Dual Quantum Geometric Tensors and Local Topological Invariant

TL;DR

This paper introduces a non-Hermitian extension of the quantum geometric tensor, revealing new topological and transport phenomena in Dirac systems through a unified geometric framework.

Contribution

It develops a non-Hermitian Zeeman quantum geometric tensor, decomposes it into normal and anomalous sectors, and links it to local Dirac-node topology and measurable transport responses.

Findings

The Zeeman QGT is generically non-Hermitian with distinct sectors.

Anomalous curvature exhibits a radial flux singularity linked to Dirac node topology.

Transport signatures like gyrotropic conductivity reflect the underlying geometric structure.

Abstract

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.