In the realm of condensed-matter physics, the quantum metric, a pivotal component of the quantum geometric tensor, has been overshadowed by the Berry curvature in terms of research attention. This study investigates the contribution of the quantum metric to the longitudinal conductivity in dispersive two-band systems. We discover that when the density of states vanishes at semimetallic band crossing points, the quantum metric dominates the longitudinal conductivity. By constructing three gapless examples in one-dimensional (1D), 2D, and 3D systems, respectively, we calculate the quantum-metric-induced longitudinal conductivity. In the 1D system with two intersecting bands, the quantum metric successfully explains the discontinuity of intraband conductivity when the band gap opens. For the 2D Dirac system, it accounts for the minimum conductivity at the Dirac point. In the 3D system with a degenerate nodal ring, the longitudinal conductivity is entirely contributed by the quantum metric and is proportional to the length of the nodal ring. Significantly, our results are independent of the scattering rate and are valid in the clean limit, offering a solid theoretical basis for experimental observation of quantum metric effects in longitudinal conductivity, and propelling forward the research on quantum geometry in condensed-matter physics.

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• Berry curvature
• Electrical conductivity
• Electrical properties
• Fractional dimension
• 1-dimensional systems
• 3-dimensional systems