  The main features in iron-pnictide superconductors are summarized as (i) the
orthorhombic transition accompanied by remarkable softening of shear modulus,
(ii) high-Tc superconductivity close to the orthorhombic phase, and (iii)
stripe-type magnetic order induced by orthorhombicity. To present a unified
explanation for them, we analyze the multiorbital 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
($\lambda ~ 0.2$) is enough to produce large orbital (=charge quadrupole)
fluctuations. The most divergent susceptibility is the
$O_{xz}$-antiferro-quadrupole (AFQ) susceptibility, which causes the s-wave
superconductivity without sign reversal (s_{++}-wave state). At the same time,
divergent development of $O_{x2-y2}$-ferro-quadrupole (FQ) susceptibility is
brought 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
$C_{66}$ shear modulus, and its long-range-order not only triggers the
orthorhombic structure transition, but also induces the instability of
stripe-type antiferro-magnetic state. In other words, the condensation of
composite bosons made of two orbitons gives rise to the FQ order and structure
transition. The theoretically predicted multi-orbital-criticality presents a
unified explanation for abovementioned features of iron pnictide
superconductors.
