  Topologically ordered systems exhibit large-scale correlation in their ground
states, which may be characterized by quantities such as topological
entanglement entropy. We propose that the concept of irreducible many-body
correlation, the correlation that cannot be implied by all local correlations,
may also be used as a signature of topological order. In a topologically
ordered system, we demonstrate that for a part of the system with holes, the
reduced density matrix exhibits irreducible many-body correlation which becomes
reducible when the holes are removed. The appearance of these irreducible
correlations then represents a key feature of topological phase. We analyze the
many-body correlation structures in the ground state of the toric code model in
an external magnetic field, and show that the topological phase transition is
signaled by the irreducible many-body correlations.
