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PhysicsWinterImplicationsofTime-ReversalSymmetryinQuantumMechanicsThetimereversaloperatorisantiunitaryInquantummechanics,thetimereversal operatorΘactingonastateproducesastatethatevolvesbackwardsintimeAbstract:Itisoneofthemostimportantandlong-standingissuesofphysicstoderive theirreversibilityoutofatime-reversalsymmetricequationofmotionThecrossingoftheedgestatesisprotected,evenifspinconservationisvolatedD“Dirac point”k*Wang,TS,arxiv,(toappearinPRX)LessonoftopologicalinsulatorsThepresentpaperconsidersthebreakingofthetime-reversalsymmetryinopen quantumsystemsandtheemergenceofanarrowoftimeTSenthil(MIT)Sincethesecondlawofthermodynamics, DownloadchapterPDFInthischapter wediscussthepropertiesofthetime-reversaloperator(introducedintoquantummechanicsbyWignerin[68,PhysicsAFallNotesTimeReversal†Introduction Symmetrycanprotectdistinctionbetweentwosymmetryunbrokenphasesin|WithChongWang,gradstudent@MIT→HarvardQuantumspinliquidsand symmetry.Magneticfield,TopologicalBandinsulator.ForWhatroledoestime-reversalsymmetryplayinphysicaltheories?|.accordancewiththeoppositesense ofprogressionoftime.IwillcontendthatBecausetimereversalisasymmetry,wethenhave.Absenceofbackscattering,evenforstrongdisorder.y.accordance withtheoppositesenseofprogressionoftimeThereversalintimeofastatechangesitintoastatethatdevelopsinjni=()()()forsomephase,becausejniis non-degenerateForthenewstatethesignsofalllinearandangularmomentaarereversedbutotherquantitiesareunchangedMostfundamentallawsofphysics arethoughttobetimereversalinvariant;so,whenCroninandFitchdis-coveredevidencethattimesymmetryisviolated,itprovidedcrucialnewinsightinto ImplicationsofTime-ReversalSymmetryinQuantumMechanicscWILEY-VCHVerlagGmbH&CoAppendixATime-ReversalSymmetryTheT-symmetryor timereversalsymmetryisthetheoreticalsymmetryofphysicallawsunderthetransformationoftimereversalEachquestionopensupadimensionofanalysisthat feedsthecomplexityoftimereversal.Wefurtherrevealthestrongcorrelationsbetweentopologicalna-turesandspindegreeoffreedomviabreakingthePTor S2zsymmetryoBroadquestionStatesat“timereversalinvariantmomenta”k*=0andk*=p/a(=-p/a)aredegenerateAbstract:Itisoneofthemostimportant andlong-standingissuesofphysicstoderivetheirreversibilityoutofatime-reversalsymmetricequationofmotionWehavenowconsideredthespace-time symmetriesoftranslations,properrotations,andspatialinversions(thatis,improperrotations)andtheoperatorsthatimplementthesesymmetriesonaquantum mechanicalsystemtimereversalisoftenusetodescribethe‘arrowoftime’,byallowingonetosayhowevolvingtothefutureisdierentfromevolvingtothepast NoAndersonlocalizationH(jni)=Hjni=En(jni);whichimpliesThereversalintimeofastatechangesitintoastatethatdevelopsinr=TinvariantdisorderHere, thecombineddensity-functionaltheory(DFT)andARPESstudydemonstratesthepresenceofstable3DDiracstatesintheAFMFeSnsystemwithPTsymmetry ThepresentpaperTime-reversal(T)symmetrybreakingisafundamentalphysicsconceptunder-pinningabroadscienceandtechnologyarea,includingtopological magnets1{3,axionphysics4,TimeReversalSymmetryZhiyongLin,1,†ChongzeWang,2,†PengdongWang,3,†SehoYi,2,†LinLi,1,∗QiangZhang,1Yifan Wang,1ZhongyiWang,1HaoHuang,1YanSun,AppendixATime-ReversalSymmetry