Characterization of topological insulators: Chern numbers for ground state multiplet

We propose the use of generic Chern numbers for the characterization of topological insulators. It is suitable for the numerical characterization of low-dimensional quantum liquids, in which strong quantum fluctuations prevent the development of conventional orders. Using the twisting parameters of boundary conditions, the non-Abelian Chem numbers are defined for a few low-lying states near the ground state in a finite system, which is a ground state multiplet with a possible (topological) degeneracy. We define the system as a topological insulator when energies of the multiplet are well separated from the above. Translational invariant twists up to a unitary equivalence are crucial to picking up only bulk properties without edge states. As a simple example, the setup is applied to a two-dimensional XXZ-spin system with an Ising anisotropy, in which the ground state multiplet is composed of doubly almost degenerate states. It gives a vanishing Chern number due to symmetry. Also Chern numbers for the generic fractional quantum Hall states are discussed briefly.