Dual Quantum Geometric Tensors and Local Topological Invariant

Quantum Physics [Submitted on 9 Apr 2026] Dual Quantum Geometric Tensors and Local Topological Invariant Rongjie Cui, Longjun Xiang, Fuming Xu, Jian Wang The conventional quantum geometric tensor (QGT) is Hermitian, with a real symmetric quantum metric and an imaginary antisymmetric Berry curvature. We show that the Zeeman QGT is generically non-Hermitian and admits a natural decomposition into normal and anomalous metric-curvature sectors. The normal sector reduces to the conventional Hermitian structure, whereas the anomalous sector contains an imaginary symmetric metric-like tensor and a real antisymmetric curvature-like tensor with no counterpart in the standard QGT. In a two-dimensional Dirac system, the anomalous Zeeman curvature develops a radial flux singularity that is Hodge-dual to the tangential winding field of the Dirac node. This recasts the same local \pi_1 topology into a curvature-flux language, analogous to the flux representation of global \pi_2 topology by the conventional Berry curvature. At the level of linear response, the four symmetry-resolved components of the gyrotropic conductivity are in one-to-one correspondence with the four components of the Zeeman QGT, while their distinct low-frequency scalings provide an additional diagnostic for isolating the underlying geometric sector. The reciprocal kinetic magnetoelectric response offers a complementary experimental route to probe the same structure. These results establish a unified framework connecting non-Hermitian Zeeman quantum geometry, local Dirac-node topology, and measurable transport signatures. Subjects: Quantum Physics (quant-ph); Mesoscale and Nanoscale Physics (cond-mat.mes-hall); Materials Science (cond-mat.mtrl-sci) Cite as: arXiv:2604.09725 [quant-ph]   (or arXiv:2604.09725v1 [quant-ph] for this version)   https://doi.org/10.48550/arXiv.2604.09725 Focus to learn more Submission history From: Longjun Xiang [view email] [v1] Thu, 9 Apr 2026 08:00:50 UTC (139 KB) Access Paper: HTML (experimental) view license Current browse context: quant-ph < prev   |   next > new | recent | 2026-04 Change to browse by: cond-mat cond-mat.mes-hall cond-mat.mtrl-sci References & Citations INSPIRE HEP NASA ADS Google Scholar Semantic Scholar Export BibTeX Citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer (What is the Explorer?) Connected Papers Toggle Connected Papers (What is Connected Papers?) Litmaps Toggle Litmaps (What is Litmaps?) scite.ai Toggle scite Smart Citations (What are Smart Citations?) Code, Data, Media Demos Related Papers About arXivLabs Which authors of this paper are endorsers? | Disable MathJax (What is MathJax?)