Excitation Spectrum of $d$-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.

Figure 1

Normal state. (a) $\omega$ dependence of $\mathrm{I}\mathrm{m}{\kappa }^{11}(\mathbf{q},\omega )$ for several choices of $T$ at $\mathbf{q}=(0.01\times 2\pi ,0)$. The result at $T=0.13J$ is compared with $\mathrm{I}\mathrm{m}{\kappa }_0^{11}(\mathbf{q},\omega )$ in the inset. (b) $\omega$ dependence of $\mathrm{I}\mathrm{m}{\kappa }^{11}(\mathbf{q},\omega )$ for several choices of $\mathbf{q}$. (c) Excitation spectrum on the plane of $\omega$ vs $q$. The shaded region is a particle-hole continuum. The open symbols correspond to the peak energy of $\mathrm{I}\mathrm{m}{\kappa }^{11}(\mathbf{q},\omega )$ at a given $\mathbf{q}$.

Figure 2

$d$-wave pairing state. (a) $\omega$ dependence of $S^{11}(\mathbf{q},\omega )$ for several values of $T$ at $\mathbf{q}=(0.01\times 2\pi ,0)$. (b) $\omega$ dependence of $\mathrm{I}\mathrm{m}{\kappa }^{11}(\mathbf{q},\omega )$ for several choices of $\mathbf{q}$ along the $q_x$ direction ($T=0.01J$). (c) Excitation spectrum on the plane of $\omega$ vs $q$. The shaded region is a continuum of excitations. The open circle, ${\omega }_{\mathrm{r}\mathrm{e}\mathrm{s}}$, corresponds to the peak energy of $\mathrm{I}\mathrm{m}{\kappa }^{11}(\mathbf{q},\omega )$ at a given $\mathbf{q}=(q,0)$ ($T=0.01J$). ${\tilde{\omega }}_{\mathrm{r}\mathrm{e}\mathrm{s}}$ is the estimation by Eq. (10).