Quantum criticality and coexistence of spontaneous ferromagnetism and field- induced metamagnetism in triple-layered Sr$_{4}$Ru$_{3}$O$_{10}$

Results of a thermodynamic and transport study of Sr$_{4}$Ru$_{3}$O$_{10}$ as a function of temperature and magnetic field are presented. The central results of this work include growing specific heat C with increasing field B, divergent magnetic contribution to C at low temperatures, an abrupt jump and a peak in C/T at B=2.90 T and =7 T for B$\vert \vert $ab-plane and B$\vert \vert $c-axis, respectively, and corresponding changes in the power law of resistivity. All results provide not only strong evidence for metamagnetic quantum criticality but also quantum fluctuations in a \textit{spontaneously} ferromagnetic state. The novelty of this work lies in the fact that the quantum criticality occurs in a system that shows both intralayer metamagnetism and interlayer \textit{spontaneous} ferromagnetism, a feature characteristically different from all other relevant systems involving quantum criticality.