First-principles study of topological surface states and pressure induced phase transitions in a novel noncentrosymmetric superconductor PbTiSe2

•This work unveil a novel noncentrosymmetric PbTiSe2 phase at ambient pressure and reveal that this stoichiometric phase has nontrivial band topology.•The clear drumhead surface states isolated from the bulk states in noncentrosymmetric PbTiSe2 are clarified.•The phase transitions in PbTiSe2 tuned by external pressure have been clarified and a novel tetragonal PbTiSe2 phase is found at high pressure.•The tetragonal PbTiSe2 superconductor is found to be topologically nontrivial.