Saddle-point Van Hove singularities in the topological surface states are interesting because they can provide a new pathway for accessing exotic correlated phenomena in topological materials. Here, based on first-principles calculations combined with a k·p model Hamiltonian analysis, we show that the layered platinum mineral jacutingaite (Pt2HgSe3) harbors saddlelike topological surface states with associated Van Hove singularities. Pt2HgSe3 is shown to host two distinct types of nodal lines without spin-orbit coupling (SOC), which are protected by combined inversion (I) and time-reversal (T) symmetries. Switching on the SOC gaps out the nodal lines and drives the system into a topological state with nonzero weak topological invariant Z2=(0;001) and mirror Chern number nM=-2. Surface states on the naturally cleaved (001) surface are found to be nontrivial with a unique saddlelike energy dispersion with type II Van Hove singularities. We also discuss how modulating the crystal structure can drive Pt2HgSe3 into a Dirac semimetal state with a pair of Dirac points. Our results indicate that Pt2HgSe3 is an ideal candidate material for exploring the properties of topological insulators with saddlelike surface states.
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