Topological dynamics of vortex-line networks in hexagonal manganites

The two-dimensional $XY$ model is the first well-studied system with topological point defects. On the other hand, although topological line defects are common in three-dimensional systems, the evolution mechanism of line defects is not fully understood. The six domains in hexagonal manganites converge to vortex lines in three dimensions. Using phase-field simulations, we predicted that during the domain coarsening process, the vortex-line network undergoes three types of basic topological changes, i.e., vortex-line loop shrinking, coalescence, and splitting. It is shown that the vortex-antivortex annihilation controls the scaling dynamics.

Figure 1

2D and 3D domain configurations. (a) Part of the domain structures from a 2D simulation on the basal plane with a system size of $1024\mathrm{\Delta }x\times 1024\mathrm{\Delta }x\times 1\mathrm{\Delta }x$ and a grid spacing of $\mathrm{\Delta }x=0.30\mathrm{nm}$. The six colors represent six types of domains. (b) Domain structures from a 3D simulation with a system size of $128\mathrm{\Delta }x\times 128\mathrm{\Delta }x\times 128\mathrm{\Delta }x$ and a grid spacing of $\mathrm{\Delta }x=0.30\mathrm{nm}$. (c) Vortex lines in 3D spaces corresponding to the domain structures in (b). The brown dots denote the intersection of vortex lines and surfaces.

Figure 2

Demonstration of vortex-core structures. (a) and (b) Distribution of the structural trimerization magnitude $Q$ near a vortex core and an antivortex core, respectively. The domain structures are from 2D simulations on the basal plane with a grid spacing of $\mathrm{\Delta }x=0.020\mathrm{nm}$, and the height represents the value of $Q$ with an equilibrium bulk value of $9.5\times {10}^{-2}\mathrm{nm}$. (c) and (d) The corresponding atomic structures near a vortex core and an antivortex core, respectively. The vortex cores have a size of about one unit cell, and show the paraelectric phase with $Q\sim 0$. The red arrows in (c) and (d) represent the displacement directions of oxygen atoms.

Figure 3

Evolution of vortex structures during vortex-antivortex annihilation. (a)–(d) Vortex structures at different simulation time steps. The simulation is on the 2D basal plane and the height represents the magnitude of the structural order parameter $Q$. In (c) the two red domains are connected by a point with $Q\sim 0$, and the point separates the two light blue domains as well.

Figure 4

Topological evolution of vortex lines. (a)–(c) Shrinking, (d)–(f) coalescence, and (g)–(i) splitting of vortex-line loops.

Figure 5

2D and 3D coarsening dynamics. (a) Average domain area as a function of simulation steps in the 2D basal plane. The squares and circles correspond to the raw data and the data with logarithmic correlations, respectively. Different lines are fitted based on the formula $a+t^b$. (b) Total vortex-line length as a function of simulation steps in 3D simulations. The pink squares correspond to the anisotropic gradient coefficients. The yellow circles are results using the isotropic gradient coefficients as the control experiments, which are fitted by a yellow line. (a) is the average result of five parallel simulations with a system size of $2048\mathrm{\Delta }x\times 2048\mathrm{\Delta }x\times 1\mathrm{\Delta }x$ and a grid spacing of $\mathrm{\Delta }x=0.30\mathrm{nm}$, and (b) is the average of three parallel simulations with a system size of $512\mathrm{\Delta }x\times 512\mathrm{\Delta }x\times 512\mathrm{\Delta }x$ and a grid spacing of $\mathrm{\Delta }x=0.30\mathrm{nm}$.