The Eggbox Ising Model

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.

Figures (12)

• Figure 1: Iterative evolution from the $1$-RSB to the $k$-RSB Eggbox Ising model. (a) Initially ($k=1$), each $\boldsymbol{\xi}^\alpha$ undergoes random spin sampling, represented in green for clarity in subsequent stages; this process is equally applicable to other $\boldsymbol{\xi}^\beta$'s (navy blue configuration on the right). In advancing to the next ($k=2$) level, half of the spins remain unchanged with the remainder being resampled (red and blue spins). Progressing further to the third level ($k=3$), this bifurcation is repeated, with half of either the red or blue spins being held fixed while the others are resampled. (b) The Parisi overlap distribution $P(q)$ for $k=1,2,3$ (left to right) simulated on $100$ disorder realizations with $N=2048, M_0=256$ using $500,000$ samples for each disorder realization. The red traces show the Parisi overlap distribution for one realization; this histogram is almost identically covered the disorder-averaged Parisi overlap distribution. The equivalence between the latter two distributions illustrates the self-averaging for disorder realizations that are generated via the same process with fixed parameters ($M_0,c,k$).
• Figure 2: An illustrative overlap matrix computed from binarized word embeddings for eight English words. The matrix exhibits a hierarchical structure: two coarse semantic clusters (clothing vs. emotions) and finer sub-clusters within each group. This mirrors the multi-level organization of overlaps encountered in $k$-RSB structures.
• Figure 3: Normalized density of states $n(u)\equiv N\omega(Nu)$ versus the energy density $u\equiv \frac{E}{N}$ with $\int_0^\infty n(u)du= \int_0^\infty \omega(Nu)d(Nu)\overset{E=Nu}{=}1$. For each $M_0$ and each $k$, the $500,000$ configurations were sampled from a $N=512$ system. From right to left, we have $M_0=16,k=1$; $M_0=16,k=20$; $M_0=256,k=1$. The histograms for varying $M_0$ and $k$ match the theoretical curves.
• Figure 4: There is no finite temperature phase transition for $V(d)=d$ and $V(d)=d^2$. Here, we plot the internal energy density (energy per spin) as a function of the inverse temperature $\beta$. The displayed data were obtained from configurations that were sampled across systems of size $N=1024$, while fixing $M=2$ and the ratio $a\equiv \frac{d_0}{N}=\frac{1}{4}$. To ensure thermalization, at every temperature, each sample of a single disorder realization underwent $110$ Metropolis-Hasting sweeps from random configurations. Each sweep includes $N$ flipping attempts. Subsequently, the internal energy density was calculated for $100$ samples. For the thermodynamic limit $N=\infty$, the internal energy was determined through theoretical computations (Eqns. (\ref{['eq:freeminima']}, \ref{['eq:piecewise']})).
• Figure 5: Sketches for the examples of potential $V(d)$. From left to right: the first and second ones are both piecewise linear potentials (Eqn. (\ref{['eq:piecewise']})) with $\gamma<1$ (type (I)) and $\gamma>1$ (type (II)) respectively. The third one $V(d)= \frac{\pi d^2}{2N}-\frac{N}{4\pi}\cos\left( \frac{4\pi d}{N}\right)$ is more analytical, behaving similarly as type (I) piecewise linear potential in terms of the phase transition.