Overscreening of magnetic impurities indx2−y2-wave superconductors

We consider the screening of a magnetic impurity in a $d_{x^2-y^2}$-wave superconductor. The properties of the $d_{x^2-y^2}$ state lead to an unusual behavior in the impurity magnetic susceptibility, the impurity specific heat, and in the quasiparticle phase shift which can be used to diagnose the nature of the condensed state. We construct an effective theory for this problem and show that it is equivalent to a multichannel (one per node) nonmarginal Kondo problem with linear density of states and coupling constant $J.\mathrm{}$ There is a quantum phase transition from an unscreened impurity state to an overscreened Kondo state at a critical value $J_c$ which varies with ${\Delta }_0,$ the superconducting gap away from the nodes. In the overscreened phase, the impurity Fermi level ${\epsilon }_f$ and the amplitude Δ of the ground state singlet vanish at $J_c$ like ${\Delta }_0\exp (-\mathrm{const}/\Delta )$ and $J-J_c$, respectively. We derive the scaling laws for the susceptibility and specific heat in the overscreened phase at low fields and temperatures.