Department of Physics, Washington University, St. Louis, Missouri 63160, USA.
Department of Physics, Indiana University, Bloomington, Indiana 47405, USA.
Phys Rev Lett. 2018 Aug 24;121(8):080601. doi: 10.1103/PhysRevLett.121.080601.
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 (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 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.
为了建立一个统一的框架来研究离散和连续耦合分布，我们引入了二项式 spin glass，这是一类模型，其中的耦合是 m 个相同分布的伯努利随机变量的和。在 m→∞的连续极限下，该类模型简化为具有高斯耦合的模型，而 m=1 对应于±J spin glass。我们证明，对于 d 维超立方格上的短程 Ising 模型，N 个自旋的基态熵密度上限为(sqrt[d/2m]+1/N)ln2，进一步表明实际熵遵循这个界所暗示的标度行为。因此，我们揭示了热力学和连续耦合极限之间的基本非交换性，这导致了简并的存在或不存在取决于极限的精确取法。缺陷能的精确计算揭示了在交叉长度标度 L^{*}(m)∼L^{κ}以下，二项式 spin glass 与高斯系统无法区分。由于 κ=-1/(2θ)，其中θ是自旋刚性指数，对于具有有限温度 spin-glass 相的系统，离散耦合在大尺度上变得无关紧要。
