Quantum-geometry-induced intrinsic optical anomaly in multiorbital superconductors

Bloch electrons in multiorbital systems carry nontrivial quantum geometric information characteristic of their orbital composition as a function of their wave vector. When such electrons form Cooper pairs, the resultant superconducting state naturally inherits aspects of the quantum geometry. In this paper, we study how this geometric character is revealed in the intrinsic optical response of the superconducting state. In particular, due to the superconducting gap opening, interband optical transitions involving states around the Fermi level are forbidden. This generally leads to an anomalous suppression of the optical conductivity at frequencies matching the band separation—which could be significantly higher than the superconducting gap energy. We discuss how the predicted anomaly may have already emerged in two earlier measurements on an iron-based superconductor. When interband Cooper pairing is present, intraband optical transitions may be allowed and finite conductivity emerges at low frequencies right above the gap edge. These conductivities depend crucially on the off-diagonal elements of the velocity matrix in the band representation, i.e., the interband velocity—which is related to the non-Abelian Berry connection of the Bloch states.

Figure 1

(a) Schematic interband optical transitions (arrows) in a $d_{xz}\text{−}{d}_{yz}$ model with only one band crossing the Fermi level. Dashed curves sketch the normal-state band structure, and the red solid curves the Bogoliubov quasiparticle spectra of the superconducting band. The black arrows denote transitions that contribute to the intrinsic optical transitions in the normal (dotted) and superconducting (solid) states, while the green dotted arrows exemplify momentum-nonconserving transitions induced by disorder scattering in normal state. (b) The real part of the optical conductivity of normal and superconducting phases near the cutoff frequency ${\omega }_F\approx 0.75$. The dotted curve shows their difference, which exhibits a dip between ${\omega }_F$ and roughly ${\omega }_F+{\mathrm{\Delta }}_1$. In the calculation, we took ${\xi }_{a\left(b\right),\mathbf{k}}=\frac{k_x^2}{2m_{\parallel (\perp )}}+\frac{k_y^2}{2m_{\perp (\parallel )}}-\mu$ and ${\lambda }_{\mathbf{k}}=\lambda k_x{k}_y$, with $(m_{\parallel },m_{\perp },\mu ,\lambda )=(1,-2,0.5,-0.75)$. Superconducting pairing is assumed to take place only on the band that crosses the Fermi level, with pairing gap ${\mathrm{\Delta }}_1=0.02$.

Figure 2

(a) Fermi surface (white contour) and normalized magnitude of the interband velocity $|V_{x,\mathbf{k}}^{12}|$ in a square lattice $d_{xz}\text{−}{d}_{yz}$ model. (b) The real part of the optical conductivity in the normal and superconducting states, and their difference. The model has ${\xi }_{a\left(b\right),\mathbf{k}}=-2t_{\parallel (\perp )}cos{k}_x-2t_{\perp (\parallel )}cos{k}_y-4t^{\prime }cos{k}_{x}cos{k}_y-\mu$ and ${\lambda }_{ab,\mathbf{k}}=\lambda sin{k}_{x}sin{k}_y$, where $(t_{\parallel },t_{\perp },t^{\prime },\lambda ,\mu )=(1,0.8,-0.6,0.4,-2.4)$. Inset shows the band structure near the M point. Only one band crosses Fermi level with a pairing gap ${\mathrm{\Delta }}_1=0.02$.

Figure 3

(a) Schematic intraband optical transitions (black arrows) induced by interband Cooper pairing. The dashed and solid curves depict the energy dispersion in the normal and superconducting states of a continuum $d_{xz}\text{−}{d}_{yz}$ model with finite SOC. (b) Real part of the low-frequency optical conductivity induced by interband pairing. The solid (red) and dotted (blue) curves show results obtained from the numerical calculation and the perturbative expansion, respectively. The continuum Hamiltonian with SOC is given in the Supplemental Material [17] and we took $(m_{\parallel },m_{\perp },\mu ,\lambda ,\eta )=(1,0.2,0.8,0.5,0.3)$. The pairing amplitudes are $({\mathrm{\Delta }}_1,{\mathrm{\Delta }}_2,{\mathrm{\Delta }}_{12})=(0.02,0.022,0.005)$ and the constant $\gamma =1$.