Nematic order in Sr$_3$Ru$_2$O$_7$ and triplet superconductivity in Sr$_2$RuO$_4$

In an externally applied magnetic field, ultra-pure crystals of the bilayer ruthenate compound Sr$_3$Ru$_2$O$_7$ undergo a metamagnetic transition at a temperature which can be tuned towards zero as a function of the angle between B and the crystalline c-axis. This ``metamagnetic quantum critical point", however, is enveloped by a nematic fluid phase with order one resistive anisotropy in the ab plane. In this talk, I will discuss the microscopic origins of metamagnetism and the accompanying nematic order in this system. I propose that both can be understood within the framework of an orbital-ordering tendency of the material and present a phase diagram which accounts for much of the experimentally observed phenomena. By contrast, the closely related monolayer compound, Sr$_2$RuO$_4$, is a spin-triplet superconductor; it does not exhibit metamagnetism or nematic order. This drastic difference in the physical properties of the two materials - despite their structural similarity - points towards a possible microscopic mechanism of triplet superconductivity in Sr$_2$RuO$_4$. I will conclude the talk with a discussion of our recent progress in understanding the microscopic origins of superconductivity in Sr$_2$RuO$_4$.