Quantum phase transitions in the transverse Ising model with three-spin interactions

The ground-state properties of the transverse Ising model with competing three-spin interactions are investigated using exact diagonalization for lattice sizes up to 24 spins. The phase diagram in the $(J_2,J_3)$-plane for fixed magnetic field is obtained using the fidelity susceptibility method. Here, $J_2 < 0$ (AF) and $J_3$ > 0 (F) are the energy couplings for the two- and three-spin interactions, respectively. The ground-state phase diagram shows an antiferromagnetic phase near the $J_2$-axis and a <1,2>-phase near the $J_3$-axis, which are always separated by a paramagnetic phase in between them. Therefore we do not observe any tricrical point, contrary to what has been reported in the literature.