Chiral Higgs Mode in Nematic Superconductors

Nematic superconductivity with spontaneously broken rotation symmetry has recently been reported in doped topological insulators, $M_x{\mathrm{Bi}}_2{\mathrm{Se}}_3$ ($M=\mathrm{Cu}$, Sr, Nb). Here we show that the electromagnetic (EM) response of these compounds provides a spectroscopy for bosonic excitations that reflect the pairing channel and the broken symmetries of the ground state. Using quasiclassical Keldysh theory, we find two characteristic bosonic modes in nematic superconductors: the nematicity mode and the chiral Higgs mode. The former corresponds to the vibrations of the nematic order parameter associated with broken crystal symmetry, while the latter represents the excitation of chiral Cooper pairs. The chiral Higgs mode softens at a critical doping, signaling a dynamical instability of the nematic state towards a new chiral ground state with broken time reversal and mirror symmetry. Evolution of the bosonic spectrum is directly captured by EM power absorption spectra. We also discuss contributions to the bosonic spectrum from subdominant pairing channels to the EM response.

Figure 1

(a) Configurations of polarized EM waves to probe the nematic pairing gap in the ${\mathsf{D}}_{\mathsf{3}\mathsf{d}}$ crystal structure. (b) Evolution of the Fermi surface and superconducting gap in the nematic state for various $\mu$. (c) Phase diagram of $M_x{\mathrm{Bi}}_2{\mathrm{Se}}_3$ computed by the quasiclassical theory with ${\eta }_1=\mathrm{\Delta }(T,\mu )$ and ${\eta }_2=0$. We set $T_c^{(E_u,2)}=T_c$ with $T_c\equiv T_c^{(E_u,1)}$. The dashed curve shows the dynamical instability of the chiral Higgs mode beyond which the nematic state is no longer metastable.

Figure 2

(a) Mass gap of the chirality mode ($M_{E_u,2}^{-}$), the nematicity mode [$M_{E_u,2}^{+}(0,\mu )$], and the $A_{1u}$ and $A_{2u}$ modes as a function of $\mu$. We set $x_{A_{1u}}=x_{A_{2u}}=-1.5$ and $x_{E_u,2}=0$, where $x_{\mathrm{\Gamma },j}=\ln T_c^{(\mathrm{\Gamma },j)}/T_c$. The color map shows the $T$ dependence of the chirality mode. The dashed curve corresponds to the dynamical instability of the chirality mode at which the mass gap closes. (b) Mass gap of the nematicity mode as a function of extrinsic symmetry breaking of $E_u$ representation measured by $x_{E_u,2}=\ln (T_c^{(E_u,2)}/T_c)$.

Figure 3

Power absorption $P(\omega )$ in the nematic state with the nematic angle $\vartheta =0$ for $T=0.05T_c$, where we set $\mathit{A}\parallel \mathit{x}$ and $\mathit{Q}\parallel \mathit{y}$ and all parameters are the same as those in Fig. 2. The shaded area shows the contribution of the bosonic excitations $P^{\mathrm{CM}}(\omega )$. The mass gaps of the chirality mode $M_{E_u,2}^{-}$ are also shown.