Topological phases of the rainbow chain

Inhomegenous quantum many body systems exhibit novel phenomena such as the breaking of the area law of entanglement entropy. A notable example is the so called rainbow model which consists of a XX spin chain where the exchange coupling constants decrease exponentially from the center towards the edges. This system has been analyzed in great detail using the strong disorder renormalization group method and conformal field theory. These methods explain the appearance of long range valence bonds across the spin chain, having the shape of a rainbow. In this talk, we shall explain another feature which is the appearance of topological order in the rainbow chain with an odd number of sites. In the particular case when there is rotational symmetry one obtains the Haldane phase. This result indicates that the short range entanglement characteristic of the SPT phases can be related to the long range entanglement of the critical phases by considering the inhomogenous deformation of the latters.