  We propose an index for pairs of a unitary map and a clustering state on
many-body quantum systems. We require the map to conserve an integer-valued
charge and to leave the state, e.g. a gapped ground state, invariant. This
index is integer-valued and stable under perturbations. In general, the index
measures the charge transport across a fiducial line. We show that it reduces
to (i) an index of projections in the case of non-interacting fermions, (ii)
the charge density for translational invariant systems, and (iii) the quantum
Hall conductance in the two-dimensional setting without any additional
symmetry. Example (ii) recovers the Lieb-Schultz-Mattis theorem, and (iii)
provides a new and short proof of quantization of Hall conductance in
interacting many-body systems.
