DOC

Meshkov-Glick Model

By Elaine Harrison,2014-12-15 03:10
8 views 0
Meshkov-Glick Model

    Meshkov-Glick Model

    Commun.Theor.Phys.56(2011)6166Vo1.56,No.1,July15,2011

    ThermalEntanglementinLipkinMeshkov-GlickModel

    DULong(杜龙),ZHANGWen-Xin(张文新),DINGJia-Yan(丁伽焱),WANGGuo

    Xiang(下国祥),

    andHOUJingMin(侯净敏)t

    DepartmentofPhysics,SoutheastUniversity,Nanjing211189,China (ReceivedSeptember13,2010;revisedmanuscriptreceivedApril20,2011) Abstract

    ?,einvestigatethethermalentanglementintheLipkin-Meshkov-GlickrLMG)modelwhichconsists

    ofspin.l,2patticleswithXXZ

    typeexehangeinteractionsbetweenanypairofthem.Theferromagnetic(FM)and antiferromagneticlAFM)case5&recompletelyanalyzedatbothfinitetemperatureandzerotemperatureAccording

    totheresultsobtainedbyaccuratenumericalcalculation,severalinterestingphysicphenomenaandcharacteristicsof

    thermalentanglementintheLMGmodelarefound.NotonjydoweevaluatetheentanementoftheLMGmodel,but

    alsodiscoverthecorrelationsbetweenmacroscopicphysicalquantitiesandthermalentanglement.

    PACSnumbers:03.65.Ud,03.67.Mn,75.10.Jm

    Keyworas:thermalentanglement,Lipkin-MeshkovGlickmodel,concurrence

    1Introduction

    Theinitialresearchesofquantumentanglementcan

    datebacktothepropositionofParadox.lJThesekindsof

    questionshavebroughtthedisputesoftraditionalphysi

    calconceptionssuchastime,spaceandphysicalrealities infrontofus.?引Astimegoesby.moreandmorephysi.

    cistshavepaidtheirattentiontothisfield,andmanycrit

    icalconceptionsofquantumentanglementhavebeenin

    troducedintootherphysicsfieldssuchascondensedmat

    terphysics.L4-51Scientistshavealreadyfounddefiniteevi

    dencefortheexistenceofrelationbetweenquantumphase transition(QPT)andquantumentanglement...]

    Inordertoquantitativelyanalyzeentanglement,es

    peciallyinmanybodysystems,alotofentanglement

    measurementshavebeendefinedforquantifyingquantum entanglement.[1113]Oneofthesemethodscalledentan- glementofformulation(EF)isdefinedtoevaluatethe entanglementofbipartitesystemsincludingmixedstate systems.Hilland?bottershaveintroducedconcurrence intoentanglementmeasurementforthesystemoftwo qubits.l4JandEFcanbeexpressedintermsofconcur. renCeaS

    EF(p)=X/1~aC2(p)l

    n

    v/l+aC2(p)

    ,

    whereCrepresentstheconcurrence.Sinceentanglement offormulationisamonotonicfunctionofconcurrence,we canalsomeasureentanglementwithconcurrence.f?J

    Lotsofphysicistshaveinvestigatedthermalentangle

    mentinmanybodysystems.[1516]Itisakindofmea-

    surementwhichevaluatetheentanglementofmany-body systemsatfinitetemperature.Inrecentyears,thermal entanglementbecomesanattractivefieldofquantamen.

    tanglementresearchsinceitscorrelationswithmicroscopic variableofathermodynamicalsystem.1171Lotsoftheoret. icalworksinthisfieldhaveevaluatedtheconcurrenceof eachtwoqubitsofspecificsystems.[1821]

