Counting Multiple Solutions in Glassy Random Matri

arXiv:cond-mat/0303521v1[cond-mat.stat-mech]25Mar2003CountingMultipleSolutionsinGlassyRandomMatrixModelsN.Deo1,2,1PoornaprajnaInstituteforScientificResearch,Bangalore560080,India,2AbdusSalamInternationalCentreforTheoreticalPhysics,Trieste,Italy.February2,2008AbstractThisisafirststepincountingthenumberofmultiplesolutionsincertainglassyrandommatrixmodelsintroducedinrefs.[1].Weareabletodothisbyreducingtheproblemofcountingthemultiplesolutionstothatofamomentproblem.Morepreciselywecountthenumberofdifferentmomentswhenweintroduceanasymmetry(tapping)intherandommatrixmodelandthentakeittovanish.ItisshownherethatthenumberofmomentsgrowsexponentiallywithrespecttoNthesizeofthematrix.Asthesemodelsmapontomodelsofstructuralglassesinthehightemperaturephase(liquid)thismayhaveinterestingimplicationsforthesupercooledliquidphaseinthesespinglassmodels.Furtheritisshownthatthenatureoftheasymmetry(tapping)iscrutialinfindingthemultiplesolutions.Thisalsoclarifiessomeofthepuzzlesweraisedinref.[2].PACS:02.70.Ns,61.20.Lc,61.43.Fs1IntroductionRandommatrixmodelscanbeusedveryeffectivelyassimplemathematicaltoymodelswheremanynewideasinphysics,biologyandeconomicscanbe1testedanalyticallyref.[4,5,6].Herewetrytounderstandtheideaoftappingandcounting,wellstudiedinthecontextofgranularmedia,intheglassyrandommatrixmodelintroducedinref.[1].Thereitwasdemonstratedthatthematrixmodelswithgapsintheireigenvaluedistributionhadmultiplesolutionsandwererelatedtothehightemperaturephaseofcertainp-spinglassmodelsref.[7].Weapproachtheprobleminmuchthesamespiritasdoneforspinsystemsinref.[8].Thisisafirststepinunderstandingwhathappenswhenwetapthemodelieintroduceaperturbationandremoveit.Thisenablesustocountthenumberofdifferentconfigurations.Studiestounderstandthefluctuation-dissipationrelationsandtherelationsbetweenthedynamicalandEdwardstemperatureinthedynamicalmatrixmodelsawaitsfurtherwork.Thisstudywillalsohelpusunderstandsomeofthepuzzlesthatweraisedinref.[2].Oneofthepuzzlesinthesemodelsisthatthelongrangecorrelatorsfoundinref.[10]bymeanfieldcalculationsdifferfromthatfoundinref.[13,2]usingtheorthogonalpolynomialmethods.Aresolutionofthishasbeensuggestedinref.[11]whereitisclaimedthatthedifferencearisesduetodiscretenessofthenumberofeigenvaluesfordoublewellmodelswithequaldepths.Herewetrytounderstandtheseresultsusingthemethodofmoments.Mostofthestudiesandapplicationsofmatrixmodelscorrespondtoeigen-valuedistributionsonasingle-cutinthecomplexplanewheretheeigenvaluedensityisnon-zeroref.[4].Herewestudyaonehermitianmatrixmodelwithamorecomplicatedeigenvaluestructure.Thesehavefoundapplicationsintwo-dimensionalquantumgravity,stringtheory,disorderedcondensedmat-tersystems,superconductors(withcomplexvectorpotentialandwithimpu-rities)andglasses.Herewestudythesemodelswithapplicationstoglassesinmindasdiscussedinrefs.[1].Toillustratesomeofthegenericpropertieswestudyaonehermitianmatrixmodelwithtwocutsfortheeigenvaluedensity.Oneoftheimportantdifferencesobservedinthesemodelsisthattheyhavemultiplesolutionswhichshowupincertaincorrelationfunctions.Herewecountthenumberofmultiplesolutionsandexplorethepossibilitythatthesemultiplesolutionsarisebytakingdifferentpathsinphasespace(eachpathmaycorrespondtoadifferentmetastableglassystate).Itisimportanttoestablishthecorrespondencebetweenthemultiplesolutionsandmetastableglassystates.Thebarrierheightscorrespondingtothesevarioussolutionsarealsofuturegoals.Iwilldiscussherethematrixmodelwithdouble-wellpotentialtheM4model(intheGaussianPennermodelwheresimilarthingshappenwillbe2pursuedelsewhere).Atappingisintroducedwhichcorrespondstocouplingthematrixmodeltoanexternalsource.Thelimitoftakingtheexternalsourcestovanishgivesdifferentvaluesforthemomentsinthesemodels.Thismayresultindifferentvaluesforthepartitionfunctionandhencethefreeenergy.Takingdifferenttappingscorrespondstoexploringthefullspaceofconfigurations.Herewepresentthefirststepsincountingthenumberofdifferentconfigurationsandfindittobeexponentiallylarge.AfterthisworkwascompletedwefindthatinadifferentcontextresultsofexponentiallylargenumberofminimahavebeenreportedinarenormalizablematrixpotentialwithSNusingadifferentmethodbySoljacicandWilczekref.[3].2NotationsandConventionsLetMbeahermitianmatrix.ThepartitionfunctiontobeconsideredisZ=RdMe−NtrV(M)whereM=N×Nhermitianmatrix.TheHaarmeasuredM=QNi=1dMiiQijdM(1)ijdM(2)ijwithMij=M(1)ij+iM(2)ijandN2independentvariables.V(M)isapolynomialinM:V(M)=g1M+(g2/2)M2+(g3/3)M3+(g4/4)M4+.....ThepartitionfunctionisinvariantunderthechangeofvariableM′=UMU†whereUisaunitarymatrix.WecanusethisinvarianceandgotothediagonalbasisieD′=UMU†suchthatD′isthematrixdiagonaltoMwitheigenvaluesλ1,λ2,.....λN.ThenthepartitionfunctionbecomesZ=CR∞−∞QNi=1dλiΔ(λ)2e−NPNi=1V(λi)whereΔ(λ)=Qij|λi−λj|istheVandermondedeterminant.Theintegra-tionoverthegroupUwiththeappropriatemeasureistrivialandisjusttheconstantC.Byexponentiatingthedeterminantasa‘tracelog’wearriveattheDysonGasorCoulombGaspicture.ThepartitionfunctionissimplyZ=CR∞−∞QNi=1dλie−S(λ)withS(λ)=NPNi=1V(λi)−2Pi,j,i6=jln|λi−λj|.ThisisjustasystemofNparticleswithcoordinatesλiontherealline,c