AI 中文总结
研究非时间反演不变动量处范霍夫临界性，通过小群表示理论等方法，确定非简并和简并能带线性耦合情况，应用于空间群225并扩展到相关空间群，提供对称性决定、参数无关的范霍夫奇点工程框架。
AI 中文摘要
在非时间反演不变动量（non-TRIMs）处，时间反演对称性不约束能带色散的线性项。因此，$\nabla E$ 是否为零完全由小群的表示理论决定。对于非简并能带，当且仅当小群的矢量表示$\Gamma_{\mathrm{vec}}$不包含平凡表示$\Gamma_1$时，$\nabla E$ 被迫为零。对于简并能带，维格纳-埃卡特定理和克莱布施-戈尔丹系数决定线性耦合是否为零。应用于空间群225时，该判据解释了为何$W$点对所有非简并能带是临界的，简并的$E$能带通常非临界，以及$K$和$U$点具有参数依赖的临界性。支持的相图揭示了两层层次结构：对称性强制$\nabla E = 0$，而能带参数决定高阶特征。我们将此分类扩展到单群极限下所有包含non-TRIMs的空间群，为三维量子材料中工程化范霍夫奇点提供了一个由对称性决定、与参数无关的框架。
英文摘要
At non-time-reversal-invariant momenta (non-TRIMs), time-reversal symmetry does not constrain the linear term of the band dispersion. Whether $\nabla E$ vanishes is therefore determined entirely by the representation theory of the little group. For nondegenerate bands, $\nabla E$ is forced to zero if and only if the vector representation $Γ_{\mathrm{vec}}$ of the little group does not contain the trivial representation $Γ_1$. When $Γ_{\mathrm{vec}}$ does contain $Γ_1$, $\nabla E$ is not forced to vanish for any nondegenerate band; the classification instead depends on the multiplicity of $Γ_1$ in $Γ_{\mathrm{vec}}$. For degenerate bands, the Wigner-Eckart theorem and Clebsch--Gordan coefficients determine whether linear couplings vanish, with classification performed at the subband level. Applied to space group 225, the criterion explains why the $W$ point is critical for all nondegenerate bands, the degenerate $E$ bands are generically noncritical, and the $K$ and $U$ points host parameter-dependent criticality. Supporting phase diagrams reveal a two-tier hierarchy: symmetry enforces $\nabla E=0$, while band parameters determine higher-order character. We extend this classification to all space groups hosting non-TRIMs in the single-group limit, providing a symmetry-dictated, parameter-independent framework for engineering Van Hove singularities in three-dimensional quantum materials.
