Marginal Fermi liquid resonance induced by a quantum magnetic impurity in d-wave superconductors

We consider a model of an Anderson impurity embedded in a $d_{x^2-y^2}$-wave superconducting state to describe the low-energy excitations of cuprate superconductors doped with a small amount of magnetic impurities. Due to the Dirac-like energy dispersion, a sharp localized resonance above the Fermi energy, showing a marginal Fermi liquid behavior ($\omega \ln \omega$ as $\omega \to 0$) is predicted for the impurity states. The same logarithmic dependence of self-energy and a linear frequency dependence of the relaxation rate are also derived for the conduction electrons, characterizing a new universality class for the strong coupling fixed point. At the resonant energies, the spatial distribution of the electron density of states around the magnetic impurity is calculated, to be confronted with measurements of the scanning tunneling microscopy on Bi$_2$Sr$_2$Ca(Cu$_{1-x}$Ni$_x$)O$_{8+\delta }$.