Soft tilt and rotational modes in the hybrid improper ferroelectric Ca$_{3}$Mn$_{2}$O$_{7}$

Raman spectroscopy is employed to probe directly the soft rotation and tilting modes, which are two primary order parameters predicted in the hybrid improper ferroelectric material Ca$_3$Mn$_2$O$_7$. We observe a giant softening of the 107-cm$^{-1}$ octahedron tilting mode by 26~cm$^{-1}$, on heating through the structural transition from a ferroelectric to paraelectric orthorhombic phase. This is contrasted by a small softening of the 150-cm$^{-1}$ rotational mode by 6~cm$^{-1}$. In the intermediate phase, the competing soft modes with different symmetries coexist, bringing about many-faceted anomalies in spin excitations and lattice vibrations. Our work demonstrates that the soft rotation and tilt patterns, relying on a phase-transition path, are a key factor in determining ferroelectric, magnetic, and lattice properties of Ca$_3$Mn$_2$O$_7$.