本文通过解析和数值计算表明，一般的哈密顿量（例如，一个大的随机矩阵）可以近似地写成精度很高的两体相互作用项的线性组合。也就是说，哈密顿函数在一个精心选择的基中是2-局部的。此外，研究表明这些哈密顿量不是微调的，这意味着光谱对耦合常数的扰动是鲁棒的。最后，研究人员通过分析耦合Jij的邻接结构，提出了量子混沌中几何局部性产生的可能机制。
据悉，量子多体系统通常具有张量积结构。这种结构是从概率论中继承而来的，在概率论中，两个独立事件的概率是两个概率的乘积。因此，哈密顿量的张量积结构将系统自然分解为独立的较小的子系统。了解一个给定的哈密顿量是否与某个特定的张量积结构相容是非常有趣的。特别是，人们会问，是否存在一个基，其中任意哈密顿量都具有2-局部形式，即它只包含成对相互作用。
附：英文原文
Title: Unveiling order from chaos by approximate 2-localization of random matrices
Author: Loizeau, Nicolas, Morone, Flaviano, Sels, Dries
Issue&Volume: 2023-9-19
Abstract: Quantum many-body systems are typically endowed with a tensor product structure. A structure they inherited from probability theory, where the probability of two independent events is the product of the probabilities. The tensor product structure of a Hamiltonian thus gives a natural decomposition of the system into independent smaller subsystems. It is interesting to understand whether a given Hamiltonian is compatible with some particular tensor product structure. In particular, we ask, is there a basis in which an arbitrary Hamiltonian has a 2-local form, i.e., it contains only pairwise interactions Here we show, using analytical and numerical calculations, that a generic Hamiltonian (e.g., a large random matrix) can be approximately written as a linear combination of two-body interaction terms with high precision; that is, the Hamiltonian is 2-local in a carefully chosen basis. Moreover, we show that these Hamiltonians are not fine-tuned, meaning that the spectrum is robust against perturbations of the coupling constants. Finally, by analyzing the adjacency structure of the couplings Jij, we suggest a possible mechanism for the emergence of geometric locality from quantum chaos.
DOI: 10.1073/pnas.2308006120
Source: https://www.pnas.org/doi/abs/10.1073/pnas.2308006120