Abstract: | In this talk I will discuss the fate of a Dirac liquid coupled to a ferroelectric order parameter at its quantum critical point. Using the renormalization group technique I will show that the coupling is always relevant, thus rendering the transition a strong coupling problem. Using Eliashberg theory, I will show the strongly coupled problem avoids the critical point through a first order transition from a paraelerctric Dirac liquid to a superconductor coexisting with a ferroelectric state. Furthermore, I will show the ferroelectric state is a non-uniform wave, which we dub a "ferroelectric density wave". Given that this theory predicts a finite superconducting transition temperature at charge neutrality, where the charge density and density of states are zero, this result contributes to our understanding of how superconductivity emerges in doped materials such as the rocksalt alloy Sn_x Pb_{1-x} Te. |