The grammar of the Ising model: A new complexity hierarchy

TL;DR

The paper introduces a language-based complexity measure for Ising models by encoding the Hamiltonian's function graph into a formal language ${\sf L}_{\mathcal{M}}$ and associating an edge-language ${\sf E}_{\mathcal{M}}$ to the interaction graphs. It proves a complete classification: ${\sf L}_{\mathcal{M}}$ is regular iff the model is finite; constructive context free iff the model is linear and ${\sf E}_{\mathcal{M}}$ is regular; constructive context sensitive iff ${\sf E}_{\mathcal{M}}$ is context sensitive; and decidable iff ${\sf E}_{\mathcal{M}}$ is decidable. The framework is applied to 1d, circular, ladder, layerwise complete, 2d, all-to-all, and perfect binary-tree Ising models, yielding constructive context-free classifications for the former group and constructive context-sensitive classifications for the latter. This grammar-based lens reveals how local interaction structure governs a hierarchy of representational complexities, offering a new angle on universality and the grammar of physical interactions. The work lays groundwork for extending grammar-based analyses to broader spin models and graph-grammar formulations, linking physical specification to formal-language complexity.

Abstract

How complex is an Ising model? Usually, this is measured by the computational complexity of its ground state energy problem. Yet, this complexity measure only distinguishes between planar and non-planar interaction graphs, and thus fails to capture properties such as the average node degree, the number of long range interactions, or the dimensionality of the lattice. Herein, we introduce a new complexity measure for Ising models and thoroughly classify Ising models with respect to it. Specifically, given an Ising model we consider the decision problem corresponding to the function graph of its Hamiltonian, and classify this problem in the Chomsky hierarchy. We prove that the language of this decision problem is (i) regular if and only if the Ising model is finite, (ii) constructive context free if and only if the Ising model is linear and its edge language is regular, (iii) constructive context sensitive if and only if the edge language of the Ising model is context sensitive, and (iv) decidable if and only if the edge language of the Ising model is decidable. We apply this theorem to show that the 1d Ising model, the Ising model on generalised ladder graphs, and the Ising model on layerwise complete graphs are constructive context free, while the 2d Ising model, the all-to-all Ising model, and the Ising model on perfect binary trees are constructive context sensitive. This work is a first step in the characterisation of physical interactions in terms of grammars.

Figures (5)

• Figure 1: (Left and Middle) We fully classify the properties of an Ising model $\mathcal{M}$ that determines the complexity of its language ${\sf L}_{\mathcal{M}}$ in the refined Chomsky hierarchy (\ref{['thm:main']}). (Right) We apply this classification to show that the language of the 1d Ising model is constructive context free; the language of the Ising model on perfect binary trees, 2d lattices, and all-to-all interaction graphs is constructive context sensitive (\ref{['sec:examples']}).
• Figure 2: Interaction graph of a limited Ising model (\ref{['def:PropIsing']}\ref{['def:limited']}). Given any vertex $i$ not within the first or last $b$ vertices, all incident edges are either connected to a vertex $j$ with $\vert j-i\vert \leq b$, i.e. to a vertex within the same block (pale yellow block), or to one of the first or last $b$ vertices (pale purple blocks).
• Figure 3: Set diagram of the properties of Ising models. The purple shapes correspond to properties that capture the complexity of the family of interaction graphs, phrased in terms of the complexity of ${\sf E}_{\mathcal{M}}$ in the Chomsky hierarchy. The red shapes correspond to properties that capture the scaling of the connectivity and the number of long-range interactions with the system size. The right-hand side shows where several examples are located in this diagram. Note that both the 2d Ising model and the all-to-all Ising model are constructive context sensitive; yet, the all-to-all model has regular ${\sf E}_{\mathcal{M}}$ and is not linear (and thus fails to be constructive context free), whereas the 2d model has context sensitive ${\sf E}_{\mathcal{M}}$ and is linear.
• Figure 4: Interaction graphs of $\mathcal{M}_{\mathrm{1d}}$ (\ref{['fig:1d']}), $\mathcal{M}_{\mathrm{circ}}$ (\ref{['fig:circle']}), $\mathcal{M}_{\mathrm{ladder}}$ (\ref{['fig:ladder']}), and $\mathcal{M}_{\mathrm{layer}}$ (\ref{['fig:network']}). All these Ising models are constructive context free. Intuitively, this is because their interaction graphs all have one distinguished dimension along which an elementary building block (that contains a constant number $k$ spins) is repeated ($i$ times) in a periodic fashion. In (\ref{['fig:1d']}) and (\ref{['fig:circle']}) there is only one dimension, in (\ref{['fig:ladder']}) and (\ref{['fig:network']}) the distinguished dimension is indicated as"direction of growth". This property is made precise in \ref{['thm:main']}\ref{['thm:main:cf1']}: constructive context free Ising models are uniquely characterised by ${\sf E}_{\mathcal{M}}$ being regular and $\mathcal{M}$ being linear (or limited according to \ref{['ssec:proofmain2']}\ref{['thm:main:cf1:iii']}).
• Figure 5: Interaction graphs of $\mathcal{M}_{\mathrm{all}}$ (\ref{['fig:all2all']}) and $\mathcal{M}_{\mathrm{tree}}$ (\ref{['fig:tree']}). (\ref{['fig:tree']}) shows a perfect binary tree that contains $2^n-1$ vertices, and thus consists of $n$ individual levels. Increasing the system size in $\mathcal{M}_{\mathrm{tree}}$ adds entire levels to the tree. These Ising models, as well as $\mathcal{M}_{\mathrm{2d}}$, have constructive context sensitive language. $\mathcal{M}_{\mathrm{all}}$ has regular edge language, but fails to be constructive context free as it is not linear. In contrast, $\mathcal{M}_{\mathrm{tree}}$ and $\mathcal{M}_{\mathrm{2d}}$ have context sensitive edge languages and are linear. \ref{['thm:main']}\ref{['thm:main:ccs']} states that the crucial property for ${\sf L}_{\mathcal{M}}$ to be constructive context sensitive is that ${\sf E}_{\mathcal{M}}$ be context sensitive.

Theorems & Definitions (20)

• Definition 1: Ising model
• Definition 2: Language of an Ising model
• Definition 3: Constructive automaton
• Conjecture 1
• Proposition 1: Refined Chomsky hierarchy
• proof
• Definition 4: Complexity measure
• Definition 5: Edge language
• Proposition 2: Uniqueness of the edge language
• proof