Electron correlation effects on topological phases

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.