Spin Supersolid in Anisotropic Spin-One Heisenberg Chain

We consider an $S=1$ Heisenberg chain with strong exchange ($\Delta=J_z/J_{\perp}$) and single--ion uniaxial anisotropy ($D$) in a magnetic field ($B$) along the symmetry axis. The low energy spectrum is described by an effective $S=1/2$ XXZ model that acts on two different low energy sectors for a finite range of fields. The vacuum of each sector exhibits Ising-like antiferro magnetic ordering coexisting with the finite spin stiffness obtained from the exact solution of the XXZ model. In this way, we demonstrate the existence of a spin supersolid phase. We also compute the full $\Delta-B$ quantum phase d iagram using a quantum Monte Carlo (QMC) method.