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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 $m$ (${T, m \ll Δ}$) 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 $\{μ_{0},μ_ϕ\}$ where $μ_{0}$ 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 $T_{\text{c}}$ only in some special cases (high concentration of paramagnetic impurities. 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 SDW and obtain the values of $\{μ_{0}, μ_ϕ\}$ 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.