Nature and Raman signatures of the Higgs amplitude mode in the coexisting superconducting and charge-density-wave state

We investigate the behavior of the Higgs (amplitude) mode when superconductivity emerges on a preexisting charge-density-wave state. We show that the weak overdamped square-root singularity of the amplitude fluctuations in a standard BCS superconductor is converted in a sharp, undamped power-law divergence in the coexisting state, reminiscent of the Higgs behavior in Lorentz-invariant theories. This effect reflects in a strong superconducting resonance in the Raman spectra, both for an electronic and a phononic mechanism leading to the Raman visibility of the Higgs. In the latter case, our results are relevant to the interpretation of the Raman spectra measured experimentally in ${\mathrm{NbSe}}_2$.

Figure 1

Frequency dependence of the real (a) and imaginary (b) parts of the Higgs spectral function (c) at $T=0$ in the presence (SC+CDW) and in the absence (SC only) of CDW, for a quasiparticle damping $\gamma =0.016{\Delta }_0$ (solid lines) and $\gamma =0.1{\Delta }_0$ (dashed lines). Here, ${\Delta }_0=0.025$, as in Fig. 2.

Figure 2

Electronic Raman response in the $A_{1g}$ channel in the coexisting (solid lines) and pure (dashed lines) SC state as a function of temperature. Inset: temperature evolution of the SC gaps ${\Delta }_0$ and of the CDW gap $D_0$.

Figure 3

Feynman diagrams representing the electronic processes responsible for the Raman visibility of (a) the Higgs mode when $g=0$ and (b) the CDW phonon above $T_c$. Here, the dashed lines represent the incoming electromagnetic radiation, the full lines the electronic Green's functions $G_0$, the red and green wavy lines the Higgs propagator $X_{\Delta \Delta }^{-1}\left(\omega \right)$ and the phonon one $D\left(\omega \right)$, respectively. In the coexistence state for $g\ne 0$ the two processes become interconnected since both the phonon propagator [see Eq. (15)] and the ${\chi }_{RD}$ [see Eq. (17)] are renormalized by the coupling between the Higgs fluctuations and the CDW amplitude fluctuations.

Figure 4

Temperature evolution of the phonon spectral function (a) and of the Raman response (b) of the CDW phonon between $T_{\text{CDW}}$ and $T_c=0.4T_{\text{CDW}}$.

Figure 5

Renormalization of the phonon frequency (a), (c) and of the phonon spectral function $A_D$ (b), (d) due to the Higgs mode according to the phenomenological approach by LV [18], where the phonon self-energy is ${\Sigma }_{\Delta }=-2g_{\Delta }^2{\chi }_{\Delta \Delta }/U{X}_{\Delta \Delta }$. The Higgs fluctuations $X_{\Delta \Delta }$ are computed in the two cases, with (solid red lines) or without (dashed blue lines, as done in Ref. [18]) a preformed CDW, and correspond to the data of Fig. 1 for $\gamma =0.1{\Delta }_0$. Here, the CDW phonon has energy ${\Omega }_0=4{\Delta }_0$ and $\delta =0.1{\Delta }_0$, leading to the spectral function at $T>T_c$ shown by the thin orange line in panels (b) and (d). When entering the SC phase, the phonon peak at ${\Omega }_0$ gets broader and slightly displaced, while a second peak develops below $2{\Delta }_0$ due to the Higgs self-energy ${\Sigma }_{\Delta }$, that causes a second intersection for the curve ${\omega }^2-{\tilde{\Omega }}_0^2$ in the inverse phonon propagator.

Figure 6

Raman response in the $A_{1g}$ channel in the mixed CDW+SC state according to Eq. (17), both in the presence (a) and in the absence (b) of the direct Raman coupling to the Higgs, encoded in ${\chi }_{R\Delta }$. The double-peak structure found below $T_c$ in the phonon spectral function, shown in panel (c) is weighted in the Raman response by the two polarization functions ${\chi }_{R\Delta }$ and ${\chi }_{RD}$. Their value at $\omega =0$ is shown in the inset of panel (a), along with the coupling ${\chi }_{D\Delta }$ between the two order parameters responsible for the Higgs signature in the phonon mode. (d) Temperature evolution of the Raman spectral weight integrated between $\omega =0$ and $0.15$, for the two cases of panels (a) and (b).

Figure 7

Real part of the Raman response function ${\chi }_{R\Delta }$ at $T=0$ in the pure SC and in the coexistence SC+CDW state for a residual $\gamma =0.1{\Delta }_0$, as in Figs. 1 and 2. As one can see, in the mixed CDW+SC case ${\chi }_{R\Delta }^{\prime }$, that appears as a prefactor in the Raman susceptibility (11), is featureless at the frequency $2{\Delta }_0$ where the Higgs spectral function shown in Fig. 1 has a maximum.

Figure 8

Doping variation of the CDW and SC gap at $T=0$ in the case of a constant (a) or varying (b) SC coupling $U$. In (b) $U$ decreases slightly with doping with respect to the value $U_0$ at $\mu =0$ (see inset), in order to reproduce a weaker increase of $\Delta$ with doping.

Figure 9

Frequency dependence of the real (a) and imaginary (b) parts of the inverse amplitude mode at $T=0$ for different doping levels. In the CDW+SC state (solid lines) the gaps are taken from Fig. 8, while in the pure SC case (dashed lines) $U$ is tuned to have the same ${\Delta }_0$ at $T=0$. The resulting Raman response is shown in panel (c), while panel (d) shows the Raman-Higgs coupling function, that is always smaller in the coexistence state.