Typical versus averaged overlap distribution in spin glasses: Evidence for droplet scaling theory
Typical versus averaged overlap distribution in spin glasses: Evidence for droplet scaling theory
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自旋玻璃中的典型重叠分布与平均重叠分布：液滴缩放理论的证据
DOI：
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发表时间：
2013
期刊：
影响因子：
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通讯作者：
T. Garel
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文献类型：
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作者：
C. Monthus;T. Garel
我们考虑了在自旋玻璃中起序参数作用的重叠分布的无序样本的统计性质。我们证明了在接近零的温度下(I)在$-1<；q<；1$：$P^{typ}(Q)=e^{ar{ln P_{cal J}(Q)}}sim e^{-etA N^{heta}Phi(Q)}$的中心区域内，{it典型}重叠分布指数是指数小的，其中$heta$是这里定义的关于自旋总数$N$的液滴指数(为了也考虑不存在长度概念的全连通模型)；(Ii)重新标度变量$v=-(ln P_(Cal J)(Q))/N^(Heta)$仍然是描述样本到样本波动的O(1)随机正变量；(Iii)平均分布$Ar{P_(Cal J)(Q)}$是非典型的，且以罕见异常样本为主。类似的陈述适用于累积重叠分布$I_{cal J}(Q_0)Equiv int_{0}^{Q_0}dq P_{cal J}(Q)$。这些结果是对于球面平均场模型显式导出的，其中$heta=1/3$，$Phi(Q)=1-q^2$，且随机变量$v$对应于GOE随机矩阵的两个最大本征值之间的重新标度差。然后，我们数值比较了长程一维伊辛自旋玻璃和平均场SK模型(形式上对应于先前模型的$sigma=0的极限)在不同指数$sigma$的不同指数$sigma$下，随机耦合衰减为$J(R)到r^{-sigma}$的典型重叠分布和平均重叠分布。我们的结论是，未来对自旋玻璃的研究应该测量重叠分布或累积重叠分布的{It典型}值，以获得关于自旋玻璃相性质的更明确的结论。
We consider the statistical properties over disordered samples of the overlap distribution $P_{cal J}(q)$ which plays the role of an order parameter in spin-glasses. We show that near zero temperature (i) the {it typical} overlap distribution is exponentially small in the central region of $-1<q<1$: $ P^{typ}(q) = e^{ar{ln P_{cal J}(q)}} sim e^{- eta N^{ heta} phi(q)} $, where $ heta$ is the droplet exponent defined here with respect to the total number $N$ of spins (in order to consider also fully connected models where the notion of length does not exist); (ii) the rescaled variable $v = - (ln P_{cal J}(q))/N^{ heta}$ remains an O(1) random positive variable describing sample-to sample fluctuations; (iii) the averaged distribution $ar{P_{cal J}(q)} $ is non-typical and dominated by rare anomalous samples. Similar statements hold for the cumulative overlap distribution $I_{cal J}(q_0) equiv int_{0}^{q_0} dq P_{cal J}(q) $. These results are derived explicitly for the spherical mean-field model with $ heta=1/3$, $phi(q)=1-q^2 $, and the random variable $v$ corresponds to the rescaled difference between the two largest eigenvalues of GOE random matrices. Then we compare numerically the typical and averaged overlap distributions for the long-ranged one-dimensional Ising spin-glass with random couplings decaying as $J(r) propto r^{-sigma}$ for various values of the exponent $sigma$, corresponding to various droplet exponents $ heta(sigma)$, and for the mean-field SK-model (corresponding formally to the $sigma=0$ limit of the previous model). Our conclusion is that future studies on spin-glasses should measure the {it typical} values of the overlap distribution or of the cumulative overlap distribution to obtain clearer conclusions on the nature of the spin-glass phase.