自旋-1/2 Kitaev-Heisenberg模型中基分辨低能波函数的协方差几何
Covariance Geometry of Basis-Resolved Low-Energy Wavefunctions in the Spin-$1/2$ Kitaev--Heisenberg Model
AI总结：
研究自旋-1/2 Kitaev-Heisenberg模型低能多体波函数组织，开发基分辨协方差框架，用主成分分析识别主导协方差模式，发现相图上协方差几何的系统演化，为表征受挫量子多体系统提供新视角。
AI中文摘要：
我们开发了一个基分辨协方差框架，用于研究自旋-1/2 Kitaev-Heisenberg模型中低能多体波函数的组织。通过从低能态的局域自旋、键关联和晶格通量表示构建协方差矩阵，采用主成分分析（PCA）来识别主导的集体协方差模式。我们发现协方差几何在相图上有系统的演化：磁有序相由基本上一维的协方差流形描述，传统磁相边界表现出主导协方差模式之间的竞争，而Kitaev区域发展出内在的多维协方差几何。此外，同一多体波函数在不同算符表示中产生不同的协方差几何，表明协方差几何由量子态和用于探测它的物理可观测量共同决定。香农熵和参与率提供了这种演化的定量度量。本框架将基分辨协方差几何确立为一种补充统计视角，用于表征超越传统序参量的受挫量子多体系统。
英文摘要：
We develop a basis-resolved covariance framework for investigating the organization of low-energy many-body wavefunctions in the spin-$1/2$ Kitaev--Heisenberg model. By constructing covariance matrices from local-spin, bond-correlation, and plaquette-flux representations of the low-energy states, Principal Component Analysis (PCA) is employed to identify the dominant collective covariance modes. We find a systematic evolution of the covariance geometry across the phase diagram: magnetically ordered phases are described by an essentially one-dimensional covariance manifold, conventional magnetic phase boundaries exhibit competition between leading covariance modes, whereas the Kitaev regimes develop intrinsically multidimensional covariance geometry. Furthermore, the same many-body wavefunction produces distinct covariance geometries in different operator representations, demonstrating that covariance geometry is determined jointly by the quantum state and the physical observables used to probe it. Shannon entropy and the participation ratio provide quantitative measures of this evolution. The present framework establishes basis-resolved covariance geometry as a complementary statistical perspective for characterizing frustrated quantum many-body systems beyond conventional order parameters.
