  In the field of fractional Chern insulators, a great deal of effort has been
devoted to characterizing Chern bands that exhibit properties similar to the
Landau levels. Among them, the concept of the vortex function, which
generalizes the complex coordinate used for the symmetric-gauge Landau-level
basis, allows for a concise description. In this paper, we develop a theory of
constructing the vortex function and basis states of Chern insulators in the
tight-binding formalism. In the first half, we consider the optimization
process of the vortex function, which minimizes an indicator that measures the
difference from the ideal Chern insulators. In particular, we focus on the
sublattice position dependence of the vortex function or the quantum geometric
tensor. This degree of freedom serves as a discrete analog of the
non-uniformity in the spatial metric and magnetic field in a continuous model.
In the second half, we construct two types of basis sets for a given vortex
function: radially localized basis set and coherent-like basis set. The former
basis set is defined as the eigenstates of an analogy of the angular momentum
operator. Remarkably, one can always find exact zero mode(s) for this operator,
which is explained by the celebrated Atiyah-Singer index theorem. As a
byproduct, we propose an inequality rooted in the band topology. We also
discuss the subtle differences between our formalism and the previous works
about the momentum-space Landau level. The latter basis set generalizes the
concept of coherent states on von Neumann lattice. While this basis set is not
orthogonal, it is useful to compare the LLL and the given Chern insulator
directly in the Brillouin zone. These basis sets are expected to be useful for
many-body calculations of fractional Chern insulators.
