The Binomial Spin Glass

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.