Phase Separation in A-site Ordered Perovskite Manganite LaBaMn${}_2$O${}_6$ Probed by ${}^{139}$La and ${}^{55}$Mn NMR

${}^{139}$La- and ${}^{55}$Mn-NMR spectra demonstrate that the ground state of the A-site ordered perovskite manganite LaBaMn${}_2$O${}_6$ is a spatial mixture of the ferromagnetic (FM) and antiferromagnetic (AFI(CE)) regions, which are assigned to the metallic and the insulating charge ordered state, respectively. This exotic coexisting state appears below 200 K via a first-order-like formation of the AFI(CE) state inside the FM one. Mn spin-spin relaxation rate indicates that the FM region coexisting with the AFI(CE) one in LaBaMn${}_2$O${}_6$ is identical to the bulk FM phase of the disordered form La${}_{0.5}$Ba${}_{0.5}$MnO${}_3$ in spite of the absence of A-site disorder. This suggests mesoscopic rather than nanoscopic nature of FM region in LaBaMn${}_2$O${}_6$@.