Polyhexatic and polycrystalline states of skyrmion lattices

We report Monte Carlo studies of lattices of up to ${10}^5$ skyrmions treated as particles with negative core energy and repulsive interaction obtained from a microscopic spin model. Temperature dependence of translational and orientational correlations has been investigated for different experimental protocols and initial conditions. Cooling the skyrmion liquid from a fully disordered high-temperature state results in the formation of a skyrmion polycrystal. A perfect skyrmion lattice prepared at $T=0$, on raising temperature undergoes a first-order melting transition into a polyhexatic state that consists of large orientationally ordered domains of fluctuating shape. On the further increasing temperature, these domains decrease in size, leading to a fully disordered liquid of skyrmions.

Figure 1

An outward Néel skyrmion, view from above (in the negative $z$ direction).

Figure 2

The scaled plot of the skyrmion size $\lambda$ vs $H$ for different values of $A$.

Figure 3

The energy of a single skyrmion $\mathrm{\Delta }E$ with respect to the uniform state and the ground-state energy of an equilibrium skyrmion lattice per skyrmion $E_0$ vs $H$.

Figure 4

Two repelling outward Néel skyrmions.

Figure 5

Triangular lattice of skyrmions in a nearly square system with periodic boundary conditions.

Figure 6

The equilibrium value of the skyrmion-lattice period $a_S$ vs $H$ for different values of $A$, compared with the skyrmion size $\lambda$ and the magnetic length ${\delta }_H$.

Figure 7

The nearest-neighbor repulsion energy $U_0$ in the equilibrium skyrmion lattice vs $H$ for different $A$.

Figure 8

Triangular lattice of skyrmions in a nearly square and rhomboid systems with rigid boundaries.

Figure 9

Polycrystalline structure of the skyrmion lattice in the magnetic film of circular shape with $N_S=9821$ skyrmions at $T=0$, obtained by cooling from high temperature.

Figure 10

Skyrmion lattice at $T/J=0.11$. Upper panel: “disclinations” found by computing the number of nearest neighbors. Skyrmions with 5 and 7 nearest neighbors are marked by blue and red points and encircled by blue and red circles of radius $R_0=a_S(\sqrt{3}+1)/2$. Lower panel: “dislocations” found by computing the Burgers vector shown by orange squares.

Figure 11

Temperature dependence of the energy per skyrmion $E$ for the lattice and random initial conditions with different Monte Carlo sensitivities.

Figure 12

Temperature dependence of the number of Monte Carlo steps done.

Figure 13

Temperature dependence of the dispersion of the nearest-neighbor distance in the skyrmion lattice.

Figure 14

Temperature dependence of the average hexagonality value $V_6$, Eq. (18).

Figure 15

Temperature dependence of translational and orientational order parameters in a nearly-square skyrmion lattice with pbc.

Figure 16

Temperature dependence of translational and orientational order parameters in a rhomboid skyrmion lattice with rigid boundaries, Fig. 8 (lower), with different system sizes. Upper panel: Orientational order parameter; Lower panel: Translational order parameter.

Figure 17

Temperature dependence of the concentration of dislocations, computed via the Burgers vector.

Figure 18

Translational correlation function in the skyrmion lattice at different temperatures.

Figure 19

Orientational correlation function in the skyrmion lattice at different temperatures. Upper panel: The results for LIC and RIC in the linear scale. Lower panel: The RIC results in the lin-log scale.

Figure 20

Evolution of the orientational order parameter $O_6$ in the course of melting at different temperatures. See the video of melting at $T/J=0.12$ in the Supplemental Material [65].

Figure 21

Evolution of the orientational order parameter $O_6$ starting from a random state at different temperatures.

Figure 22

Polyhexatic state at $T/J=0.12$ obtained from RIC in the course of the MC simulation (see Fig. 21). Upper panel: 580 000 MCS; Lower panel: 710 000 MCS. See the video of the evolution between these states (stationary fluctuations) in the Supplemental Material [65].