A hallmark of the phase diagrams of quantum materials is the existence of multiple electronic ordered states. In many cases those are not independent, competing phases, but instead display a complex intertwinement. In this talk, we focus on a realization of intertwined orders with a fluctuation-driven vestigial phase characterized by a composite order parameter. In other words, we are investigating the order of fluctuations.
We demonstrate that this concept naturally explains the nematic state in iron-based superconductors and nematic superconductivity in doped topological insulators. In addition we propose a natural mechanism for charge 4e superconductivity with half flux quanta. We present a formalism that provides a framework to understand the complexity of quantum materials based on symmetry, largely without resorting to microscopic models.