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Lattice construction of pseudopotential Hamiltonians for Fractional Chern Insulators

Fractional Chern insulators are new realizations of fractional quantum Hall states in lattice systems without orbital magnetic field. These states can be mapped onto conventional fractional quantum Hall states through the Wannier state representation (Phys. Rev. Lett. 107, 126803 (2011)). In this paper, we use the Wannier state representation to construct the pseudopotential Hamiltonians for fractional Chern insulators, which are interaction Hamiltonians with certain ideal model wavefunctions as exact ground states. We show that these pseudopotential Hamiltonians can be approximated by short-ranged interactions in fractional Chern insulators, and that their range will be minimized by an optimal gauge choice for the Wannier states. As illustrative examples, we explicitly write down the form of the lowest pseudopotential for several fractional Chern insulator models including the lattice Dirac model and the checkerboard model with Chern number 1, and the d-wave model and the triangular lattice model with Chern number 2. The proposed pseudopotential Hamiltonians have the 1/3 Laughlin state as their groundstate when the Chern number C=1, and a topological nematic (330) state as their groundstate when C=2. Also included are the results of an interpolation between the d-wave model and two decoupled layers of lattice Dirac models, which explicitly demonstrate the relation between C=2 fractional Chern insulators and bilayer fractional quantum Hall states. The proposed states can be verified by future numerical works, and in particular provide a model Hamiltonian for the topological nematic states that have not been realized numerically.