Excitation Spectrum ofd-Wave Fermi Surface Deformation

Several instabilities competing with the $d$-wave singlet pairing were proposed for high-$T_{\mathrm{c}}$ cuprates. One of them is the $d$-wave Fermi surface deformation ($d\mathrm{F}\mathrm{S}\mathrm{D}$), which is generated by forward scattering. In this Letter, correlation functions of the $d\mathrm{F}\mathrm{S}\mathrm{D}$ are calculated within the random phase approximation. In the normal state, the excitation spectrum shows a low energy peak, which smoothly connects to critical fluctuations of the $d\mathrm{F}\mathrm{S}\mathrm{D}$ at lower temperature. The competition with the $d$-wave pairing, however, blocks the critical fluctuations. The whole spectral weight is transferred to high energy and a pronounced peak appears there in the $d$-wave pairing state. This peak is an overdamped collective mode of the $d\mathrm{F}\mathrm{S}\mathrm{D}$ and can grow to become a resonance mode at moderate finite wave vectors.