through atlas constructions, a repair transformation, and Shearer-type entropy bounds. They establish the existence and complete characterization of maximal-pressure Gibbs states: for each dominant pattern $(A,B)$ there is an extremal periodic Gibbs state $\mu_{(A,B)}$, and every periodic maximal-pressure Gibbs state is a mixture of these states; in high dimensions, the number of extremal states matches the number of dominant patterns. The results apply to a wide array of models, including the antiferromagnetic Potts, beach, clock models, Lipschitz height functions, hard-core, Widom–Rowlinson, and their multi-type extensions, and they refine topological pressure formulas in the high-dimensional limit. The methods yield quantitative thresholds and rate bounds, enabling precise phase diagrams in many cases and connecting to conjectured pressure expansions for homomorphism models. Overall, the paper provides a robust, general mechanism for proving long-range order and classifying equilibrium states in complex lattice spin systems, with broad applicability and sharp high-dimensional predictions.

Abstract

We establish long-range order for discrete nearest-neighbor spin systems on $\mathbb{Z}^d$ satisfying a certain symmetry assumption, when the dimension $d$ is higher than an explicitly described threshold. The results characterize all periodic, maximal-pressure Gibbs states of the system. The results further apply in low dimensions provided that the lattice $\mathbb{Z}^d$ is replaced by $\mathbb{Z}^{d_1}\times\mathbb{T}^{d_2}$ with $d_1\ge 2$ and $d=d_1+d_2$ sufficiently high, where $\mathbb{T}$ is a cycle of even length. Applications to specific systems are discussed in detail and models for which new results are provided include the antiferromagnetic Potts model, Lipschitz height functions, and the hard-core, Widom--Rowlinson and beach models and their multi-type extensions. We also establish a formula conjectured by Jenssen and Keevash for the topological pressure in the high-dimensional limit.

Figures (7)

• Figure 1: Graph representations of the three models discussed as first applications. The edges correspond to the pairs of states $\{i,j\}$ with maximal pair interaction $\lambda_{i,j}$.
• Figure 2: Graph representations of the hard-core and Widom--Rowlinson models. The edges correspond to the pairs of states $\{i,j\}$ with maximal pair interaction $\lambda_{i,j}$.
• Figure 3: The multi-type models.
• Figure 4: Some parameter values for various models.
• Figure 5: The phase diagram of the high-dimensional Ising antiferromagnet at positive external magnetic field. Uniqueness of the Gibbs state is known in the green region, defined by \ref{['eq:Dobrushin uniqueness for AF Ising']} and \ref{['eq:disagreement percolation for AF Ising']}. The existence of multiple Gibbs states is known in the blue region, defined by \ref{['eq:Dobrushin multiplicity for AF Ising']}, while our results prove it also in the orange region (see \ref{['eq:new multiplicity regime for AF Ising']}), establishing that for $h$ just above $2d$ the model undergoes two phase transitions as the temperature increases (the "bulging phenomenon"). A possibility for the region of multiplicity is sketched by the diagonal lines.

Theorems & Definitions (94)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 2.1
• Theorem 2.2
• Theorem 2.3
• Lemma 4.1: feldheim2016growth
• Lemma 4.2
• Lemma 4.3