Relative hybridization textures as local coordinates for band geometry and topology
arXiv:2607.25673
The paper introduces a projector-level diagnostic called the relative hybridization coordinate Z, which links global band geometry quantities such as Berry curvature and quantum metric to local orbital and spin degrees of freedom, and demonstrates its use in diagnosing Chern and Z₂ topological phases in the QWZ and BHZ lattice models.
Abstract
Global diagnostics such as Berry curvature and quantum metrics characterize the geometry and topology of an occupied Bloch subspace, leaving the microscopic sectors that carry this structure implicit. We introduce the relative hybridization coordinate as a projector-level diagnostic connecting these global quantities to local degrees of freedom. As the Grassmann graph coordinate relative to a chosen sector, reconstructs the local projector and retains the phase and matrix orientation absent from ordinary weight or fat-band descriptions. On valid chart patches, its momentum-space texture encodes Berry curvature, quantum metric, Berry phases, and Wilson loops, while chart obstructions appear as rank-drop defects whose balanced-chart winding of gives the first Chern number. In the QWZ model this defect inventory reproduces the Chern phase diagram. In the lattice BHZ model, matrix diagnoses the orbital partition as a robust matched chart for the QSH geometry, while the spin partition remains essential to the block and interpretation and shows rank deficiency as a matched chart in the spin-conserving limit. The relative hybridization coordinate thus provides a sector-resolved framework for relating band geometry and topology to microscopic structure.