Time-dependent equation for the magnetic order parameter near the quantum critical point in multiband superconductors with a spin-density wave

Using a simple two-band model for Fe-based pnictides and the generalized Eilenberger equation, we present a microscopic derivation of a time-dependent equation for the amplitude of the spin-density wave near the quantum critical point where it turns to zero. This equation describes the dynamics of the magnetic (m), as well as the superconducting order parameter (Δ). It is valid at low temperatures T and small m (T,mΔ) in a region of coexistence of both order parameters, m and Δ. The boundary of this region is found in the space of the nesting parameter μ₀,μ_φ, where μ₀ describes the relative position of the electron and the hole pockets on the energy scale and μ_φ accounts for the ellipticity of the electron pocket. At low T the number of quasiparticles is small due to the presence of the energy gap Δ, and therefore the quasiparticles do not play a role in the relaxation of m. This circumstance allows one to derive the time-dependent equation for m in contrast to the case of conventional superconductors for which the time-dependent Ginzburg-Landau equation can be derived near Tc only in some special cases (high concentration of paramagnetic impurities) [L. P. Gor'kov and G. M. Eliashberg, Sov. Phys. JETP 27, 328 (1968)]. In the stationary case the derived equation is valid at arbitrary temperatures. We find a solution of the stationary equation which describes a domain wall in the magnetic structure. In the center of the domain wall the superconducting order parameter has a maximum, which means a local enhancement of superconductivity. Using the derived time-dependent equation for m, we investgate also the stability of a uniform commensurate spin-density wave (SDW) and obtain the values of μ₀,μ_φ at which the first-order transition into the state with $m=0$ takes place or the transition to the state with an inhomogeneous SDW occurs.