    Manytheoreticalmanybodysystemshavebeen

    investigated.11ljHowever.thesemodelsmainlyconcen

    tratedinone-dimensionalcaseswhich1ustconsidered thenearestornextnearestinteractions.[2325InthisDa_

    per.westudythethermalentanglementintheLMG mode1.[26271inwhichidenticalXXZtypeexchangein

    teractionexistsbetweeneverytwoparticles.Wenumer

    icallyevaluatetheconcurrenceasafunctionoftemper

    atureandexternalmagneticfield.Itsbehaviorsat nitetemperatureandzerotemperaturearealsocalculated. Someinterestingphenomenaandbehaviorsarefound. 2ModelandNumericCalculationSummary

    Here,weconsidertheLMGmodelwithXXZtypeex-

    changeinteractionsunderanexternalmagneticfield.The Hamiltonianforthismodelcanbewrittenas

    N

    H=?[(+yuJY)+()]+?^,(1)

    i<j

    where(=x,Y,z)arethePaulimatrices,Jand

    representthe,y-componentandz-componentexchange interactionsbetweenspins,respectively.histhestrength ofmagneticfieldalongthezdirection.Sinceonlythe ratiosof3z/Jandh/Jareconcernedduringtheprocess ofcalculation.sowesetJ=1fortheAFMcaseandset J=1fortheFMcase.

    SupportedbytheNationalNaturalScienceFoundationofChinaunderGrantNo.11004028,t

heScienceandTechnologyFoundation

    ofSoutheastUniversityunderGrantNo.KJ2010417,andtheTeachingandResearchFoundat

    ionfortheOutstandingYoungFacultyof

    SoutheastUniversity

    tE-mail:jmhou~seu.edu.cn

    @2011ChinesePhysicalSocietyandIOPPublishingLtd http://www.iop.org/EJ/journal/ctphttp://ctp.itp.ac.cn ?一

    62CommunicationsinTheoreticalPhysicsVo1.56 Inamanybodysystem,theconcurrenceofeverytwo qubitsforthei-thandthethspin-12particlesisdefined

    as[14]

    C=max{AiA23A4,0),(2)

    where.(oL=1,2,3,4),satisfyingA123A4,

    arethesquarerootsoftheeigenvaluesofmatrixR R=%(@y)p(0y),(3)

    wherePjisthereduceddensitymatrixofthei-thand j-thspinsubsystem,(rn=i,J)representsthecorre

    spondingYcomponentPaulioperatorofeachspin.Since RisapositivedefiniteHermitianmatrix.itseigenvalues areallpositiverealnumbers.Thereduceddensitymatrix ,.

    canbeobtainedbypartiallytracingoutthedensity matrixPofthewholesystem.i.e.

    ,

    j=trAllbuti,JP.(4)

    Accordingtotheprincipleofmicrocanonicalensemblein quantumstatisticmechanics,densitymatrixPcanbeex

    pressedas

    1

=e一衄

    ,(5P)e?,J

    where=1/kBTwithks

    andTbeingtemperature,

    tionfunction.

    beingtheBoltzmannconstant

    andZ=tr(e一卢,istheparti

    WecanobtainadensitymatrixP0atzerotempera

    oo(i.e.,T__?0)inthe turethroughtakingthelimit__?

    densitymatrix(5)as

    P0=lira

    _oo

    e-8H

    z

    Thezero??temperaturedensitymatrixP0canalsobeex-- pressedinequalmixtureofallthegroundstatesasfollows, IG,

    Ji=1

    wheregrepresentsdegeneratenumberofthegroundstates andlGt)indicatesthei-thgroundstate.Thiszero

    temperaturedensitymatrixP0isalsonamedasthether

    malgroundstate..]Certainly,

    whenthegroundstateis

    n0ndegenerate,Porepresentstheonlygroundstate. Withtheprinciplesofsolutionforemetioned,suchas concurrenceoftwoparticlesanddensityorreducedden

    sitymatrixofourmodel,theprocessesofnumericcalcu- lationarecompliedwithfollowingsteps.

