The Binomial Spin Glass

To establish a unified framework for studying both discrete and continuous coupling distributions, we introduce the {\it binomial} spin glass. In this model, the couplings are the sum of $m$ identically distributed Bernoulli random variables. In the continuum limit $m \to \infty$, this system reduces to a model with Gaussian couplings, while $m=1$ corresponds to the $\pm J$ spin glass. We show that for short-range Ising models on $d$-dimensional hypercubic lattices the ground-state entropy density for $N$ spins is bounded from above by $(\sqrt{d/2m} + 1/N) \ln2$ confirming that for Gaussian couplings the degeneracy is not extensive. We further argue that the entropy density scales, for large $m$, as $1/\sqrt{m}$. Exact calculations of the defect energies reveal the presence of a crossover length scale $L^\ast(m)$ below which the binomial spin glass is indistinguishable from the Gaussian system. These results highlight the non-commutativity of the thermodynamic and continuous coupling limits.