  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.

Mesoscale and Nanoscale Physics