具有稀疏相互作用的一维Hamilton链中的次扩散
Subdiffusion in one-dimensional Hamiltonian chains with sparse interactions
摘要：
我们严格地建立了一类无序一维哈密顿链的输运比扩散慢。这是通过导出能量或粒子流的平衡方差的定量界限作为时间的函数来完成的。缓慢的传输源于稀有绝缘区域（格里菲斯区域）的存在。在多体无序量子链中，它们对应于高度无序的区域，在那里系统处于定域相。相反，我们处理量子和经典无序链的相互作用，通常被称为非谐耦合在经典系统中，是稀疏的。因此，该系统拥有没有相互作用的稀有区域，并且由于链在没有相互作用的情况下是安德森局域化的，因此非相互作用的稀有区域是绝缘的。我们的模型的数学兴趣的一部分是，它是为数不多的不可积模型，其中扩散常数可以严格证明不是无限的。
英文摘要：
We establish rigorously that transport is slower than diffusive for a class of disordered one-dimensional Hamiltonian chains. This is done by deriving quantitative bounds on the variance in equilibrium of the energy or particle current, as a function of time. The slow transport stems from the presence of rare insulating regions (Griffiths regions). In many-body disordered quantum chains, they correspond to regions of anomalously high disorder, where the system is in a localized phase. In contrast, we deal with quantum and classical disordered chains where the interactions, usually referred to as anharmonic couplings in classical systems, are sparse. The system hosts thus rare regions with no interactions and, since the chain is Anderson localized in the absence of interactions, the non-interacting rare regions are insulating. Part of the mathematical interest of our model is that it is one of the few non-integrable models where the diffusion constant can be rigorously proven not to be infinite.
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