  We propose a gauge-invariant system of the Chern-Simons-Schrodinger type on a
one-dimensional lattice. By using the spatial gauge condition, we prove local
and global well-posedness of the initial-value problem in the space of square
summable sequences for the scalar field. We also study the existence region of
the stationary bound states, which depends on the lattice spacing and the
nonlinearity power. A major difficulty in the existence problem is related to
the lack of variational formulation of the stationary equations. Our approach
is based on the implicit function theorem in the anti-continuum limit and the
solvability constraint in the continuum limit.
