The Binomial Spin Glass

TL;DR

This paper introduces the binomial spin glass model, unifying discrete and continuous couplings, and explores how the order of limits affects ground-state degeneracies and the relevance of discrete couplings at large scales.

Contribution

It defines the binomial spin glass model and analyzes the impact of the thermodynamic and continuum limits on ground-state entropy and degeneracies.

Findings

Ground-state entropy density is bounded by a function of dimension, coupling parameter, and system size.

A crossover length scale determines when discrete couplings become indistinguishable from Gaussian couplings.

Discrete couplings are irrelevant at large scales in systems with a finite-temperature spin-glass phase.

Abstract

To establish a unified framework for studying both discrete and continuous coupling distributions, we introduce the {\it binomial} spin glass, a class of models where the couplings are sums of $m$ identically distributed Bernoulli random variables. In the continuum limit $m \to \infty$, the class reduces to one with Gaussian couplings, while $m=1$ corresponds to the $\pm J$ spin glass. We demonstrate that for short-range Ising models on $d$-dimensional hypercubic lattices the ground-state entropy density for $N$ spins is bounded from above by $(\sqrt{d/2m} + 1/N)\ln2$, and further show that the actual entropies follow the scaling behavior implied by this bound. We thus uncover a fundamental non-commutativity of the thermodynamic and continuous coupling limits that leads to the presence or absence of degeneracies depending on the precise way the limits are taken. Exact calculations of defect energies reveal a crossover length scale $L^\ast(m) \sim L^\kappa$ below which the binomial spin glass is indistinguishable from the Gaussian system. Since $\kappa = -1/(2\theta)$, where $\theta$ is the spin-stiffness exponent, discrete couplings become irrelevant at large scales for systems with a finite-temperature spin-glass phase.