Hierarchical single-ion anisotropies in spin-1 Heisenberg antiferromagnets on the honeycomb lattice

We consider two-dimensional spin-1 antiferromagnets with distinct single-ion anisotropies along two perpendicular axes on the 2D honeycomb lattice. For a single easy-plane anisotropy, a Berezinskii-Kosterlitz-Thouless (BKT) transition is known to appear, below which quasi long-range order emerges. Instead, in the case of different anisotropies along two perpendicular axes, a thermal Ising transition with (true) long-range magnetic order appears. Recent inelastic neutron scattering experiments on the spin-1 compound BaNi${}_2$V${}_2$O${}_8$ indicate, however, that near to the critical point reminiscent BKT scaling still prevails. We investigate this scenario by addressing basic spin-1 models for BaNi${}_2$V${}_2$O${}_8$ with weak anisotropies, which we probe over the full range of experimentally relevant correlation length scales using large-scale quantum Monte Carlo simulations.