Covariance Geometry of Basis-Resolved Low-Energy Wavefunctions in the Spin-$1/2$ Kitaev--Heisenberg Model

Abstract

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.