Crystal-symmetry preserving Wannier states for fractional Chern insulators

Abstract

Recently, many numerical evidences of fractional Chern insulator, i.e., the fractional quantum Hall states on lattices, are proposed when a Chern band is partially filled. Some trial wave functions of fractional Chern insulators can be obtained by mapping the fractional quantum Hall wave functions defined in the continuum onto the lattice through the Wannier state representation [Phys. Rev. Lett. 107, 126803 (2011)] in which the single particle Landau orbits in the Landau levels are identified with the one-dimensional Wannier states of the Chern bands with Chern number $C=1$. However, this mapping generically breaks the lattice point group symmetry. In this paper we discuss a general approach of modifying the mapping to accommodate the lattice rotational symmetry. The wave functions constructed through this modified mapping should serve as better trial wave functions in the thermodynamical limit and on the rotationally invariant finite lattice. Also these wave functions will form a good basis for the construction of lattice symmetry preserving pseudopotential formalism for fractional Chern insulators. The focus of this paper shall be mainly on the ${C}_{4}$ rotational symmetry of square lattices. Similar analysis can be straightforwardly generalized to triangular or hexagonal lattices with ${C}_{6}$ symmetry. We also generalize the discussion to the lattice symmetry of fractional Chern insulators with high Chern number bands.