Fig. 3: Optical Hall conductivity in a continuum model of superconductivity from a chiral metal.
From: Circular-polarization-selective perfect reflection from chiral superconductors
a, b Energy bands of the Bogoliubov-de Gennes Hamiltonian. Line colors indicate the electron \(\hat{c}\) (red) vs hole \({\hat{c}}^{{\dagger} }\) (blue) characters. Shown here is the case with M > 0. The case with M < 0 has a similar band structure. Superconductivity does not open the gap in (a) because the crossing between the electron and hole bands is protected by PC symmetry, where (PC)2 = 1. Although the band structure in (a) resembles that of the normal-state metal, the new optical excitation channel relating electron and hole states develops by superconducting pairing Δx. In (b) where (PC)2 = − 1, a full gap opens. Similarly, Δa,b pairings also open the full gap (not shown). In these cases, the PC-symmetric optical excitations are forbidden within electric-dipole approximation. c Chern number of the occupied BdG states for M > 0. This number equals the number of left-chiral Majorana edge states minus the number of right-chiral Majorana edge states. d Hall conductivity in the zero-frequency limit. M > 0 for solid and M < 0 for dashed lines. Gap functions are taken to produce the superconducting gap of about 20 meV. e Optical Hall conductivity for Δy = 10 meV. Δa,b pairing leads to similar spectra. We take a Lorentzian broadening Γ = 1 meV of resonant transitions. The optical Hall conductivity develops a finite imaginary value above the optical excitation gap Eg meV. Resonant transitions are forbidden for Eg≤ ℏω < Eexc because of the PC-symmetry of the BdG Hamiltonian. For the Δx pairing in (f) a resonant optical transition occurs down to zero frequency, i.e., Eg = Eexc = 0. For all plots, we take \(\hslash v/{a}_{\parallel }=A/{a}_{\parallel }^{2}=0.5\) eV, ∣M∣ = 20 meV, and EF = ∣μ∣ − ∣M∣ = 20 meV in Eq. (7) and consider superconducting pairing of Eq. (8).
