短程自旋玻璃模型的Griffiths型定理
Griffiths-type theorems for short-range spin glass models
AI总结：
本文将Griffiths定理推广到短程自旋玻璃模型，证明复制品重叠分布展宽、两复制品自由能对耦合参数的不可微性、非零Edwards-Anderson序参量以及三复制品置换对称性自发破缺之间的等价关系，并推广到长程模型。
摘要：
我们建立了自旋玻璃模型中序的不同刻画之间的关系。我们首先证明，复制品重叠分布由复制品重叠$R^{1,2}$的非零标准差所指示的展宽，意味着两复制品自由能对复制品耦合参数$λ$的不可微性。在诸如标准Edwards-Anderson模型这样的$\mathbb Z_2$不变模型中，这种不可微性等价于由非零Edwards-Anderson序参量所刻画的自旋玻璃序。这一Griffiths定理的推广被证明适用于任何具有经典有界自旋的短程自旋玻璃模型。我们还证明，上述两复制品自由能的不可微性意味着字面意义上的复制品对称破缺，即在具有三个复制品的模型中置换对称性的自发破缺。这是一个一般性结果，适用于一大类随机自旋模型，包括诸如Sherrington-Kirkpatrick模型和随机能量模型这样的长程模型。
英文摘要：
We establish relations between different characterizations of order in spin glass models. We first prove that the broadening of the replica overlap distribution indicated by a nonzero standard deviation of the replica overlap $R^{1,2}$ implies the non-differentiability of the two-replica free energy with respect to the replica coupling parameter $λ$. In $\mathbb Z_2$ invariant models such as the standard Edwards-Anderson model, the non-differentiability is equivalent to the spin glass order characterized by a nonzero Edwards-Anderson order parameter. This generalization of Griffiths' theorem is proved for any short-range spin glass models with classical bounded spins. We also prove that the non-differentiability of the two-replica free energy mentioned above implies replica symmetry breaking in the literal sense, i.e., a spontaneous breakdown of the permutation symmetry in the model with three replicas. This is a general result that applies to a large class of random spin models, including long-range models such as the Sherrington-Kirkpatrick model and the random energy model.
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