Effects of an in-plane magnetic field on c-axis sum rule and superfluid density in high-$T_c$ cuprates

In layered cuprates, the application of an in-plane magnetic field $({\bf H})$changesthec-axisopticalsumruleandsuperfluiddensity$\rho_{s}$. For pure incoherent c-axis coupling, $\mathbf{H}$ has no effect on either quantities but it does if an additional coherent component is present. For the coherent contribution, different characteristic variations on $\mathbf{H}$ and on temperature result from the constant part $(t_{\perp })$ of the hopping matrix element and from the part $(t_{\varphi })$ which has zero on the diagonal of the Brillouin zone. Only the constant part $(t_{\perp })$ leads to a dependence on the direction of $\mathbf{H}$ as well as on its magnitude.