一个简单的一维玻璃态Kac模型
A simple one dimensional glassy Kac model
摘要：
我们定义了一个新的家庭的随机自旋模型的一维结构，有限范围的多自旋相互作用，和有界的平均度（在每个自旋参与的相互作用的数量）。无阻基态可以描述为稀疏的带对角线性系统的解，从而允许有效的数值分析。在无限大的相互作用范围内，我们恢复了所谓的XORSAT（稀释p自旋）模型，即已知的经历一个随机的一级相变的平均度的增加。在这里，我们调查一个大的，但有限的相互作用范围内的最重要的后果：（一）涨落引起的修正热力学量;（二）需要一个非均匀的（位置相关）序参量;（三）出现一个有限的马赛克长度尺度。特别是，我们研究的相关长度发散（平均场）玻璃化转变。
英文摘要：
We define a new family of random spin models with one-dimensional structure, finite-range multi-spin interactions, and bounded average degree (number of interactions in which each spin participates). Unfrustrated ground states can be described as solutions of a sparse, band diagonal linear system, thus allowing for efficient numerical analysis. In the limit of infinite interaction range, we recover the so-called XORSAT (diluted p-spin) model, that is known to undergo a random first order phase transition as the average degree is increased. Here we investigate the most important consequences of a large but finite interaction range: (i) Fluctuation-induced corrections to thermodynamic quantities; (ii) The need of an inhomogeneous (position dependent) order parameter; (iii) The emergence of a finite mosaic length scale. In particular, we study the correlation length divergence at the (mean-field) glass transition.
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