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1、Almost-conservedoperatorsinnearlymany-bodylocalizedsystemsNicolaPancotti,1MichaelKnap,2DavidA.Huse,3J.IgnacioCirac,1andMariCarmenBa~nuls11Max-Planck-Institutf•urQuantenoptik,Hans-Kopfermann-Str.1,85748Garching,Germany2DepartmentofPhysicsandInstituteforAdvancedStudy,Tec
2、hnicalUniversityofMunich,85748Garching,Germany3PhysicsDepartment,PrincetonUniversity,Princeton,NewJersey08544,USA(Dated:April12,2018)Weconstructalmostconservedlocaloperators,thatpossessaminimalcommutatorwiththeHamiltonianofthesystem,nearthemany-bodylocalizationtransiti
3、onofaone-dimensionaldisor-deredspinchain.Wecollectstatisticsoftheseslowoperatorsfordi erentsupportsizesanddisorderstrengths,bothusingexactdiagonalizationandtensornetworks.OurresultsshowthatthescalingoftheaverageofthesmallestcommutatorswiththesupportsizeissensitivetoGri
4、thse ectsinthethermalphaseandtheonsetofmany-bodylocalization.Furthermore,wedemonstratethattheprobabilitydistributionsofthecommutatorscanbeanalyzedusingextremevaluetheoryandthattheirtailsrevealthedi erencebetweendi usiveandsub-di usivedynamicsinthethermalphase.I.INTROD
5、UCTIONspondingoperator.45Wecomputetheslowestoperatorsfordi erentsupportsizesanddisorderstrengths,usingbothexactdiagonalizationandtensornetworks.46,47OurIsolatedinteractingquantumsystemscanundergoanwork,ratherthanfocusingonthephasetransition,itunconventionaltransitionfr
6、omanergodicthermalphaseaimsatcharacterizethesub-di usiveregioninitsvicin-inwhichthesystemapproachesthermalequilibriumirre-ity.Speci cally,wecollectstatisticsfromnumerousdis-spectiveoftheinitialconditionstoamany-bodylocalized1{4orderrealizationsandshowthatnotonlythemean
7、value(MBL)phaseinwhichitdoesnot.Thereexists rm5,67ofthesmallestcommutator,butalsotheprobabilitydis-evidence,fromnumerical,rigorousmathematical,and8{15tributionsaresensitivetotheunderlyingphase.Thetailsexperimentalwork,thattheMBLphaseisreal-ofthedistributionsarea ectedb
8、ytheappearanceofrareizedinstronglydisorderedandinteractingquantumsys-17,21Grithsregions,whichactasbottlenecksfortran
