Linear Response Theory Around a Localized Impurity in the Pseudogap Regime of an Anisotropic Superconductor

We compare and contrast the polarizability of a d-wave superconductor in the pseudogap regime, within the precursor pairing scenario (dPG), and of a d-density-wave (dDW) state, characterized by a d-wave hidden order parameter, but no pairing. Our study is motivated by STM imaging experiments around an isolated impurity, which may in principle distinguish between precursor pairing and dDW order in the pseudogap regime of the high-$$T_{c}$$ superconductors. In both cases, the $${\bf q}$$-dependence of the polarizability is characterized by an azimuthal modulation, consistent with the d-wave symmetry of the underlying state. However, only the dDW result shows the fingerprints of nesting, with nesting wave vector $${{\bf Q}}=(\pi,\pi)$$, albeit imperfect, due to a nonzero value of the hopping ratio $$t^\prime /t$$ in the band dispersion relation. As a consequence of nesting, the presence of hole pockets is also reflected by the $$({\bf q},\omega)$$ dependence of the retarded polarizabilit