Directional Criticality and Higher-Order Flatness: Designing Van Hove Singularities in Three Dimensions

TL;DR

This paper introduces a comprehensive classification of Van Hove singularities in three-dimensional systems, highlighting directional criticality and higher-order flatness as key design principles for electronic properties.

Contribution

It establishes a unified framework for classifying VHSs based on criticality and flatness, and demonstrates their emergence in a specific lattice model through tuning parameters.

Findings

All singularity classes appear at high-symmetry points with tuning.

Noncritical VHSs exhibit directional quenching with finite density-of-states.

Directional criticality and higher-order flatness can be engineered in 3D materials.

Abstract

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 ($M$-type), 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 dependencies. 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.