Conductivity in flat bands from the Kubo-Greenwood formula

Conductivity in a multiband system can be divided into intra- and interband contributions, and the latter further into symmetric and antisymmetric parts. In a flat band, intraband conductivity vanishes and the antisymmetric interband contribution, proportional to the Berry curvature, corresponds to the anomalous Hall effect. We investigate whether the symmetric interband conductivity, related to the quantum metric, can be finite in the zero frequency and flat band limit. Starting from the Kubo-Greenwood formula with a finite scattering rate $\ensuremath{\eta}$, we show that the DC conductivity is zero in a flat band when taking the clean limit $(\ensuremath{\eta}\ensuremath{\rightarrow}0)$. If commonly used approximations involving derivatives of the Fermi distribution are used, finite conductivity appears at zero temperature $T=0$; we show however that this is an artifact due to the lack of Fermi surfaces in a (partially) flat band. We then analyze the DC conductivity using the Kubo-Streda formula, and note similar problems at $T=0$. The predictions of the Kubo-Greenwood formula (without the approximation) and the Kubo-Streda formula differ significantly at low temperatures. We illustrate the results within the Su-Schrieffer-Heger model where one expects vanishing DC conductivity in the dimerized limit as the unit cells are disconnected. We discuss the implications of our results to previous work which has proposed the possibility of finite flat band DC conductivity proportional to the quantum metric. Our results also highlight that care should be taken when applying established transport and linear response approaches in the flat band context, since many of them utilize the existence of a Fermi surface and assume scattering to be weak compared to kinetic energy.