  We study the relation between Chern numbers and Quantum Phase Transitions
(QPT) in the XY spin-chain model. By coupling the spin chain to a single spin,
it is possible to study topological invariants associated to the coupling
Hamiltonian. These invariants contain global information, in addition to the
usual one (obtained by integrating the Berry connection around a closed loop).
We compute these invariants (Chern numbers) and discuss their relation to QPT.
In particular we show that Chern numbers can be used to label regions
corresponding to different phases.
