### Abstract

To establish a unified framework for studying both discrete and continuous coupling distributions, we introduce the binomial spin glass, a class of models where the couplings are sums of m identically distributed Bernoulli random variables. In the continuum limit m → ∞, the class reduces to one with Gaussian couplings, while m = 1 corresponds to the ±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 (p d/2m + 1/N) ln 2, and further show that the actual entropies follow the scaling behavior implied by this bound. We thus uncover a fundamental noncommutativity 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 ∗ (m) ∼ L

κ below which the binomial spin glass is indistinguishable from the Gaussian system. Since κ = −1/(2θ), where θ is the spin-stiffness exponent, discrete couplings become irrelevant at large scales for systems with a finite temperature

spin-glass phase.

for short-range Ising models on d-dimensional hypercubic lattices the ground-state entropy density for N spins is bounded from above by (p d/2m + 1/N) ln 2, and further show that the actual entropies follow the scaling behavior implied by this bound. We thus uncover a fundamental noncommutativity 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 ∗ (m) ∼ L

κ below which the binomial spin glass is indistinguishable from the Gaussian system. Since κ = −1/(2θ), where θ is the spin-stiffness exponent, discrete couplings become irrelevant at large scales for systems with a finite temperature

spin-glass phase.

Original language | English |
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Article number | 080601 |

Number of pages | 6 |

Journal | Physical Review Letters |

Volume | 121 |

Issue number | 8 |

DOIs | |

Publication status | Published - 22 Aug 2018 |

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## Cite this

Vaezi, M-S., Ortiz, G., Weigel, M., & Nussinov, Z. (2018). Binomial Spin Glass.

*Physical Review Letters*,*121*(8), [080601]. https://doi.org/10.1103/PhysRevLett.121.080601