Fig. 8

Snapshots of spin configurations at different temperatures for a suspended phosphorene lattice (no strain) with \({\text{J}}/{\text{J}}^{\prime } = 3^{\frac{3}{2}} = 3.22\) (a to f) and for a phosphorene on a substrate (strained phosphorene) with \({\text{J}}/{\text{J}}^{\prime } = 1.5\) (g to l), where a dark blue square denotes a spin up atom and a yellow square denotes a spin down atom. At very low temperatures for the suspended phosphorene (a and b) there are small ordered regions separated by frozen defects. This is a glassy state. There are no such small ordered regions separating by frozen defects in the low-temperature configurations of strained phosphorene (g and h). It means that there is no glassy state at \({\text{J}}/{\text{J}}^{\prime } = 1.5\). At intermediate temperatures (c, d, i and j) the spin configuration becomes partially ordered. This ordered state disappears at high temperatures. At high temperatures (e, f, k and l) the spin directions become random. There are no clear ordered regions separating by domain boundaries. This disordered state consists of many short-range defects whose creation is due to thermal fluctuations. To further visualize, understand, and track the spin behavior of the phosphorene lattice—particularly near the phase transition points—we have added to the supplementary information a continuous series of spin configuration snapshots for temperatures ranging from 0.1 to 4.9, in steps of 0.2. These snapshots are provided for two different coupling ratios, \(J/J^{\prime } = 1.5\) and \(J/J^{\prime } = 3.2\), each representing a regime on either side of the transition point at \(J/J^{\prime } = 2\), as shown in Figures S5 and S6. This set of visualizations offers valuable insight into the behavior of different spin phases in the phosphorene lattice, as well as the formation and annihilation of defects across the temperature range. We also highlighted the defects in one of the snapshots of the phosphorene lattice, as shown in Figure S7.