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A large number of recent works point to the emergence of intriguing analogues of fractional quantum Hall (FQH) states in two-dimensional lattice models due to effective interactions in nearly flat bands with Chern number C=1. In principle, this opens up a number of intriguing possibilities including lattice realizations of quantum Hall phenomena at room temperature and zero magnetic field. Here, I will summarize some of our recent work in the area. First, I will discuss how the most prominent fractional Chern insulators (FCIs) in C=1 bands quantitatively compare to Abelian and non-Abelian FQH states by using a Wannier basis representation of the Chern band. Second, I will give a heuristic explanation of the numerically observed FCIs and their relative stability by relating the lattice interactions to Haldane's pseudopotentials in the continuum, as well as discuss a new interaction driven instability to incompressible states that is absent in continuum systems and is responsible for the absence of some FCIs. Third, I will explain how flat bands with higher Chern number, C>1, naturally form in quasi-two-dimensional slabs of pyrochlore oxides and demonstrate the existence of qualitatively new correlated states within these bands. |
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