Pomeranchuk Instability Induced by an Emergent Higher-Order van Hove Singularity on the Distorted Kagome Surface of Co$_3$Sn$_2$S$_2$

Materials hosting flat bands at the vicinity of the Fermi level promote exotic symmetry broken states. Common to many of these are van Hove singularities at saddle points of the dispersion or even higher-order van Hove singularities where the dispersion is flattened further. The band structure of kagome metals hosts both a flat band and two regular saddle points flanking a Dirac node. We investigate the kagome ferromagnetic metal Co$_3$Sn$_2$S$_2$ using scanning tunneling spectroscopy. We identify a new mechanism by which a triangular distortion on its kagome Co$_3$Sn surface termination considerably flattens the saddle point dispersion, and induces an isolated higher-order van Hove singularity (HOvHS) with algebraically divergent density of states pinned to the Fermi energy. The distortion-induced HOvHS precipitates a Pomeranchuk instability of the Fermi surface, resulting in the formation of a series of nematic electronic states. We visualize the nematic order across an energy shell of about 100 meV in both real-, reciprocal-, and momentum-spaces, as a cascade of wavefunction distributions which spontaneously break the remaining rotational symmetry of the underlying distorted kagome lattice, without generating any additional translational symmetry breaking. It signifies the spontaneous removal of a subset of saddle points from the Fermi energy to lower energies. By tracking the electronic wavefunction structure across the deformed Fermi surface we further identify a charge pumping-like evolution of the wavefunction center of mass. The mechanism we find for the generation of higher-order saddle points under a kagome distortion may be common to other kagome materials, and potentially other lattice structures, suggesting a generic new avenue for inducing unconventional electronic instabilities towards exotic states of matter.