Melting of Néel skyrmion lattice

Melting of a particle lattice in two dimensional systems substantially differs from the one in three dimensions. The most prominent theory of the lattice melting in 2D, formulated by Kosterlitz, Thouless, Halperin, Nelson, and Young (KTHNY), describes the solid-liquid phase transition as a two-step process, where the solid phase turns into the liquid via the hexatic phase. The microscopic nature of this process lies in the excitation and unbinding of topological defects in the 2D lattice. The KTHNY melting scenario has been observed in number of 2D physical systems. Here, we theoretically analyze the melting process of a Néel skyrmion lattice by means of numerical simulations using a model of lacunar spinel ${\mathrm{GaV}}_4{\mathrm{S}}_8$ hosting hexagonal skyrmion lattice at low temperatures. We show that topological defects are excited in such skyrmion lattice causing its melting via the hexatic phase in agreement with the KTHNY theory.

Figure 1

Hexagonal 2D lattice and definition of the angle ${\theta }_{nm}$ of a bond connecting two neighboring lattice points, $n$ and $m$, with respect to a fixed vector (in blue).

Figure 2

Skyrmion configurations at low temperatures $T\to 0\mathrm{K}$ calculated for different values of the applied magnetic field $B$. Figures show map of the magnetization $z$ component in a $(200\times 200)a^2$ sized square fragment of the simulated lattice, where $a$ is the lattice constant.

Figure 3

Translational correlation functions calculated with applied magnetic field $B=1.2\mathrm{T}$ at different temperatures $T$. For comparison, the dashed line is the critical translational correlation decay given by $G\left(r\right)\propto r^{-1/3}$.

Figure 4

Normalized diffuse scattering intensity calculated for the skyrmion systems at applied magnetic field $B=1.2\mathrm{T}$ and temperatures (a) $2\mathrm{K}$, (b) $3.01\mathrm{K}$, (c) $3.03\mathrm{K}$, and (d) $4\mathrm{K}$.

Figure 5

Global orientational order parameter as a function of temperature calculated for different values of the applied magnetic field.

Figure 6

Orientational correlation functions calculated for applied magnetic field $B=1.2\mathrm{T}$ for different temperatures $T$. The grey points are the calculated correlation functions while the color lines are the fits of their local maxima to power-law (for temperatures $2.0-3.03\mathrm{K}$) or exponential (for temperatures $3.2-4.0\mathrm{K}$) function. For comparison, the black dashed curve corresponds to power-law decay $\propto r^{-1/4}$.

Figure 7

(a) Parameter ${\eta }_6$ as a function of temperature for magnetic field $B=1.2\mathrm{T}$. The red solid line is a linear fit of the points in the interval $\langle 3.0,3.1\rangle$, where ${\eta }_6$ increases. The dashed line crosses the function in ${\eta }_6=1/4$. (b) The open circles show the total average number of defects in the skyrmion lattice as a function of temperature, while the full circles shows average number of isolated disclinations.

Figure 8

Finite-size scaling of the global orientational order parameter calculated in sub-blocks of size $L_{\mathrm{n}}=L/n$ for $n=1,2$, and 4 calculated for different temperatures. Temperatures and colors of the dashed lines correspond to those in Fig. 6. The black solid lines show functions with slopes $-1/4$ and $-2$.

Figure 9

Topological defects and local orientational order parameter calculated at applied magnetic field $B=1.2\mathrm{T}$ at different temperatures; [(a), (b)] $T=2\mathrm{K}$, [(c), (d)] $T=3.01\mathrm{K}$, [(e), (f)] $T=4.5\mathrm{K}$. In the first column we show connections between nearest neighboring skyrmions obtained by Delaunay triangulation. Each point of the net corresponds to one skyrmion center. Moreover, in the graph we emphasize the topological defects in the skyrmion lattice. Each orange point corresponds to a skyrmion with 7 neighbors while the blue one has just 5 neighbors. The green dots correspond to skyrmions with different numbers of neighbors than 5, 6, or 7. The second column shows the spatial distribution of the local orientational order parameter. Each disk in the plots (b), (d), and (f) corresponds to one skyrmion, its color and arrow direction are related to the argument of the complex local orientational order parameter.