Thermoelectric Transport Driven by the Hilbert-Schmidt Distance

The geometric characteristics of Bloch wavefunctions play crucial roles in the properties of electronic transport. Within the Boltzmann equation framework, we demonstrate that the thermoelectric performance of materials is significantly influenced by the Hilbert-Schmidt distance of Bloch wavefunctions. The connection between the distribution of quantum distance on the Fermi surface and the electronic transport scattering rate is established in the presence of magnetic and nonmagnetic impurities. The general formulation is applied to isotropic quadratic band-touching semimetals, where one can concentrate on the role of quantum geometric effects other than the Berry curvature. It is verified that the thermoelectric power factor can be succinctly expressed in terms of the maximum quantum distance, d_max. Specifically, when d_max reaches one, the power factor doubles compared to the case with trivial geometry (d_max = 0). These findings highlight the significance of quantum geometry in understanding and improving the performance of thermoelectric devices.