Hierarchy of Electromagnetic Gauge Field Quantum Geometries
Abstract
The conventional Bloch quantum geometric tensor (QGT) describes electric-dipole geometry for spatially uniform perturbations and vertical Bloch transitions. We develop a broader framework linking quantum geometry to local and nonlocal electromagnetic response in gauge-field space, placing this tensor within a hierarchy of electric and magnetic multipole quantum geometries. The intrinsic antisymmetric adiabatic response in every electromagnetic channel is a Berry curvature in the space of its conjugate fields. The field-dependent projector uniquely fixes the QGT for that channel, pairing the response curvature with a quantum metric. This construction extends to spatially nonuniform perturbations, whose finite momentum transfer induces nonvertical Bloch transitions. We explicitly construct the current--current QGT connecting Bloch states at k±q/2, providing a unified metric and curvature description of nonlocal current response. In the doubled Brillouin zone, tangential derivatives generate conventional momentum-space QGT multipoles, whereas normal derivatives connect finite-momentum geometry to uniform electric- and magnetic-multipole-pair QGTs. We derive this connection through second order in q. The finite-momentum curvature governs the intrinsic nonlocal Hall response, while an inverse-frequency moment of symmetric optical absorption probes the index-symmetric metric sector.
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BibTeX
@misc{wang2026hierarchy,
title = {{Hierarchy of Electromagnetic Gauge Field Quantum Geometries}},
author = {Bin Wang and Fuming Xu and Lei Zhang and Jian Wang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.41610},
eprint = {2610.41610},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.41610},
}
RIS
TY - UNPB
TI - Hierarchy of Electromagnetic Gauge Field Quantum Geometries
AU - Bin Wang
AU - Fuming Xu
AU - Lei Zhang
AU - Jian Wang
PY - 2026
DA - 2026/10/30
PB - arXiv
DO - 10.48550/arxiv.2610.41610
UR - https://arxiv.org/abs/2610.41610
AB - The conventional Bloch quantum geometric tensor (QGT) describes electric-dipole geometry for spatially uniform perturbations and vertical Bloch transitions. We develop a broader framework linking quantum geometry to local and nonlocal electromagnetic response in gauge-field space, placing this tensor within a hierarchy of electric and magnetic multipole quantum geometries. The intrinsic antisymmetric adiabatic response in every electromagnetic channel is a Berry curvature in the space of its conjugate fields. The field-dependent projector uniquely fixes the QGT for that channel, pairing the response curvature with a quantum metric. This construction extends to spatially nonuniform perturbations, whose finite momentum transfer induces nonvertical Bloch transitions. We explicitly construct the current--current QGT connecting Bloch states at k±q/2, providing a unified metric and curvature description of nonlocal current response. In the doubled Brillouin zone, tangential derivatives generate conventional momentum-space QGT multipoles, whereas normal derivatives connect finite-momentum geometry to uniform electric- and magnetic-multipole-pair QGTs. We derive this connection through second order in q. The finite-momentum curvature governs the intrinsic nonlocal Hall response, while an inverse-frequency moment of symmetric optical absorption probes the index-symmetric metric sector.
ER -