▁intr insic ▁magnetic ▁topological ▁ins ulator ▁M n Bi Te ▁has ▁the ▁quant izing ▁symmetry ▁. ▁ ▁Sur face ▁anom alous ▁hall ▁conduct ivity ▁ ▁So ▁far ▁we ▁have ▁discussed ▁the ▁mathematical ▁properties ▁of ▁the ▁ax ion ▁coupling . ▁Phys ically , ▁a ▁non - t rivial ▁ax ion ▁coupling ▁() ▁will ▁result ▁in ▁a ▁half - quant ized ▁surface ▁anom alous ▁Hall ▁conduct ivity ▁() ▁if ▁the ▁surface ▁states ▁are ▁g apped . ▁To ▁see ▁this , ▁note ▁that ▁in ▁general ▁ ▁has ▁two ▁contribution . ▁One ▁comes ▁from ▁the ▁ax ion ▁coupling ▁, ▁a ▁quantity ▁that ▁is ▁determined ▁from ▁bulk ▁consider ations ▁as ▁we ▁have ▁seen , ▁while ▁the ▁other ▁is ▁the ▁Ber ry ▁phase ▁ ▁of ▁the ▁surface ▁states ▁at ▁the ▁F erm i ▁level ▁and ▁therefore ▁depends ▁on ▁the ▁surface . ▁In ▁summary ▁for ▁a ▁given ▁surface ▁term ination ▁the ▁per pend icular ▁component ▁of ▁the ▁surface ▁anom alous ▁Hall ▁conduct ivity ▁to ▁the ▁surface ▁will ▁be ▁. ▁ ▁The ▁expression ▁for ▁ ▁is ▁defined ▁ ▁because ▁a ▁surface ▁property ▁() ▁can ▁be ▁determined ▁from ▁a ▁bulk ▁property ▁() ▁up ▁to ▁a ▁quantum . ▁To ▁see ▁this , ▁consider ▁a ▁block ▁of ▁a ▁material ▁with ▁some ▁initial ▁ ▁which ▁we ▁wrap ▁with ▁a ▁ 2 D ▁quantum ▁anom alous ▁Hall ▁ins ulator ▁with ▁Ch ern ▁index ▁. ▁As ▁long ▁as ▁we ▁do ▁this ▁without ▁closing ▁the ▁surface ▁gap , ▁we ▁are ▁able ▁to ▁increase ▁ ▁by ▁ ▁without ▁alter ing ▁the ▁bulk , ▁and ▁therefore ▁without ▁alter ing ▁the