Entanglement spectrum of a topological phase in one dimension

We propose a scheme to classify gapped phases of one dimensional systems in terms of properties of the entanglement spectrum. We show that the Haldane phase of $S=1$ chains is characterized by a double degeneracy of the entanglement spectrum which is protected by any one of the following three symmetries: (i) the dihedral group of $\pi$-rotations about $x,y$ and $z$ axes; (ii) time-reversal symmetry $S^{x,y,z} \rightarrow - S^{x,y,z}$; (iii) link inversion symmetry. The degeneracy cannot be lifted unless either a phase boundary to another, ``topologically trivial'', phase is crossed, or the symmetry is broken. Physically, the degeneracy of the entanglement spectrum can be observed by adiabatically weakening a bond to zero, which leaves the two disconnected halves of the system in a finitely entangled state.