Nonrelativistic Dynamics of the Amplitude (Higgs) Mode in Superconductors

Despite the formal analogy with the Higgs particle, the amplitude fluctuations of the order parameter in weakly coupled superconductors do not identify a real mode with a Lorentz-invariant dynamics. Indeed, its resonance occurs at $2{\mathrm{\Delta }}_0$, which coincides with the threshold $2E_{\mathrm{gap}}$ for quasiparticle excitations that spoil any relativistic dynamics. Here we investigate the fate of the Higgs mode in the unconventional case where $2E_{\mathrm{gap}}$ becomes larger than $2{\mathrm{\Delta }}_0$, as due to strong coupling or strong disorder. We show that also in this situation, the amplitude fluctuations never identify a real mode at $2{\mathrm{\Delta }}_0$, since such a “bosonic” limit is always reached via strong mixing with the phase fluctuations, which dominate the low-energy part of the spectrum. Our results have direct implications for the interpretation of the subgap optical absorption in disordered superconductors.

Figure 1

Intensity map of the spectral function of the amplitude and phase modes in the homogeneous case at $U=8t$ for $n=0.1$ (upper panels) and $n=1$ (lower panels). The solid red lines mark the value of $2{\mathrm{\Delta }}_0$, and the dashed black lines mark the value of $2E_{\mathrm{gap}}$. At $n=0.1$ the bosonic limit $E_{\mathrm{gap}}>{\mathrm{\Delta }}_0$ is reached, but the amplitude mode is strongly mixed with the phase, which dominates any subgap structure of the Higgs (with zero spectral weight at $\omega =0$). At $n=1$ the system is always particle-hole symmetric, so the amplitude and phase sectors remain decoupled. However, here $E_{\mathrm{gap}}={\mathrm{\Delta }}_0$ and the Higgs mode has a broad spectral weight only above $2E_{\mathrm{gap}}$. The bending back of the phase mode at $\mathbf{Q}=(\pi ,\pi )$ (where its weight is zero) is a signature of the degenerate CDW instability present in the model (3) at $n=1$ [54].

Figure 2

Spectral functions of the amplitude and phase modes at $\mathbf{q}=0$ for $n=0.875$ obtained on a $20\times 20$ lattice after averaging over 50 disorder configurations for two values of SC coupling ($U/t$) and disorder ($V_0/t$). The vertical dashed lines and green arrows mark the spectral gap $2E_{\mathrm{gap}}$ and $2⟨{\mathrm{\Delta }}_0⟩$, respectively, whose disorder dependence is shown in the insets. For comparison, we also show $\sigma (\omega )$ (blue line) from Ref. [56].

Figure 3

Leading-order contributions of the SC collective modes to the conductivity, from the current given in Eq. (11). Wavy, solid, and dashed lines represent insertion of the electromagnetic field, the SC phase propagator, and the SC amplitude propagators, respectively [20, 22]. (a) Is the one-phason process while (b) is the convoluted phase-amplitude response, see discussion in the text.