AI 中文总结
本文对具有特定对称性的三重Q金兹堡-朗道理论的7个稳定相的体缺陷进行固定晶格同伦分类，明确缺陷类型及性质，区分自由内部缺陷、晶态畴壁与壁束缚复合物。
AI 中文摘要
多Q磁序将连续自旋旋转与平移、点群变换相关的离散晶态部分结合，产生比传统单Q磁体更丰富的缺陷结构。我们对具有(𝖵₄⋊𝖣₃)×𝖮(N)对称性的M点三重Q金兹堡-朗道理论中N=2和3的所有7个稳定相的体缺陷进行分类，其中𝖵₄是平移生成的克莱因四群。原子晶格被视为规定背景，排除晶格位错和向错，三个傅里叶场保留其物理M点标签。母群中与恒等元连续连通的变换构成G₀={e}×𝖲𝖮(N)，对于参考态稳定子H，包含参考态的连通分支为G₀/(H∩G₀)，而非通过将H投影到自旋空间得到的商，该区别使正交三重Q相具有完整流形𝖮(3)，并出现手性壁和阿贝尔ℤ₂框架涡旋，而非非阿贝尔二元多面体涡旋。𝖮(2)相的每个连通分支均支持整数2π涡旋，而分数缠绕仅在连接离散畴壁时闭合，且在非零壁张力下线性受限。平移对称性进一步禁止交叉梯度双线性项，使二次弹性部分简化为各向同性和M点锁定的各向异性刚度。该分类区分了三重Q磁体中的自由内部缺陷、晶态畴壁及壁束缚复合物。
英文摘要
Multiple-$Q$ magnetic orders combine continuous spin rotations with discrete crystalline sectors associated with translations and point-group transformations, producing a richer defect structure than conventional single-$Q$ magnets. We classify the bulk defects of all seven stable phases for $N=2$ and $3$ in the $M$-point triple-$Q$ Ginzburg--Landau theory with $(\Vfour\rtimes\Dthree)\times\OO(N)$ symmetry, where $\Vfour$ is the translation-generated Klein four-group. The atomic lattice is treated as a prescribed background, with lattice dislocations and disclinations excluded and the three Fourier fields retaining their physical $M$-point labels. The parent-group transformations continuously connected to the identity form $G_0=\{e\}\times\SO(N)$. For a reference-state stabilizer $H$, the connected component containing the reference state is $G_0/(H\cap G_0)$, not the quotient obtained by projecting $H$ onto spin space. This distinction gives the orthogonal triple-$Q$ phase the full manifold $\OO(3)$, with chirality walls and Abelian $\ZZ_2$ frame vortices rather than non-Abelian binary-polyhedral vortices. Every connected component of the $\OO(2)$ phases supports an integer $2π$ vortex, whereas fractional windings close only when attached to a discrete-domain wall and are linearly confined at nonzero wall tension. Translation symmetry further forbids cross-gradient bilinears, reducing the quadratic elastic sector to an isotropic and an $M$-point-locked anisotropic stiffness. The classification separates free internal defects, crystalline domain walls, and wall-bound composites in triple-$Q$ magnets.
