Van Hove singularities (VHSs) play a pivotal role in driving correlated electronic phenomena. Traditional classifications focus only on critical points where the band gradient vanishes in all directions. Here we establish a unified classification of VHSs in three-dimensional systems, characterized by the number of vanishing gradient components and Hessian eigenvalues: ordinary $\left(M\right)$, higher-order $(T_1$, $T_2$, $T_3)$, noncritical ordinary $(N_0$, $N_1$, $N_2)$, and noncritical higher-order $(S_1$, $S_2)$ types. Noncritical VHSs exhibit directional quenching: The gradient vanishes in a two-dimensional subspace while remaining finite along the orthogonal direction, yielding finite density-of-states enhancements with distinct energy dependences. Using an $s$-orbital tight-binding model on the pyrochlore lattice with spin-orbit coupling, we demonstrate that all singularity classes emerge at distinct high-symmetry points through controlled tuning of the hopping ratio. This work establishes directional criticality and higher-order flatness as design principles for tailoring density-of-states enhancements in three-dimensional quantum materials.

Close

PrevNext

View figure in article

• Density of states
• Fermi surface
• Van Hove singularity
• Electronic structure
• Tight-binding model