Spin dynamical properties and orbital states of the layered perovskite ${\mathrm{La}}_{2-2x}{\mathrm{Sr}}_{1+2x}{\mathrm{Mn}}_2{\mathrm{O}}_7\hspace{0.5em}(0.3\mathrm{}<~x<\mathrm{}0.5)$

Low-temperature spin dynamics of the double-layered perovskite ${\mathrm{La}}_{2-2x}{\mathrm{Sr}}_{1+2x}{\mathrm{Mn}}_2{\mathrm{O}}_7$ (LSMO327) was systematically studied in a wide hole concentration range $(0.3\mathrm{}<~x<\mathrm{}0.5).\mathrm{}$ The spin-wave dispersion, which is almost perfectly two-dimensional, has two branches due to a coupling between layers within a double-layer. Each branch exhibits a characteristic intensity oscillation along the out-of-plane direction. We found that the in-plane spin stiffness constant and the gap between the two branches strongly depend on x. By fitting to calculated dispersion relations and cross sections assuming a Heisenberg model, we have obtained the in-plane ${(J}_{\Vert }),$ intra-bilayer ${(J}_{\perp })$ and inter-bilayer ${(J}^{\prime })$ exchange interactions at each x. At $x=0.30,$ $J_{\Vert }=-4\mathrm{meV}$ and $J_{\perp }=-5\mathrm{meV},$ namely almost isotropic and ferromagnetic. Upon increasing x, $J_{\perp }$ rapidly approaches zero while $|J_{\Vert }|$ increases slightly, indicating an enhancement of the planar magnetic anisotropy. At $x=0.48,$ $J_{\Vert }$ reaches $-9\mathrm{meV},$ while $J_{\perp }$ turns to $+1\mathrm{}\mathrm{meV},$ indicating an antiferromagnetic interaction. Such a drastic change of the exchange interactions can be ascribed to the change of the relative stability of the $d_{x^2-y^2}$ and $d_{{3z}^2-r^2}$ orbital states upon doping. However, a simple linear combination of the two states results in an orbital state with an orthorhombic symmetry, which is inconsistent with the $I4/mmm$ tetragonal symmetry of the crystal structure. We thus propose that an “orbital liquid” state realizes in LSMO327, where the charge distribution symmetry is kept tetragonal around each Mn site. Orbital liquid states are formulated in a theoretical model which takes into account strong electron correlations. The calculated results satisfactorily explain the systematic changes of the exchange interactions in LSMO327 observed in the experiments.