  Recent theory and experiment have revealed that strong spin-orbit coupling
can have dramatic qualitative effects on the band structure of weakly
interacting solids. Indeed, it leads to a distinct phase of matter, the
topological band insulator. In this paper, we consider the combined effects of
spin-orbit coupling and strong electron correlation, and show that the former
has both quantitative and qualitative effects upon the correlation-driven Mott
transition. As a specific example we take Ir-based pyrochlores, where the
subsystem of Ir 5d electrons is known to undergo a Mott transition. At weak
electron-electron interaction, we predict that Ir electrons are in a metallic
phase at weak spin-orbit interaction, and in a topological band insulator phase
at strong spin-orbit interaction. Very generally, we show that with increasing
strength of the electron-electron interaction, the effective spin-orbit
coupling is enhanced, increasing the domain of the topological band insulator.
Furthermore, in our model, we argue that with increasing interactions, the
topological band insulator is transformed into a "topological Mott insulator"
phase, which is characterized by gapless surface spin-only excitations. The
full phase diagram also includes a narrow region of gapless Mott insulator with
a spinon Fermi surface, and a magnetically ordered state at still larger
electron-electron interaction.
