Critical behavior of a quantum chain with four-spin interactions in the   presence of longitudinal and transverse magnetic fields
  We study the ground-state properties of a spin-1/2 model on a chain containing four-spin Ising-like interactions in the presence of both transverse and longitudinal magnetic fields. We use entanglement entropy and finite-size scaling methods to obtain the phase diagrams of the model. Our numerical calculations reveal a rich variety of phases and the existence of multi-critical points in the system. We identify phases with both ferromagnetic and anti-ferromagnetic orderings. We also find periodically modulated orderings formed by a cluster of like-spins followed by another cluster of opposite like-spins. The quantum phases in the model are found to be separated by either first or second order transition lines. 
B. Boechat, J. Florencio, A. Saguia, O. F. de Alcantara Bonfim
cond-mat.stat-mech