Fig. 1. Spin structures and field-induced topological transition of the thermodinamically phases in monolayer NiI₂.

a, b Snapshots of spin configurations at T = 1 K from MC simulations on a 24 × 24 supercell obtained for B_z/J^1iso = 0 and B_z/J^1iso ≃ 1.5, displaying the antibiskyrmion and Bloch-skyrmion lattice, respectively. Black arrows represent in-plane components of spins, colormap indicates the out-of-plane (s_z) spin components. The magnetic unit cell is shown with dashed lines; dashed arrows in a are guidelines for the eyes to visualize the spins orientation and directions defining the antibiskyrmion. Insets show correspondent topological charge densities Ω_i in the selected magnetic unit cell. c The spin structure factor S(q), corresponding to a triple-q state with q₁ = (δ, δ), q₂ = (δ, − 2δ), and q₃ = ( − 2δ, δ) in the hexagonal setting, as highlighted by green arrows, with δ ≃ 0.125. S(q) on the left refers to the A2Sk state; S(q) on the right refers to the Sk state with the additional peak at Γ reflecting the ferromagnetic component induced by the applied magnetic field. d Topological charge ∣Q∣ (closed circles) per magnetic unit cell and corresponding topological susceptibility χ_Q (closed squares) as a function of temperature for B_z/J^1iso = 0, pointing to a transition temperature T_c ≃ 28 K within the used DFT approximations. e Evolution of ∣Q∣, normalized to the number of skyrmionic objects in the magnetic unit cell and of magnetization M_z as a function of the magnetic field B_z at T = 1 K. f Phase diagram in the temperature-field plane. Colormap indicates the topological charge densities ∣Q∣ as defined in e. Critical temperatures (T_c) for topological phase transitions are also reported in circles.