Quantum phase transition between cluster and antiferromagnetic states
  We study a Hamiltonian system describing a three spin-1/2 cluster-like interaction competing with an Ising-like exchange. We show that the ground state in the cluster phase possesses symmetry protected topological order. A continuous quantum phase transition occurs as result of the competition between the cluster and Ising terms. At the critical point the Hamiltonian is self-dual. The geometric entanglement is also studied. Our findings in one dimension corroborate the analysis of the two dimensional generalization of the system, indicating, at a mean field level, the presence of a direct transition between an antiferromagnetic and a valence bond solid ground state. 
Wonmin Son, Luigi Amico, Rosario Fazio, Alioscia Hamma, Saverio   Pascazio, Vlatko Vedral
quant-ph cond-mat.stat-mech