String order and adiabatic continuity of Haldane chains and band insulators

The ground state of spin-1 Haldane chains is characterized by the so-called string order. We show that the same hidden order is also present in ordinary one-dimensional band insulators. We construct a family of Hamiltonians which connect adiabatically band insulators to two topologically nonequivalent spin models, the Haldane chain and the antiferromagnetic spin-$1∕2$ ladder. We observe that the localized spin-$1∕2$ edge-state characteristic of spin-1 chains is smoothly connected to a surface-bound state of band insulators and its emergence is not related to any bulk phase transition. Furthermore, we show that the string order is absent in any dimensions higher than 1.