Proposal of a spin-one chain model with competing dimer and trimer interactions

A new kind of spin-one chain Hamiltonian consisting of competing dimer and trimer projection operators is proposed. As the relative strengths and signs of the interactions are varied, the model exhibits a number of different phases including the gapped dimer phase and the gapless trimer phase with critical correlations described by a conformal field theory with central charge $c=2$. A symmetry-protected topological phase also exists in this model, even though the microscopic interactions are not the simple adiabatic extensions of the well-known Heisenberg and Affleck-Kennedy-Lieb-Tasaki models and contain both two- and three-particle permutations. A fourth phase is characterized by macroscopically degenerate ground states. While bearing almost a one-to-one resemblance to the phase diagram of the bilinear-biquadratic spin-one chain Hamiltonian, our model is rooted in a very different physical origin, namely, the two competing tendencies of spin-one particles to form singlets through either dimer or trimer formation.