A Berry curvature is an imaginary component of the quantum geometric tensor (QGT) and is well studied in many branches of modern physics; however, the quantum metric as a real component of the QGT is less explored. Here, by using tunable superconducting circuits, we experimentally demonstrate two methods to directly measure the quantum metric tensor for characterizing the geometry and topology of underlying quantum states in parameter space. The first method is to probe the transition probability after a sudden quench, and the second one is to detect the excitation rate under weak periodic driving. Furthermore, based on quantum metric and Berry-curvature measurements, we explore a topological phase transition in a simulated time-reversal-symmetric system. The work opens up a unique approach to explore the topology of quantum states with the QGT.
In this paper, we extend the quantum geometric tensor for parameter-dependent curved spaces to higher dimensions, and introduce an equivalent definition that generalizes the Zanardi, et al, formulation of the tensor. The parameter-dependent metric modifies the behavior of both the quantum metric tensor and Berry curvature in a purely geometric way. Our focus is on understanding the distinctions in higher dimensions that emerge when using the generalized tensor compared to the conventional one. Through a comparative analysis, illustrated with examples in two dimensions, we highlight unique quantum geometric properties for both the quantum metric tensor and the Berry curvature. Additionally, we explore differences between analytical and perturbative approaches in solving the problems.
Local Dirac-node topology is usually described by a winding number or Berry phase, whereas curvature-flux formulations are commonly associated with global band topology. Here we show that the non-Hermitian Zeeman quantum geometric tensor (QGT) provides a curvature-flux representation of local
π1 topology. Unlike the conventional Hermitian QGT, whose metric and Berry curvature are fixed respectively by its real symmetric and imaginary antisymmetric parts, the Zeeman QGT must be decomposed according to symmetry under index exchange. This decomposition yields normal and anomalous metric-curvature sectors. The normal sector has the same metric-curvature structure as the conventional Hermitian QGT. The anomalous sector, by contrast, contains two components absent in the conventional QGT: an imaginary symmetric metric-like tensor and a real antisymmetric curvature-like tensor. In a two-dimensional Dirac system, the anomalous Zeeman curvature forms a radial flux field Hodge-dual to the tangential winding field of the Dirac node, thereby converting the local winding invariant into a Gauss-type flux invariant. We further show that the four Zeeman-geometric sectors map one-to-one onto frequency- and symmetry-resolved gyrotropic conductivity channels, with reciprocal kinetic magnetoelectric response providing a complementary probe. These results establish non-Hermitian Zeeman quantum geometry as a measurable framework connecting local Dirac-node topology and transport.