Unusual heavy-mass nearly ferromagnetic state with a surprisingly large Wilson ratio in the double layered ruthenates (Sr$_{1-x}$Ca$_{x}$)$_{3}$Ru$_{2}$O$_{7}$

We report an unusual nearly ferromagnetic, heavy-mass state with a surprisingly large Wilson ratio $R_{textrm{w}}$ (e.g., $R_{textrm{w}}sim$ 700 for $x =$ 0.2) in double layered ruthenates (Sr$_{1-x}$Ca$_{x}$)$_{3}$Ru$_{2}$O$_{7}$ with 0.08 $< x <$ 0.4. This state does not evolve into a long-range ferromagnetically ordered state despite considerably strong ferromagnetic correlations, but freezes into a cluster-spin-glass at low temperatures. In addition, evidence of non-Fermi liquid behavior is observed as the spin freezing temperature of the cluster-spin-glass approaches zero near $x approx$ 0.1. We discuss the origin of this unique magnetic state from the Fermi surface information probed by Hall effect measurements.