Collective mode at Lifshitz transition in iron-pnictide superconductors.

We obtain the exact low-energy spectrum of two mobile holes in a t-J model for an isolated layer in an iron-pnictide superconductor. The minimum d xz and d yz orbitals per iron atom are included, with no hybridization between the two. After tuning the Hund coupling to a putative quantum critical point (QCP) that separates a commensurate spin-density wave from a hidden-order antiferromagnet at half filling, we find an s-wave hole-pair groundstate and a d-wave hole-pair excited state. Near the QCP, both alternate in sign between hole Fermi surface pockets at the Brillouin zone center and emergent electron Fermi surface pockets at momenta that correspond to commensurate spin-density waves (cSDW). The dependence of the energy splitting with increasing Hund coupling yields evidence for a true QCP in the thermodynamic limit near the putative one, at which the s-wave and d-wave Cooper pairs are degenerate. A collective s-to-d-wave oscillation of the macroscopic superconductor that couples to orthorhombic shear strain is also identified. Its resonant frequency is predicted to collapse to zero at the QCP in the limit of low hole concentration. This implies degeneracy of Cooper pairs with s, d and [Formula: see text] symmetry in the corresponding quantum critical state. We argue that the critical state describes Cooper pairs in hole-doped iron superconductors at the Lifshitz transition, where electron bands first rise above the Fermi level. We thereby predict that the s-to-d-wave collective mode observed by Raman spectroscopy in Ba1-x K x Fe2As2 at optimal doping should also be observed at higher doping near the Lifshitz transition.