Abstract

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 order with a fluctuation-driven vestigial phase, characterized by a composite order parameter. 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 states of algebraic order in frustrated magnets and a mechanism for charge-4e superconductivity with half flux quanta, accompanied by Majorana bound states in triplet superconductors. The formalism, based on a symmetry classification of vestigial order, provides a general framework to understand the complexity of quantum materials. Electronic states with scalar and vector chiral order, spin-nematic order, Ising-nematic order, time-reversal symmetry-breaking order, and algebraic vestigial order emerge from one underlying principle.