Superfluid density and competing orders in $d$-wave superconductors

We derive expressions for the superfluid density $\rho_s$ in the low-temperature limit $T \to 0$ in $d$-wave superconductors, taking into account the presence of competing orders such as spin-density waves, $is$-pairing, etc. Recent experimental data for the thermal conductivity and for elastic neutron scattering in La$_{2-x}$Sr$_x$CuO$_4$ suggest there are magnetic field induced anomalies that can be interpreted in terms of competing orders. We consider the implications of these results for the superfluid density and show in the case of competing spin-density wave order the usual Volovik-like $\sqrt{H}$ depletion of $\rho_s(H)$ is replaced by a slower dependence of $\rho_s$ on the applied magnetic field $H$. We show that it is crucial to include the competing order parameter in the self-consistent equation for the impurity scattering rate.