Exact dimer phase with anisotropic interaction for one dimensional magnets

We report the exact dimer phase, in which the ground states are described by product of singlet dimer, in the extended XYZ model by generalizing the isotropic Majumdar-Ghosh model to the fully anisotropic region. We demonstrate that this phase can be realized even in models when antiferromagnetic interaction along one of the three directions. This model also supports three different ferromagnetic (FM) phases, denoted as $x$-FM, $y$-FM and $z$-FM, polarized along the three directions. The boundaries between the exact dimer phase and FM phases are infinite-fold degenerate. The breaking of this infinite-fold degeneracy by either translational symmetry breaking or $\mathbb{Z}_2$ symmetry breaking leads to exact dimer phase and FM phases, respectively. Moreover, the boundaries between the three FM phases are critical with central charge $c=1$ for free fermions. We characterize the properties of these boundaries using entanglement entropy, excitation gap, and long-range spin-spin correlation functions. These results are relevant to a large number of one dimensional magnets, in which anisotropy is necessary to isolate a single chain out from the bulk material. We discuss the possible experimental signatures in realistic materials with magnetic field along different directions and show that the anisotropy may resolve the disagreement between theory and experiments based on isotropic spin-spin interactions.