Magnetic Phases in Ca$2$Ru${1-x}$Mn$_x$O$_4$

The evolution of magnetic order in \textbf{Ca$_2$Ru$_{1-x}$Mn$_x$O$_4$} ($x = 0.03$--$0.14$) has been investigated using neutron diffraction. Mn doping preserves the low-temperature orthorhombic symmetry (\textit{Pbca}) but weakens the orthorhombic distortion by reducing the difference between the $a$ and $b$ axes. It also systematically enhances the antiferromagnetic transition temperature and modifies the balance between two competing magnetic structures: the \textit{A}-centered and \textit{B}-centered phases. With increasing Mn content, the \textit{B}-centered phase progressively expands at the expense of the \textit{A}-centered one, which becomes short-ranged at $x = 0.14$. These results underscore the crucial role of orbital degrees of freedom in driving the magnetic phase competition in this layered ruthenate.