Symmetry Criterion for Van Hove Criticality at Non-Time-Reversal-Invariant Momenta
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 $\Gamma_{\mathrm{vec}}$ of the little group does not contain the trivial representation $\Gamma_1$. When $\Gamma_{\mathrm{vec}}$ does contain $\Gamma_1$, $\nabla E$ is not forced to vanish for any nondegenerate band; the classification instead depends on the multiplicity of $\Gamma_1$ in $\Gamma_{\mathrm{vec}}$. 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.