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Saddle-point van Hove singularities in the topological surface states areinteresting because they can provide a new pathway for accessing exoticcorrelated phenomena in topological materials. Here, based on first-principlescalculations combined with a $\mathbf {k \cdot p}$ model Hamiltonian analysis,we show that the layered platinum mineral jacutingaite (Pt$_2$HgSe$_3$)harbours saddle-like topological surface states with associated van Hovesingularities. Pt$_2$HgSe$_3$ is shown to host two distinct types of nodallines without spin-orbit coupling (SOC) which are protected by combinedinversion ($I$) and time-reversal ($T$) symmetries. Switching on the SOC gapsout the nodal lines and drives the system into a topological insulator statewith nonzero weak topological invariant $Z_2=(0;001)$ and mirror Chern number$n_M=2$. Surface states on the naturally cleaved (001) surface are found to benontrivial with a unique saddle-like energy dispersion with type II van Hovesingularities. We also discuss how modulating the crystal structure can drivePt$_2$HgSe$_3$ into a Dirac semimetal state with a pair of Dirac points. Ourresults indicate that Pt$_2$HgSe$_3$ is an ideal candidate material forexploring the properties of topological insulators with saddle-like surfacestates.

Saddle-point von Hove singularity and dual topological insulator state in Pt$_2$HgSe$_3$