Vortex lattice in two-dimensional chiral $XY$ ferromagnets and the inverse Berezinskii-Kosterlitz-Thouless transition

In this Rapid Communication we will show that, in the presence of a properly modulated Dzyaloshinskii-Moriya (DM) interaction, a $U\left(1\right)$ vortex-antivortex lattice appears at low temperatures for a wide range of the DM interaction. Even more, in the region dominated by the exchange interaction, a standard Berezinskii-Kosterlitz-Thouless (BKT) transition occurs. In the opposite regime, the one dominated by the DM interaction, a kind of inverse BKT transition takes place. As temperature rises, the vortex-antivortex lattice starts melting by the annihilation of vortex-antivortex pairs, in a sort of “inverse” BKT transition.

Figure 1

(a) shows the direction of the $D$ vectors, represented by red and light blue out-of-plane arrows of a given plaquette, when circulating the lattice in the positive direction of the axes. The colors of the bonds indicate the corresponding directions for the remaining $D$ vectors. (b) shows a vortex lattice of $10\times 10$ spins obtained through a standard Metropolis Monte Carlo method, where the $X$ and $Y$ axes represent the direction on the lattice in units of the lattice spacing. Green plaquettes hold a counterclockwise vortex while blue plaquettes hold a clockwise vortex, and spins in each sites are represented with blue arrows.

Figure 2

Illustrative: The figure shows a configuration in which only one vortex, a positive one, has been eliminated from the lattice, and the remaining vortex-antivortex lattice has not been affected; $D$-dominated regime.

Figure 3

The BKT behavior of $\mathrm{{\rm Y}}\left(T\right)$ for the chiral $XY$ model in the $D$-dominated regime is shown. A jump in $\mathrm{{\rm Y}}$ is appreciated at each value of $\mathcal{J}$, and the theoretical prediction for $J\to 0$ is shown in purple. In the inset, the characteristic $XY$ $\mathrm{{\rm Y}}\left(T\right)$ behavior for the $J$-dominated regime is also shown, and the curves with a lower intercept correspond to lower values of $D/J$.

Figure 4

The picture shows vortex and antivortex densities as temperature rises, for $D\gg J$. In the inset, the same densities show that standard phenomenology for $D=0$.

Figure 5

Qualitative phase diagram of the studied model. The vertical axis represents the variable $\phi$ that goes from $\phi =0$ ($J$-dominated regime) up to $\phi =\pi /4$ ($D$-dominated regime). The value $\phi =\pi /4$ (where $D=J$) is depicted with a dashed horizontal line. We identify three regions: The region VL where the system stays in a vortex lattice configuration, the FM where the system stays in a ferromagnetic configuration, and the decorrelated high-temperature phase. As mentioned in the text, the possibility of a coexistence of both VL and FM phases at the dashed line cannot be discarded.