influence of the disorder distribution on ground s

arXiv:cond-mat/0411191v1[cond-mat.dis-nn]8Nov2004Spin-glasschaininamagneticfield:influenceofthedisorderdistributionongroundstatepropertiesandlow-energyexcitationsC´ecileMonthusandThomasGarelServicedePhysiqueTh´eorique,Unit´ederechercheassoci´eeauCNRS,DSM/CEASaclay,91191Gif-sur-Yvette,FranceForthespin-glasschaininanexternalfieldh,anon-zeroweightattheoriginofthebonddistributionρ(J)isknowntoinduceanon-analyticalmagnetizationatzerotemperature:forρ(J)∼A|J|μ−1nearJ→0,themagnetizationfollowstheChen-MascalingM∼hμ/(2+μ).Inthispaper,wenumericallyrevisitthismodeltoobtaindetailedstatisticalinformationonthegroundstateconfigurationandonthelow-energytwo-levelexcitationsthatgovernthelowtemperatureproperties.Thegroundstateconsistsoflongunfrustratedintervalsseparatedbyweakfrustratedbonds:Weaccordinglycomputethestrengthdistributionofthesefrustratedbonds,aswellasthelength-andmagnetization-distributionsoftheunfrustratedintervals.Wefindthatthelow-energyexcitationsareoftwotypes(i)onefrustratedbondofthegroundstatemayhavetwopositionsthatarenearlydegenerateinenergy(ii)twoneighboringfrustratedbondsofthegroundstatemaybeannihilatedorcreatedwithnearlyzeroenergycost.Foreachexcitationtype,wecomputeitsprobabilitydensityasafunctionofitslength.Wemoreovershowthatthecontributionsoftheseexcitationstovariousobservables(specificheat,Edwards-Andersonorderparameter,susceptibility)areinfullagreementwithdirecttransfermatrixevaluationsatlowtemperature.Finally,followingthespecialbimodalcase±J,whereaMa-DasguptaRGprocedurehasbeenpreviouslyusedtocomputeexplicitlytheaboveobservables,wediscussthepossibilityofanextendedRGprocedure:wefindthatthegroundstatecanbeseenastheresultofahierarchical‘fragmentation’procedurethatwedescribe.I.INTRODUCTIONInthispaper,weconsidertheonedimensionalspin-glasschaininasmallexternalfieldh0H=−XiJiσiσi+1−hXiσi(1)toobtaindetailedresultsonthegroundstateandthelow-energyexcitations,asafunctionoftheexponentμ0characterizingtheweightofthecouplingdistributionforsmallcouplingsρ(J)≃J→0A|J|μ−1(2)Asiswell-known,thepreviousmodelisequivalenttoarandom-bondandrandom-fieldferromagneticchain[4]H=−Xi|Ji|SiSi+1−hXixiSi(3)wherexi=Qij=1sgn(Jj).A.BimodaldistributionJi=±J:Imry-Maargumentandreal-spaceRGForthespecialcaseofthebimodaldistributionJi=±Jwithprobabilities(1/2,1/2),themodel(3)correspondstoapureIsingchain|Ji|=Jinabimodalrandomfieldhi=hxi=±h.TheImry-Maargument[1]fortherandomfieldIsingchaincanbeimmediatelytranslatedforthespin-glassinexternalfield,sincethedomainwallsoftheRFIMnowbecomesfrustratedbondsforthespin-glass:therandommagnetizationmofanunfrustrateddomainoflengthlisoforderm∼√l,i.e.itgivesrisetoanenergyoforder2h√lintheexternalfieldh,whereasapairoftwofrustratedbondshasforenergycost4J.Asaconsequence,thegroundstateismadeofunfrustrateddomainshavingthetypicalImry-MalengthLIM∼4J2/h2.Thereal-spaceMa-DasguptaRG[2]allowstoconstructexplicitlythepositionsoffrustratedbondsandtocomputevariousstatisticalproperties,suchasthedistributionofthedomainlengths.Thisapproachmoreoveryieldsthestatisticsoflow-energytwo-levelexcitations[3].Anaturalquestionisthen:whatarethecorrespondingresultsforageneraldistributionρ(J)thatisnotbimodal?Itturnsoutthatadifferentbehavioroccursifρ(J)hassomeweightatsmallcouplingsJ∼0.Thiscase,whichincludestheGaussiandistribution,completelychangesthephysicsofthemodel,aswenowdiscuss.2B.Distributionswithsmallcouplingsρ(J)≃A|J|μ−1:Chen-MaargumentFordistributionspresentingsomeweightatsmallcouplings(2)theaboveImry-MaargumentforthebimodalcaseisreplacedbythefollowingChen-Maargument[4].Theessentialideaisthatfrustratedbondswillbenowlocatedonweakbonds,incontrastwiththebimodalcasewherethecostofafrustratedbondisthesameeverywhere.Moreprecisely,theChen-Ma(CM)argumentisasfollows:thebondsJiweakerthansomecut-off|Ji|≤JCMareseparatedbyatypicaldistanceoforderlCM∼J−μCM(4)ThemagnetizationoftheunfrustrateddomainbetweentwosuchweakbondsisofordermCM∼plCM∼J−μ/2CM(5)TheflippingofsuchadomainthusinvolvesatypicalenergyoforderJCMforthecreationoftwoweakfrustratedbonds,butallowstogainamagneticenergyoforderhmCM∼hJ−μ/2CM.Thebalancebetweenthetwotermsyieldsanoptimalcut-offoforder[4]JCM∼h2/(2+μ)(6)sothatthemagnetizationperspinMspresentsthefollowingnon-analyticalbehaviorMs∼mCMlCM∼hμ(2+μ)(7)Thezero-temperaturesusceptibilityχ(T=0)∼Msh∼h−2/(2+μ)(8)thusdivergesatzerofieldh→0.Forinstance,theGaussiandistributionρ(J),whichcorrespondstotheexponentμ=1,leadstothebehaviorMs∼h1/3.Thecriticalexponentμ(2+μ)forthemagnetizationinexternalfieldwasfoundtobeexactviatransfermatrixcalculationsbyGardnerandDerrida[5],wheretheprefactorwasmoreovercomputed.Thepresenceofsmallcouplingsdoesinduceinterestingnewpropertiesforthegroundstatewithrespecttothebimodalcase.ChenandMahavealsoanalyzedthelowtemperatureproperties,inparticularintheregimewherethetemperatureTismuchsmallerthanthetypicalenergyJCMofadomain.InthisregimeT≪JCM(6),onlyasmallfractionoftwo-levelexcitationswillbeexcited:thedensityρ(E=0)ofexcitationsnearzeroenergycanbeestimatedtoscaleas[4]ρ(E=0)∼1lCM×1JCM∼Jμ−1CM∼h2(μ−1)/(2+μ)(9)where1/lCMrepresentsthedensityoffrustratedbondsinth