Wigner crystallization at large fine structure constant

We consider the fate of the Wigner crystal state in a two dimensional system of massive Dirac electrons as the effective fine structure constant $\alpha$ is increased. In a Dirac system, larger $\alpha$ naively corresponds to stronger electron-electron interactions, but it also implies a stronger interband dielectric response that effectively renormalizes the electron charge. We calculate the critical density and critical temperature associated with quantum and thermal melting of the Wigner crystal state using two independent approaches. We show that at $\alpha \gg 1$, the Wigner crystal state is best understood in terms of logarithmically-interacting electrons, and that both the critical density and the melting temperature approach a universal, $\alpha$-independent value. We discuss our results in the context of recent experiments in twisted bilayer graphene near the magic angle.