▁ax ion ▁coupling ▁. ▁ ▁One ▁of ▁the ▁most ▁dram atic ▁effects ▁occurs ▁when ▁ ▁and ▁time - re vers al ▁symmetry ▁is ▁present , ▁i . e . ▁non - mag net ic ▁topological ▁ins ulator . ▁Since ▁ ▁is ▁a ▁pseud ove ctor ▁on ▁the ▁surface ▁of ▁the ▁cry stal , ▁it ▁must ▁respect ▁the ▁surface ▁symmet ries , ▁and ▁▁ ▁is ▁one ▁of ▁them , ▁but ▁ ▁resulting ▁in ▁. ▁This ▁forces ▁ ▁on ▁every ▁surface ▁resulting ▁in ▁a ▁Di rac ▁cone ▁( or ▁more ▁generally ▁an ▁odd ▁number ▁of ▁Di rac ▁con es ) ▁on ▁every ▁surface ▁and ▁therefore ▁making ▁the ▁boundary ▁of ▁the ▁material ▁conduct ing . ▁ ▁On ▁the ▁other ▁hand , ▁if ▁time - re vers al ▁symmetry ▁is ▁absent , ▁other ▁symmet ries ▁can ▁quant ize ▁ ▁and ▁but ▁not ▁force ▁ ▁to ▁van ish . ▁The ▁most ▁extreme ▁case ▁is ▁the ▁case ▁of ▁in version ▁symmetry ▁( I ). ▁In version ▁is ▁never ▁a ▁surface ▁symmetry ▁and ▁therefore ▁a ▁non - zero ▁ ▁is ▁valid . ▁In ▁the ▁case ▁that ▁a ▁surface ▁is ▁g apped , ▁we ▁have ▁ ▁which ▁results ▁in ▁a ▁half - quant ized ▁surface ▁A HC ▁. ▁ ▁A ▁half ▁quant ized ▁surface ▁Hall ▁conduct ivity ▁and ▁a ▁related ▁treatment ▁is ▁also ▁valid ▁to ▁understand ▁topological ▁ins ul ators ▁in ▁magnetic ▁field ▁ ▁giving ▁an ▁effective ▁ax ion ▁description ▁of ▁the ▁elect rod ynam ics ▁of ▁these ▁materials . ▁ ▁This ▁term ▁leads ▁to ▁several ▁interesting ▁predictions ▁including ▁a ▁quant