When we consider the response of a quantum lattice model with Hamiltonian $H=H_{kin}+H_{int}$ to an applied vector potential $\mathbf{A}(\mathbf{r},t)$ we obtain the current operator $\mathbf{j}(\mathbf{r})$ by expanding the kinetic energy $H_{kin}(\mathbf{A})$ to second order in $\mathbf{A}$ and taking the derivative, $j_x=-\frac{\delta H_{kin}(\mathbf{A})}{\delta A_x}$, see for example PRL 68 2830, 1992.
Since we have expanded $H_{kin}$ to the second order, this gives us two terms for the current operator, $j_x=j_x^p+j_x^d A_x$, the paramagnetic and the diamagnetic current, respectively.
What is the physical meaning of the diamagnetic current $\mathbf{j}^d$? What does it describe and why/when is it important? What does the fact that it arises from the second order in $\mathbf{A}$ tell us?
Two extra questions: In quantum lattice systems this procedure to find the current operator seems to be somewhat questionable (PRL 66 365, 1991), why? Why does the diamagnetic current play an important role in superconductors (Phys. Rev. 117 648, 1960), does it have something to do with the fact that a superconductor is a perfect diamagnet?