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

The gist: Ferrichiral skyrmions emerge in an extended Kitaev model on a triangular lattice, characterized by a sublattice-resolved scalar chirality where two sublattices carry unit skyrmion charge and the third remains nonchiral, in sharp contrast to conventional skyrmion-hosting systems.

Introduction and Motivation

Magnetic skyrmions are particle-like vortex spin textures characterized by an integer-valued topological invariant, the skyrmion number Their nontrivial topology gives rise to emergent electrodynamic phenomena such as the topological Hall effect and the skyrmion Hall effect. Magnetic skyrmions were first experimentally realized in the chiral magnet MnSi, where they are stabilized by the Dzyaloshinskii–Moriya (DM) interaction arising from the interplay of spin–orbit coupling and inversion-symmetry breaking. More recently, it has become clear that inversion symmetry breaking is not a prerequisite for skyrmion formation. Skyrmion phases have been theoretically and experimentally identified in centrosymmetric systems stabilized by geometric frustration, RKKY interactions, and competing orbital-dependent exchange processes. The role of frustrated exchange interactions alone in generating skyrmion textures, particularly in the absence of magnetic fields and DM interactions, remains relatively unexplored.

Model Mapping and Phase Identification

The study focuses on an extended Kitaev model on the triangular lattice including Heisenberg (J), Kitaev (K) and off-diagonal Γ and Γ′ interactions in the Γ = Γ′ limit, which maps to XXZ + Kitaev model as we will show below. When setting Γ = Γ′, the last two terms in Eq.(1) adds up to a bond-independent term; Sαi + Sβi + Sγi = √3S Zi, where Z is one of the crystal axes, and the two terms of Γ and Γ′ in the Hamiltonian becomes: (4). Then the Hamiltonian reduces to an anisotropic XXZ model in addition to the Kitaev interaction. This parametrization allows us to study the effect of both the bond-dependent Kitaev interaction and XXZ type anisotropy.

Classical Phase Diagram Analysis

The classical Monte Carlo (MC) simulations were performed on a triangular lattice of 36 × 36 sites with periodic boundary conditions. The results were summarized in Fig. 2. When ϕ is set to 0, the model becomes the XXZ model, whose AFM region was known to be dominated by two coplanar phases: the Y-phase in easy-axis region (ϵ 1). As the ϕ is turned into an intermediate regime, a vortex phase emerges in the center of the four ordered phases, as demonstrated in Fig. 2(f). This phase has been studied in the Heisenberg-Kitaev model (ϵ = 1), where it was identified as the Z2 vortex phase. The vortex density, nvortices, defined the number of vortices per site, can be determined from the deviation of these satellite peaks, Qpeak, from the K-points as follows: nvortices = Nvortices / Nsites = δK2 / K2, (6).

Sublattice-Resolved Skyrmion Characterization

Inspired by this observation, we propose a sublattice-dependent skyrmion number as a characterization of the vortex phase in the XXZ + Kitaev model. The skyrmion number is then defined as the summation of Qs on the three sublattices: Q = Σs=A,B,C Qs = 1/4π Z s S·(∂xS s × ∂yS s)dxdy ≡ 1/4π Z qs(x, y)dxdy (7). To reveal the non-trivial topological property of the vortex phase, we compute the sublattice-resolved skyrmion number per vortex Q/Nvortices defined in Eq. (9): Q = Σs=A,B,C Qs = Σs 1/X ⟨ijk⟩∈s qs,⟨ijk⟩ (8). The calculation is performed along a specific path in the phase diagram, as indicated in Fig. 2(a) for a 36 × 36 lattice. Along this cut, we find Q/Nvortices ≈ 0 within both the Stripe-I and the 120◦ in-plane ordered phases. In contrast, a finite value emerges inside the vortex phase, featuring two distinct plateaus. The first plateau attains the maximal value of 2, corresponding to a state where the cores of all vortices are uniformly aligned.

Ferrichiral Skyrmion State Emergence

This quantized behavior demonstrates that each individual vortex of sublattice A and B contributes ±1 to the total skyrmion number Q, which can be understood from its internal structure. Since the polarization of C sublattice in the central region is anti-correlated with the Sz component of sublattices A and B due to AFM couplings, the core polarization is thereby binded to the sign of the vortex’s topological charge Q. The quantization of Q/Nvortices to ±2 reveals an emergent binary, or Z2, degree of freedom. This uniform Z2 order coupled with the chiral vortex texture, we term this phase the ferrichiral skyrmion state.

Conclusion and Material Candidates

Our findings naturally motivate a discussion of material realizations. Since both XXZ anisotropy and Kitaevtype interactions arise from the same spin–orbit-coupling mechanism, materials previously described as XXZ magnets are expected to host finite Kitaev interactions as well. Consequently, triangular-lattice XXZ systems, such as the Yb-delafossites NaYbCh2(Ch = O, S, Se) and their doped variant NaYb1−xLuxCh2, Cs2CuBr4 and Ba3CoSb2O9, are promising candidates for realizing this physics. Potential Z2 vortex crystal signatures have been reported in materials such as NaCrO2 and (CD3ND3)2NaRuCl6.

Summary and Discussion

In this work, we study the XXZ + Kitaev model and demonstrate the emergence of a sublattice-dependent skyrmion structure within the Z2 vortex phase. Specifically, two of the sublattices carry a unit skyrmion charge, while the remaining sublattice hosts zero skyrmion charge. Our theory reveals an unconventional mechanism for stabilizing skyrmion phases, distinct from the conventional routes that rely on inversion-symmetry breaking and the resulting Dzyaloshinskii–Moriya interaction, or on explicit time-reversal-symmetry breaking either spontaneously broken or induced by external magnetic fields. Notably, this phase emerges at relatively high temperatures, as demonstrated by classical Monte Carlo simulations. The phase is also robust across a broad parameter range in the studied case (Γ = Γ′), hinting at its possible stability for Γ ≠ Γ′.

Acknowledgments

We thank Haoting Xu for useful discussion. This work is supported by the NSERC Discovery Grant No. 2022-04601 and NSERC CREATE program No. 575280-2023. H. Y. K. acknowledges support from the Canada Research Chairs Program No.

1. textbfbold sublattice-resolved topological characterization of vortex cores: The improved AI can precisely quantify skyrmion charge distribution by calculating sublattice-resolved scalar chirality using Equation (9), which reveals that each individual vortex of sublattice A and B contributes ±1 to the total skyrmion number Q.

2. textbfferromagnetic correlation sensing in topological phases: The improved system can detect the emergent binary degree of freedom by analyzing the saturation plateau, where |Q|/Nvortices saturates at 2, indicating that the Z2 degrees of freedom order ferromagnetically, corresponding to the |Q|/Nvortices = 2 plateau.

3. textbfunconventional skyrmion physics discovery: The AI can identify and predict the emergence of novel topological structures by recognizing that a topologically nontrivial ferrichiral state emerges at zero magnetic field and in the absence of DM interactions, establishing a mechanism driven purely by frustrated exchange interactions.

4. textbfpredictive material screening for Kitaev systems: The system can screen candidate materials for hosting this physics by checking if they are triangular-lattice XXZ systems, such as the Yb-delafossites NaYbCh2(Ch = O, S, Se)[54–56] and their doped variant NaYb1−xLuxCh2[57–59], and Cs2CuBr4[60, 61] and Ba3CoSb2O9[62–64].

5. textbffinite-temperature phase stability assessment: The AI can assess the robustness of the ferrichiral state by analyzing its stability across thermal fluctuations, noting that this phase emerges at relatively high temperatures, as demonstrated by classical Monte Carlo simulations.

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 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.