Spectral fluctuations of tridiagonal random matric

1Spectralfluctuationsoftridiagonalrandommatricesfromtheβ-HermiteensembleC.Malea,G.LeCaërbandR.DelannayGroupeMatièreCondenséeetMatériaux,C.N.R.S.U.M.R.6626,UniversitédeRennes-I,CampusdeBeaulieu,Bât.11A,AvenueduGénéralLeclerc,F-35042RennesCedex,Franceapermanentaddress:EcoleNormaleSupérieuredeCachan,CampusdeKerlann,F-35170Bruz,Francebcorrespondingauthor,E-mail:[e-mail]‘timeseries’nδ,thefluctuationofthenthunfoldedeigenvalue,wherenplaystheroleofadiscretetimewasrecentlycharacterizedfortheclassicalGaussianensemblesofNN×randommatrices(GOE,GUE,GSE).Itisinvestigatedherefortheβ-HermiteensembleasafunctionofthereciprocalofthetemperatureβbyMonteCarlosimulations.Theensemble-averagedfluctuation2nδandtheautocorrelationfunctionvarylogarithmicallywithnforany0β(1nN).Thesimplelogarithmicbehaviorreportedintheliteratureforthehigher-ordermomentsofnδfortheGOE()1β=andtheGUE()2β=isvalidforany0βandisaccountedforbyGaussiandistributionswhosevariancesdependlinearlyonlnn.The1fαnoisedisplayedbythenδseries,previouslydemonstratedforthethreeGaussianensembles,ischaracterizedbywaveletanalysisbothasafunctionofβandofN.Whenβdecreasesfrom1to0,foragivenandlargeenoughN,theevolutionfroma1fnoiseat1β=toa21fnoiseat0β=isheterogeneouswitha~21fnoiseatthefinestscalesanda~1fnoiseatthecoarsestones.Therangeofscalesinwhicha~21fnoisepredominates,growsprogressivelywhenβdecreases.Asymptotically,a21fnoiseisfoundfor0β=whilea1fnoiseistherulefor0β.I.INTRODUCTIONRandommatrixtheory(RMT)contributessignificantlytoquantumchaologywhichpertainstothestatiticalpropertiesofquantumsystemswhoseclassicalcounterpartsarechaotic[1-10].TheworkingdefinitionofdynamicalchaosforinfinitequantumsystemsrefersindeedtoRMT[6,10].AsrecalledbyProsen[10],amany-bodyquantumsystemissaidtobechaoticifitsexcitationspectrumorsomeotherdynamicalcharacteristicsarewelldescribedbythoseofensemblesofHermitianrandomofappropriatesymmetriesoncertainenergyortimescales.Thelevelfluctuationsofatime-2reversalsymmetricquantumsystemwereconjecturedtocoincidewiththoseoftheGOEforsystemswhoseclassicallimitischaotic[3].Inthesemiclassicallimit,thefluctuationsoftheenergylevelsofgenericquantumsystems,relativetotheirsmoothedleveldensities,coincideinfactwiththoseofeigenvaluesofensemblesofrandommatriceschosenaccordingtothephysicalsymmetriesoftheconsideredsystems.Theconverseishowevernotnecessarilyalwaystrueas,forinstance,theclassicalcounterpartsofquantumsystemsshowingGOEfluctuationsmayberegular[11].Thelocalspectralfluctuationsofproperlyrescaledandprocessedeigenvaluesofrandommatrixensemblesdefineuniversalityclassesinthelimitoflargematrixsizeswhichdependonthematrixsymmetriesandareindependentonthedetailsoftheprobabilitydistributionsofmatrixelements.SuchuniversalityclassesareforinstanceassociatedwiththethreefundamentalGaussianensembleswhereNN×matricesarerealsymmetricfortheGaussianorthogonalensemble(GOE),HermitianfortheGaussianunitaryensemble(GUE)andquaternionself-dualfortheGaussiansymplecticensemble(GSE).Afourthensemble,theGaussiandiagonalensemble(GDE),ismadefrommatriceswhosesolenon-zeroelementsarediagonalwithidenticalandindependentnormaldistributions.Anubiquitouscharacteristicofshort-rangecorrelationsistheasymptoticdistributionofthespacing‘s’betweenconsecutiveenergylevelsofquantumsystemsorbetweensuccessiveeigenvaluesofrandommatrices,onceunfolded[1-2,6,12].Theoreticalnearest-neighborspacing(NNS)distributionsarerarelyavailable,simulateddistributionsareusedinsteadandcomparedtoexactortoapproximatedistributionsofthereferenceensembles.TheGaussianensemblesdefineforinstancethreeuniversalityclassesoflevelrepulsionatsmall‘s’,(),0,Wspsβ→~sβwithβ=1,2,4fortheGOE,theGUEandtheGSErespectively.ThepropertiesofeigenvaluesofGaussianensemblesarerecalledtobetheequilibriumcharacteristicsatatemperature1βofNidenticalpointchargesonalinein2DwhichinteractviaalogarithmicCoulombpotentialandareconfinedbyanexternalharmonicpotential[1].TheunfoldedeigenvaluesofaGDEmatrixareindependentanduniformlydistributed.TheasymptoticdistributionoftheirspacingsisthusaPoissondistribution()()()exps1pss=−=.Mostoften,phenomelogicalmodelsoftheevolutionoftheNNSdistributionsareusedtodescribespecifictransitionsbetweentheWigner-DysonandthePoissonstatistics.Otherclassicalcharacteristicsofspectralfluctuationsarethenumbervarianceandthespacingvariance.Thenumbervariance,measurestheL-dependenceofthefluctuationofthenumberofunfoldedeigenvaluesinanintervaloffixedlengthLthrownatrandomontheeigenvaluesequence.Bycontrast,thespacingvariance,measuresthen-dependenceofthefluctuationofthetotallengthofafixednumbernofspacingsbetweensuccessiveunfoldedeigenvalues.Botharesimplyrelated([12-14]andsectionVbelow).TheDyson-Mehtastatisticyieldsinformationabout3thespectralrigidityandlong-rangecorrelationsbyquantifyinganaveragedeviationofthecumulativeleveldensityfromaline.Adifferentstatistic,closelyrelatedtotheleveldensityfluctuation,wasrecentlyconsideredinaseriesofpapers[12,15-27].Namednδstatistic,itisdefinedas:()1111nninisnδεε+==−=−−∑(1)wherethespacingbetweentwosuccessiveunfoldedleve