The Eggbox Ising Model

TL;DR

The Eggbox Ising model is a flexible framework for creating complex energy landscapes with controllable hierarchical structures, enabling the study of phase transitions and metastability in disordered systems.

Contribution

It introduces a novel tunable Ising model with rugged landscapes that can realize arbitrary replica-symmetry-breaking structures and analyze phase transition behaviors.

Findings

Realizes arbitrary k-step replica-symmetry-breaking structures.

Induces discontinuous finite-temperature transitions with metastability.

Provides a systematic expansion to Hopfield-type couplings.

Abstract

We introduce the Eggbox Ising model, a tunable construction of rugged energy landscapes defined by distances to a prescribed set of patterns. Correlated pattern ensembles realize arbitrary k-step replica-symmetry-breaking structures and controllable Parisi overlap distributions p(q), consistent with the hierarchical overlap structure observed in a simple word-embedding example from empirical data. A softened variant allows a systematic expansion leading to Hopfield-type couplings (and higher-body terms). We analyze the density of states and show that suitable potentials induce discontinuous finite-temperature transitions with metastability and hysteresis.