Incommensurate Charge and Spin Fluctuations in d-wave Superconductors

We show analytic results for the irreducible charge and spin susceptibilities, ${\chi }_0(\omega ,\mathbf{Q})$, where $\mathbf{Q}$ is the momentum transfer between the nodes in d-wave superconductors. Using the BCS theory and a circular Fermi surface, we find that the singular behavior of the irreducible charge susceptibility leads to the dynamic incommensurate charge collective modes. The peaks in the charge structure factor occur at a set of wave vectors which form an ellipse around ${\mathbf{Q}}_{\pi }=(\pi ,\pi )$ and ${\mathbf{Q}}_0=(0,0)$ in momentum space with momentum dependent spectral weight. It is also found that, due to the non-singular irreducible spin susceptibility, an extremely strong interaction via random phase approximation is required to support the magnetic peaks near ${\mathbf{Q}}_{\pi }$. Under certain conditions, the peaks in the magnetic structure factor occur near $\mathbf{Q}=(\pi ,\pi (1\pm \delta ))$ and $(\pi (1 \pm \delta),\pi)$.