Frustrated Antiferromagnetic Triangular Lattice with Dzyaloshinskii-Moriya Interaction: Ground States, Spin Waves, Skyrmion Crystal, Phase Transition

We study in this article a triangular lattice with Heisenberg spins interacting with each other via an antiferromagnetic exchange interaction $J$ and a Dzyaloshinskii-Moriya (DM) interaction $D$, between nearest-neighbors (NN). We consider two cases: the first case in which the DM vector $\mathbf D$ is perpendicular to the lattice plane and the second case where it lies in the plane. A magnetic field $H$ is applied perpendicular to the spin plane in both cases. The ground state (GS) of this system is calculated by minimizing the energy using the very fast steepest-descent method in the two cases. In the case of perpendicular $\mathbf D$ with $H=0$, the GS is periodic. We analytically determine the GS configuration which is characterized by two well-defined angles. We calculate the spin-wave spectrum in this case which shows that for small wave vectors the spin waves are forbidden in the system. When $H\neq 0$, the GS in the perpendicular $\mathbf D$ case shows no skyrmions. However, in the case of in-plane $\mathbf D$ with $H\neq 0$, we find a crystal of skyrmions at $T=0$ composed of three interpenetrating skyrmion sublattice crystals, in agreement with low-$T$ spin textures found in earlier works. We show by Monte Carlo simulations that this skyrmion crystal is stable at finite temperatures below a critical temperature.