  Topological insulators are found in materials that have elements with strong
spin orbit interaction. However, electron Coulomb repulsion also potentially
generates the topological insulators as well as Chern insulators by the
mechanism of spontaneous symmetry breaking, which is called topological Mott
insulators. The quantum criticality of the transition to the topological Mott
insulators from zero-gap semiconductors follows unconventional universality
distinct from the Landau-Ginzburg-Wilson scenario. On the pyrochlore lattice,
the interplay of the electron correlation and the spin orbit interaction
provides us in a rich phase diagram not only with simple topological insulators
but also with Weyl semimetal and topologically distinct antiferromagnetic
phases. Magnetic domain wall of the all-in-all-out type antiferromagnetic order
offers a promising candidate of magnetically controlled transport, because,
even when the Weyl points disappears, the domain wall maintains robust gapless
excitations with Fermi surfaces around it embedded in the bulk insulator and
bears uniform magnetization simultaneously. The ingap state is protected by a
mechanism similar to the solitons in polyacetylene. Puzzling experimental
results of pyrochlore iridates are favorably compared with the prediction of
the domain wall theory.
