The photon absorption edge in superconductors and gapped 1D systems

Opening of a gap in the low-energy excitations spectrum affects the power-law singularity in the photon absorption spectrum $A(\Omega)$. In the normal state, the singularity, $A(\Omega)\propto [D/(\Omega-\Omega_{\rm th})]^\alpha$, is characterized by an interaction-dependent exponent $\alpha$. On the contrary, in the supeconducting state the divergence, $A(\Omega)\propto (D/\Delta)^\alpha(\Omega-\tilde{\Omega}_{\rm th})^{-1/2}$, is interaction-independent, while threshold is shifted, $\tilde{\Omega}_{\rm th}=\Omega_{\rm th}+\Delta$; the ``normal-metal'' form of $A(\Omega)$ resumes at $(\Omega-\tilde{\Omega}_{\rm th})\gtrsim \Delta\exp(1/\alpha)$. If the core hole is magnetic, it creates in-gap states; these states transform drastically the absorption edge. In addition, processes of scattering off the magnetic core hole involving spin-flip give rise to inelastic absorption with one or several {\it real} excited pairs in the final state, yielding a structure of peaks in $A(\Omega)$ at multiples of $2\Delta$ above the threshold frequency. The above conclusions apply to a broad class of systems, e.g., Mott insulators, where a gap opens at the Fermi level due to the interactions.