▁ ▁The ▁topological ▁nature ▁of ▁the ▁ax ion ▁coupling ▁is ▁evident ▁if ▁one ▁cons iders ▁gauge ▁transformations . ▁In ▁this ▁cond ensed ▁matter ▁setting ▁a ▁gauge ▁transformation ▁is ▁a ▁unit ary ▁transformation ▁between ▁states ▁at ▁the ▁same ▁ ▁point ▁. ▁ ▁Now ▁a ▁gauge ▁transformation ▁will ▁cause ▁ ▁, ▁. ▁Since ▁a ▁gauge ▁choice ▁is ▁arbitrary , ▁this ▁property ▁tells ▁us ▁that ▁ ▁is ▁only ▁well ▁defined ▁in ▁an ▁interval ▁of ▁length ▁ ▁e . g . ▁. ▁ ▁The ▁final ▁ing red ient ▁we ▁need ▁to ▁ac quire ▁a ▁ ▁classification ▁based ▁on ▁the ▁ax ion ▁coupling ▁comes ▁from ▁obser ving ▁how ▁cry st all ine ▁symmet ries ▁act ▁on ▁. ▁ ▁Fra ction al ▁lattice ▁transl ations ▁, ▁n - fold ▁rot ations ▁: ▁. ▁ ▁Time - re vers al ▁, ▁in version ▁: ▁. ▁The ▁consequence ▁is ▁that ▁if ▁time - re vers al ▁or ▁in version ▁are ▁symmet ries ▁of ▁the ▁cry stal ▁we ▁need ▁to ▁have ▁ ▁and ▁that ▁can ▁only ▁be ▁true ▁if ▁( t rivial ),( non - t rivial ) ▁ ▁( note ▁that ▁ ▁and ▁ ▁are ▁identified ) ▁giving ▁us ▁a ▁ ▁classification . ▁Furthermore , ▁we ▁can ▁combine ▁in version ▁or ▁time - re vers al ▁with ▁other ▁symmet ries ▁that ▁do ▁not ▁affect ▁ ▁to ▁ac quire ▁new ▁symmet ries ▁that ▁quant ize ▁. ▁For ▁example , ▁mirror ▁symmetry ▁can ▁always ▁be ▁expressed ▁as ▁ ▁giving ▁rise ▁to ▁cry st all ine ▁topological ▁ins ul ators , ▁while ▁the ▁first