Ferrichiral skyrmions with sublattice-resolved chirality in extended Kitaev model in triangular lattice

TL;DR

Can skyrmion textures arise from exchange frustration alone on a triangular lattice with Kitaev-type interactions? The authors perform classical Monte Carlo simulations of the XXZ $+$ Kitaev model in the $\Gamma=\Gamma'$ limit to map the phase diagram and characterize the $\mathbb{Z}_2$ vortex regime. They find a ferrichiral skyrmion phase where two sublattices carry unit skyrmion charge per vortex, giving $Q=\pm 2$ per vortex, with two Bloch-type skyrmions of opposite helicities and a nonchiral background on the third sublattice; this phase remains stable up to relatively high temperatures, indicating robust exchange-frustration-driven skyrmion physics. The work suggests a new route to topological spin textures in spin-orbit-coupled triangular magnets and points to XXZ magnets as promising platforms for ferrichiral chirality without external fields or DM interactions.

Abstract

We study an extended Kitaev model on the triangular lattice in a limit where the symmetric off-diagonal bond-dependent and Heisenberg interactions together map onto an XXZ model, in addition to the Kitaev interaction. Within the previously identified $\mathbb{Z}_2$ vortex regime, we uncover a ferrichiral skyrmion phase characterized by a sublattice-resolved scalar chirality: two of the three sublattices carry unit skyrmion charge, while the third remains nonchiral. Using classical Monte Carlo simulations, we show that this ferrichiral skyrmion phase emerges at zero temperature and in the absence of both an external magnetic field and Dzyaloshinskii-Moriya interactions, in sharp contrast to conventional skyrmion-hosting systems. The phase is stable over a wide parameter window and persists to relatively high temperatures. Our results reveal an unconventional route to skyrmion physics driven purely by frustrated exchange interactions and highlight the emergence of rich topological structures. Since both XXZ anisotropy and Kitaev interactions originate from the same spin-orbit-coupling mechanism, materials traditionally classified as XXZ magnets are expected to host finite Kitaev interactions as well. The potential for ferrichirality in these systems therefore warrants further investigation.

Figures (5)

• Figure 1: (a) Global coordinates $(X,Y,Z)$ and local octahedral coordinates $(x,y,z)$. (b) Spins at the center of the vortex from Fig.\ref{['fig:phase_diagram']}(f), colored according to their sublattice belonging, A-sublattice (blue), B-sublattice (green), and C-sublattice (red). (c) 3D plot of spins in different sublattices. It can be found that in A-sublattice (blue) and B-sublattice (green), spins are forming vortex, and they both contribute a positive value to the sublattice-dependent skyrmion number, whereas spins in C-sublattice (red) are forming non-chiral background.
• Figure 2: (a) Classic phase diagram obtained via Monte Carlo down to $T/J_0 = 10^{-6}$ in a $36 \times 36$ sites triangular lattice. The solid lines between phases represent second order phase transitions. The dashed area in Stripe-I phase represent a frustrated region that is unstable in Monte Carlo simulation and requires a future study. The dashed line corresponds to the line parameters used in Fig.\ref{['fig:phi_cut']}. (b)-(e) Real space spin configuration of the four ordered phases surrounding the vortex phase, calculated at parameters indicated by the diamonds in (a). They are: (b) the Stripe-I phase, (c) the Stripe-II phase, (d) the Y-phase, (e) the $120^\circ$ in-plane phase. The arrow represents the projections of spins in $xy$-plane, whereas the color represents the $S_z$ components. (f) Real space spin configuration of the vortex phase, calculated at the diamond point in (a), where $\phi/\pi = 0.198$, $\epsilon=0.833$. Different colors represents different sublattices; whereas the brightness corresponds to the $z$-component of spins. In this case, the spins in the two sublatticies forming vortices (colored with blue and green) have mostly positive $S_z$, whereas in the other sublattice spins have mostly negative $S_z$.
• Figure 3: Static structure factors of the vortex phase. (a) The static structure factor $\braket{\vec{S}\cdot\vec{S}}(\mathbf{q})$ in momentum space, with triple-q satellite peaks (black dots) around the K points. The dashed line outlines the first Brillouin zone where the special momentum points such as $K$, $M$, and $\Gamma$ are denoted by the blue dots. The inset shows the zoom-in area near the K-point to indicate the $\delta{\bf K}$ characterizing the vortex density $n_{\rm vortices}$ defined in Eq. (6). (b) The real-space spin correlation $\braket{\vec{S}_{\mathbf{R}+\mathbf{r}}\cdot\vec{S}_{\mathbf{R}}}_{\mathbf{R}}$, revealing the vortex superlattice. The dashed line indicates the lattice boundary with periodic boundary conditions.
• Figure 4: The sublattice-resolved skyrmion number per vortex, $|Q|/N_{\text{vortices}}$, computed along the dashed line in Fig. \ref{['fig:phase_diagram']}(a) with $\phi=0.25\pi$ and $\epsilon \in (0.2,1.8)$, using MC method is shown as the black dots. The dashed red line is a guide to the eye. Two sharp transitions from regions where $|Q|/N_{\text{vortices}} \sim 0$ to a non-zero region occur at $\epsilon = 0.69$ and $\epsilon = 1.31$, in agreement with the phase boundaries in Fig. \ref{['fig:phase_diagram']}(a). For $\epsilon < 0.69$, small finite values $\sim0.09$ were found, which is attributed to the adjacent unstable frustrated region. A narrow plateau at $|Q|/N_{\text{vortices}} \approx 1.2$ emerges within $\epsilon \in (0.69, 0.90)$, indicating that the vortices are not yet fully ordered in this region. Within $\epsilon\in(0.90, 1.31)$, $|Q|/N_{\text{vortices}}$ saturates at $2$, indicating that the vortices are aligned ferromagnetically in the entire lattice.
• Figure 5: Calculation using parallel tempering on a $64\times64$ lattice, with parameters set to be $\phi = 0.25\pi$, $\epsilon=1.1$, and temperatures $T$ ranging from $0.02\sim 0.5$, in unit of $J_0$. $2\times10^6$ sweeps were done at each single temperature point. (a) $Q/N_{\text{vortices}}$ as a function of temperature $T$ obtained by Monte Carlo calculations is denoted by black dots. The blue dashed line is a guide to the eye. The vertical dotted line indicats the transition temperature $T_c$. $Q/N_{\rm vortices}$ saturates to $\pm 2$ at low temperatures. (b) Energy $E$ (blue) and specific heat $C$ (red) as a function of $T$. $T_c = 0.16J_0$ can be determined by the maximum of specific heat, which also corresponds to the onset of $Q/N_{\text{vortices}}$ as shown in (a).