Inter-surface interaction via phonon in three-dimensional topological insulator

We theoretically study the phonon mediated intersurface electron-electron interactions on the pseudo two-dimensional metallic states at the two surfaces of a three-dimensional topological insulator. From a model of a three-dimensional topological insulator including the phonon excitation in it, we derive the effective Lagrangian which describes the two surface metallic states and the interaction between them. The intersurface electron-electron interactions can be either repulsive or attractive depending on parameters such as temperature, the speed of the phonon, and the Fermi velocity of the surface states. The attractive interaction removes the Dirac nodes from the two surface states as a result of the spontaneous symmetry breaking. On the basis of the calculated results, we also discuss how to tune the inter-surface interaction.