Research on replica symmetry breaking in sparsely connected spin glass models

这项研究的目的是探索在稀疏连接的自旋玻璃模型中观察到的复制对称破缺(RSB)现象。特别地，我们寻找了表示RSB局域不稳定性的De Almeida-Thouless(AT)条件的一般表达式，并考察了稀疏连接模型的RSB相的性质。所获得的结果概括如下：1.利用信念传播算法对自旋玻璃磁化率进行了计算，得到了AT条件的一般表达式。我们还发现AT条件是由一个服从大偏差统计的随机过程决定的，并发展了一种评估表征随机过程的大偏差率函数的方法。提出了一种高效搜索Bethe逼近解的算法。我们使用该算法对Bethe自由能进行了Hessian分析，发现当外场相对较小时，Bethe近似的解很可能符合预期的有限尺寸标度关系，尽管临界点的数值确定很困难。对于基于马尔可夫链蒙特卡罗的临界点的评估，我们考察了三个度量的性质，即磁化率矩阵A的主本征值、自旋玻璃磁化率矩阵ASG和自旋玻璃磁化率XSG的主本征值。我们发现，对于A和ASG，由于有限尺寸效应，某些偏差校正是必不可少的，尽管ASG在数据精度方面是最受欢迎的。用复制法考察了其在消失温度范围内的性能。分析表明，基态具有一步RSB解的特征。这一性质与全连通模型不同，全连通模型的零温态一般用全RSB解来描述。

The objective of this research is to explore the replica symmetry breaking (RSB) phenomena observed in sparsely connected spin glass models of Ising spins. In particular, we seek for a general expression of the de Almeida-Thouless (AT) condition which signals the local instability for RSB and examine properties of the RSB phase of the sparsely-connected models. The obtained results are summarized as follows.1. A general expression of the AT condition was obtained by evaluation of the spin glass susceptibility utilizing the belief propagation algorithm. We also found that the AT condition is determined by a stochastic process that is subject to a large deviation statistics, and developed a methodology to assess the large deviation rate function that characterizes the stochastic process.2. An algorithm to efficiently search Bethe approximation solutions is developed. We performed the Hessian analysis of the Bethe free energy using this algorithm and found that an expected finite size scaling relation is likely to hold for solutions of the Bethe approximation when the external field is relatively small although numerical determination of the critical point is difficult.3. For Markov Chain Monte Carlo-based assessment of the critical point, we examined properties of three measures, the principal eigen value of the susceptibility matrix A, that of the spin glass susceptibility matrix ASG and the spin glass susceptibility XSG. We found that for A and ASG, certain bias corrections are indispensable due to finite-size effect although ASG is most preferred in terms of accuracy of data.4. Properties in the limit of vanishing temperature was examined by using the replica method. The analysis indicates that the ground states are characterized by the one step RSB solution. This property is in contrast with that of the fully connected models, the zero temperature states of which are generally described by the full RSB solution.