Three-Spin Interactions in Optical Lattices and Criticality in Cluster Hamiltonians

We demonstrate that in a triangular configuration of an optical lattice of two atomic species a variety of novel spin-$1/2$ Hamiltonians can be generated. They include effective three-spin interactions resulting from the possibility of atoms tunneling along two different paths. This motivates the study of ground state properties of various three-spin Hamiltonians in terms of their two-point and $n$-point correlations as well as the localizable entanglement. We present a Hamiltonian with a finite energy gap above its unique ground state for which the localizable entanglement length diverges for a wide interval of applied external fields, while at the same time the classical correlation length remains finite.