We study the statistics of matrix elements of local operators in the basis of energy eigenstates in a paradigmatic, integrable, many-particle quantum theory, the Lieb-Liniger model of bosons with repulsive delta-function interactions. Using methods of quantum integrability, we determine the scaling of matrix elements with system size. As a consequence of the extensive number of conservation laws, the structure of matrix elements is fundamentally different from, and much more intricate than, the predictions of the eigenstate thermalization hypothesis for generic models. We uncover an interesting connection between this structure for local operators in interacting integrable models and the one for local operators that are not local with respect to the elementary excitations in free theories. We find that typical off-diagonal matrix elements ⟨μ|O|λ⟩ in the same macrostate scale as exp(−cOLln(L)−LMμ,λO), where the probability distribution function for Mμ,λO is well described by Fréchet distributions and cO depends only on macrostate information. In contrast, typical off-diagonal matrix elements between two different macrostates scale as exp(−dOL2), where dO depends only on macrostate information. Diagonal matrix elements depend only on macrostate information up to finite-size corrections. Published by the American Physical Society 2024

中文翻译：

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可积模型中局部算子的矩阵元素统计


我们在范式、可积、多粒子量子理论中基于能量特征态研究局部算子矩阵元素的统计，即具有排斥 δ 函数相互作用的玻色子的 Lieb-Liniger 模型。使用量子可积性方法，我们确定矩阵元素随系统大小的缩放。由于大量的守恒定律，矩阵单元的结构与通用模型的特征态热化假说的预测有着根本的不同，并且比一般模型的预测复杂得多。我们揭示了交互可积模型中局部算子的这种结构与自由理论中非局部激励的局部算子的结构之间的有趣联系。我们发现典型的非对角矩阵元素 ⟨μ|O|λ⟩ 与 exp（−cOLln（L）−LMμ，λO） 相同的宏观状态尺度上，其中 Mμ，λO 的概率分布函数由 Fréchet 分布很好地描述，而 cO 仅取决于宏观状态信息。相比之下，两个不同宏态之间典型的非对角矩阵元素缩放为 exp（−dOL2），其中 dO 仅取决于宏态信息。对角矩阵元素仅依赖于宏观状态信息，最高可达有限大小的校正。 美国物理学会 2024 年出版