Temperature evolution of polar states in GdMn$_2$O$_5$ and Gd$_{0.8}$Ce$_{0.2}$Mn$_2$O$_5$

The polar order along the $c$ axis is revealed in GdMn$_2$O$_5$ and Gd$_{0.8}$Ce$_{0.2}$Mn$_2$O$_5$ at $T\leq T_{C1}\approx 160$ K for the first time. This polar order is induced by the charge disproportion in the 2D superstructures emerged due to phase separation. The dynamic state with restricted polar domains of different sizes is found at $T > T_{C1}$ which is typical of the diffuse ferroelectric phase transition. At the lowest temperatures ($T < 40$ K) two polar orders of different origins with perpendicular orientations (along the $b$- and $c$-axes) coexist. The 1D superlattices studied by us earlier in the set of RMn$_2$O$_5$ multiferroics are the charged domain walls which separate of these polar order domains.