Emil J. Bergholtz, Zhao Liu
Topological insulators are accompanied by exotic edge states that are protected by a bulk single-particle band gap once the filled bands are characterized by non-trivial topological invariants. Interactions can have profound effects and lead to entirely new insulating phases, with an altogether much richer and less explored phenomenology, as is particularly clear in the case of partial filling of weakly dispersive bands. Most saliently, lattice generalizations of fractional quantum Hall states, dubbed fractional Chern insulators, have recently been predicted to be stabilized by interactions within nearly dispersionless bands with non-zero Chern number, $C$. Contrary to their continuum analogues, these states do not require an external magnetic field and may potentially persist at high temperatures, which make these systems very interesting in the context of applications such as topological quantum computation. This review recapitulates the basics of tight-binding models hosting nearly flat bands with non-trivial topology, $C\neq 0$, and summarizes the present understanding of interactions and strongly correlated phases within these bands. Emphasis is put on the analogy with continuum Landau level physics, as well as qualitatively new, lattice specific, aspects including Berry curvature fluctuations, competing instabilities as well as novel collective states of matter emerging in bands with $|C|>1$. Possible experimental realizations, including oxide interfaces and cold atom implementations as well as generalizations to flat bands characterized by other topological invariants are also discussed.
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http://arxiv.org/abs/1308.0343
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