Magnetoelectric response arising from Zeeman quantum geometry

The quantum metric is a foundational quantity for Bloch states, yet it has garnered far less attention than its counterpart, i.e., the Berry curvature. Here, we investigate the magnetoelectric effects using Zeeman quantum geometry, which is defined in [Phys. Rev. Lett. 134, 116301 (2025)], aiming to unveil the role of the quantum metric in the context of linear response phenomena. At first, we establish the magnetic field-induced derivative formulas of current density and charge polarization in the framework of Zeeman quantum geometry. Then, in two-band systems, we determine the geometric relations between the Zeeman quantum geometry and the quantum geometric tensor. Furthermore, in two-band models with linear spin-orbit coupling, we derive the integral relation linking the magnetoelectric conductivity to the quantum metric. Finally, we put forward experimental measurement criteria for the Zeeman quantum geometry across a series of magnetic systems, including two-dimensional (2D) spiral skyrmions, 3D isotropic materials, and 2D magnetic topological insulators. Our work bridges the Zeeman quantum geometry and quantum geometry tensor, underscores the role of quantum metric in linear transport, and provides testable signatures to guide the design of topological magnetoelectric devices.