  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 $\eta$, we show that the DC conductivity is zero in a flat band when
taking the clean limit ($\eta \rightarrow 0$). 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-Heeger 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.
