Origin of orthorhombic transition, magnetic transition, and shear-modulus softening in iron pnictide superconductors: Analysis based on the orbital fluctuations theory

Abstract

The main features in iron pnictide superconductors are summarized as (i) the orthorhombic transition accompanied by a remarkable softening of the shear modulus, (ii) high-${T}_{c}$ superconductivity close to the orthorhombic phase, and (iii) stripe-type magnetic order induced by orthorhombicity. To present a unified explanation for these features, we analyze the multi-orbital Hubbard-Holstein model with Fe-ion optical phonons based on the orbital fluctuation theory. In the random-phase approximation (RPA), a small electron-phonon coupling constant ($\ensuremath{\lambda}~0.2$) is enough to produce large orbital (charge quadrupole) fluctuations. The most divergent susceptibility is the ${O}_{\mathit{xz}}$-antiferroquadrupole (AFQ) susceptibility, which causes $s$-wave superconductivity without sign reversal (${s}_{++}$-wave state). At the same time, divergent development of ${O}_{{x}^{2}\ensuremath{-}{y}^{2}}$-ferroquadrupole (FQ) susceptibility is brought about by the ``two-orbiton process'' with respect to the AFQ fluctuations, which is absent in the RPA. The derived FQ fluctuations cause the softening of the ${C}_{66}$ shear modulus, and its long-range order not only triggers the orthorhombic structure transition, but also induces the instability of the stripe-type antiferromagnetic state. In other words, the condensation of composite bosons made of two orbitons gives rise to the FQ order and structure transition. Therefore, the theoretically predicted multi-orbital criticality presents a unified explanation for the above-mentioned features of iron pnictide superconductors.