The Kubo formula for linear response is given as $$\sigma_{ab}(\omega)=\frac{i}{\omega}[\Pi_{ab}(\omega)-\Pi_{ab}(\omega=0)]$$ with $\Pi$ the current-current correlation. It is often claimed that the second part, the diamagnetic term, must be diagonal. For instance, in Mahan's book 'Many-particle Physics' Eq. (3.388), it is derived to be $\frac{e^2n}{m}\delta_{ab}$ for quadratic electrons with mass $m$ and density $n$.
But is it diagonal always? If not, any example?
