Oscillatory magnetoresistance periodic in $1/B$ and $B$ and specific heat of Ca$_{3}$Ru$_{2}$O$_{7}$*

We report magnetoresistance oscillations in high magnetic fields, B, up to 45 T and over a wide range of temperature in the Mott-like system Ca$_{3}$Ru$_{2}$O$_{7}$ [1]. For B rotating within the \textit{ac}-plane, slow and strong Shubnikov-de Haas (SdH) oscillations periodic in 1/B are observed for T$\le $1.5 K in the presence of metamagnetism. These oscillations are highly angular dependent and intimately correlated with the spin-polarization of the ferromagnetic state. For B$\vert \vert $ [110], oscillations are also observed \textit{but periodic in B }(rather than 1/B)\textit{ which persist up to 15 K. } While the SdH oscillations are a manifestation of the presence of small Fermi surface (FS) pockets in the Mott-like system, the B-periodic oscillations, an exotic quantum phenomenon, may be a result of anomalous coupling of the magnetic field to the $t_{2g}$-orbitals that makes the extremal cross-section of the FS field-dependent. In addition, measurements of the specific heat show sharp anomalies at 48 K and 56 K, the former with a strong frequency dependence consistent with a sluggish first-order transition. Much less entropy is removed at the upper, second-order transition. All results will be presented and discussed along with comparisons drawn from impurity doped Ca$_{3}$Ru$_{2}$O$_{7}$. [1] XN Lin,et al.,Phys.Rev.Lett.95,017203(2005)