Spatially resolved collective modes in d-wave superconductors

TL;DR

The paper investigates how long-range Coulomb interactions modify the collective modes in a clean $d$-wave superconductor, focusing on both transverse (phase) and longitudinal (amplitude) excitations. Using complementary quasiclassical (Keldysh-Nambu) and diagrammatic techniques, the authors derive the susceptibilities and extract mode dispersions, revealing that the transverse plasma mode at $T=0$ matches the $s$-wave case, while at finite $T$ nodal quasiparticles soften and broaden it. The longitudinal mode exhibits a direction-dependent dispersion and remains damped within the continuum, with a resonance near $2elta_{ m max}$ at $q=0$ but spectral weight extending for all frequencies, and its decay rate follows a $1/t^2$ law independent of momentum. Overall, the work clarifies the momentum-direction and temperature dependence of both transverse and longitudinal collective modes in $d$-wave superconductors, with implications for optical and Raman spectroscopy in cuprates.

Abstract

We analyze the dispersion of collective modes in a superconductor with $d-$wave symmetry of the order parameter in the presence of long-range Coulomb interaction. We use diagrammatic technique and quasiclassical theory in Keldysh-Nambu formalism to compute longitudinal and transverse pair susceptibilities and extract from them the dispersion of the longitudinal and transverse collective mode. We show that at T=0, the dispersion of the transverse (plasma) mode is the same as in an s-wave superconductor, but at a finite temperature it is softer and has a much larger decay rate due to the partial screening of the Coulomb potential by nodal quasiparticles. We show that the dispersion of the longitudinal mode depends on the direction of momentum with respect to the positions of the nodes of the d-wave gap, while the decay rate of this mode does not depend on momentum. We discuss experimental implications of our results.

Figures (7)

• Figure 1: Dyson equation for the pair-pair susceptibility $\chi_M(\mathbf q,\Omega_m)$ with momentum $\mathbf q$ and Matsubara frequency $\Omega_m$. The single arrow solid line represents normal Green's function $G$, and the double-headed arrow solid lines represent anomalous Green's function $F$ and $F^+$. The red and blue triangular vertices represent renormalized particle-particle vertices $\Gamma_\mathbf k$ with two incoming fermion momenta $k_+=k+q/2$ and $-k_-=-k+q/2$ and $\bar{\Gamma}_\mathbf k$ with two outgoing fermion momenta $k_-=k-q/2$ and $-k_+=-k-q/2$ respectively. The double wavy green line represents fully renormalized Coulomb interaction, $\tilde{V}_q$. The intermediate momenta are labeled by $k_\pm=k\pm q/2$ and $p_\pm=p\pm q/2$ where $k=(\mathbf k,\omega_k),p=(\mathbf p,\omega_p)$, and $q=(\mathbf q,\Omega_m)$. The black dot on the right vertex represents the d-wave form factor $\gamma(\theta_\mathbf k)= \sqrt{2}\cos2\theta_{\mathbf k}$. $\uparrow$ and $\downarrow$ represent up and down spin components.
• Figure 2: Dyson equation for the renormalized particle-particle vertices (a) $\Gamma_\mathbf k$ (blue triangle with two incoming fermion momenta $k_+=k+q/2$ and $-k_-=-k+q/2$), (b) $\bar{\Gamma}_\mathbf k$ (red triangle with two outgoing fermion momenta $k_-=k-q/2$ and $-k_+=-k-q/2$), and (c) renormalized Coulomb interaction, $\tilde{V}_q$. In the diagrams shown here, the black single wavy line represents the pairing interaction, the green single wavy line represents the long-range Coulomb interaction $V_\mathbf q$, and the double wavy green line stands for the renormalized Coulomb potential $\tilde{V}_q$. (c) Diagrammatic form of the Dyson equation for the renormalized Coulomb potential. The single arrow solid lines represent normal Green's function $G$, and the double-headed arrow solid lines represent anomalous Green's function $F$ and $F^+$. The intermediate momenta are labeled by $k_\pm=k\pm q/2$, $p_\pm=p\pm q/2$, $\ell_\pm=\ell\pm q/2$ where $k=(\mathbf k,\omega_k),p=(\mathbf p,\omega_p)$, $\ell=(\boldsymbol{\ell} ,\omega_\ell)$ and $q=(\mathbf q,\Omega_m)$. The black dot represent the d-wave form factor $\gamma(\theta_\mathbf k)= \sqrt{2}\cos2\theta_{\mathbf k}$.
• Figure 3: Left panel: real part of transverse pair susceptibility $\chi_{\textrm{AB}}(\mathbf q,\omega)$ for $d$-wave superconductor plotted as a function of frequency and at a fixed value of momentum $v_Fq=0.125\Delta$ for different values of temperatures. All energies are given in the units of the pairing amplitude at zero temperature. Function $\chi_{\textrm{AB}}(\mathbf q,\omega)$ exhibits a peak at $\omega=\Omega_{\mathrm{peak}}$ which corresponds to the energy of the collective transverse mode. Right panel: the dependence of $\Omega_{\mathrm{peak}}$ on temperature.
• Figure 4: Frequency dependence of the imaginary and real parts of the longitudinal susceptibility, evaluated at different values of momentum $\mathbf q$ for both $s$-wave (right panel) and $d$-wave superconductors (left panel). The results for the $d$-wave case were obtained for $\mathbf q$ pointing along the anti-nodal direction $\mathbf q=q(1,0)$. We remind the reader that $\Delta_{\mathrm{max}}=\sqrt{2}\Delta$.
• Figure 5: Frequency dependence of the imaginary part of the $d$-wave longitudinal susceptibility evaluated for five different values of momentum. At small values of momentum $v_Fq<\Delta$, the difference between these two directions is negligible, however it becomes pronounced for $v_Fq>\Delta$.