    Firstofall,weexpressoperatorssuchasHamiltonian

(Eq.(1))anddensitymatrixasnumericmatricesinacer

    tainrepresentation.Here,wechoosePaulirepresentation forconvenience,i.e.,

    {IT1-??TN-1T),ItlTN-11),,I1…一1i)).(8)

    WithregularorderofPaulirepresentationabove,Hamil- tonian(Eq.(1))isobtainedby

    N

    H=0?Jo~6rl(Tm+N0,.

    (9)

    Z<ma=,Y,z=1

    Subsequently,thedensitymatrixofwholesystemcanbe gotthroughEq.(5).Itisnecessaryfornumericcomputa

    tiontofindoutadiagonaltransformationmatrixUthat satisfyH=TwhereHcontainsnothingbuteigen-

    valuesinitsdiagonalline.So,densitymatrix(Eq.(5))is computedbyP=[Uexp(ZU*HU)U*]/Zinthecaseof

    finitetemperaturewhileitiscomputedbyeigenstatesof HamiltonianwithEq.(7)inthecaseofzerotemperature. Macroscopicphysicalquantityiscalculatedbytheaver

    ageofquantumoperatorinnumericcomputation.For example,themethodofcalculatingtotalspinmagnetic momentjs

    (s):(0s)=rn0s),t=1i

    wherestzisthespinoperatorofi-thparticle.Numericcal

    culationresultsandconclusionsareshowninthefollowing sections.

    3ThermalEntanglementofLMGModel

    Accordingtothepreviousdiscussion,wenumericcal

    culatetheconcurrenceofeachtwoqubitsandmagnetic

moment(S)bothatfinitetemperatureandzerotempera

    ture.Severalbehaviorsoftherma1entanglementandsome correlationsbetweenthermalentanglementandmagnetic momentarefound.

    3.1FiniteTemperature

    Theconcurrencebetweeneachtwoparticlesatfinite temperatureunderanexternalmagneticfieldisevaluated bynumericalcomputation.Wledepicttheresultsofthe AFMandFMcasesinFigs.1and2respectively.The concurrencedecreaseswithanincreasingtemperaturein theentangledregion(C?0)inbothAFMandFMcase

    asitshowsinFigs.1and2respectively.Thisbehavior isconsistencewiththermalentanglementinmanyother models.suchasonedimensionalHeisenbergmode1.[17,22] Clearly,theconcurrenceissymmetricabouttheexternal magneticfieldsincetheformofHamiltonian. DuetothesignoftheparametersofJandJz,the distributionofconcurrenceforeachcasehasitsownchar

    acteristics.Theentangledareaisconcentratedonust twosmallseparatedregionson,Tplaneandonlytwo

    peaksarepositedsymmetricabouth=0inAFMmode1. Thereisnothermalentanglementexceptcertainareawith specia1combinationofmagneticfieldandtemperature.In morespecificterms,asthestrengthofmagneticfieldin

    creasestoacertainstrength,entanglementappearssud

    denly.Figure1isshownforillustratingallcharacteris

    ticsabove.Wlecal1thisvalueofmagneticfieldthreshold magneticstrength.AsshowninFig.1(d),whenthepar

    ticlenumberincreases,thethresholdmagneticstrength andthepeaksofconcurrencedeviatefromh:0butthe

maximumofconcurrencedescends.

    No.1CommunicationsinTheoreticalPhysics

    8

    0

    0

    0

    8

    2

    

    0

    O.4 N3.J1,1

    0.06 ^

    ?=7.1=1,,2=l

    Ic

    J

    00 10

    S

    -J

    

    0

    0

    8

    2

    5

    

    0

    0

N==:5..,:1..,2=1

    O.3 0.2 0,

    0.0 05

    ^

    0

    06

    0.4 0.2

    0

    0.O O

    8

    

    U

    

    o

    0

    j

    N=3,J=1.,.=-0.3

    N=5,.,=-1,,:0.3

    O

    8

    

    

    0

    0

0

    0

    N4,.,=:l,z---0.3

    0.3

    8

    

    0.2

    ;

    0

    g().1

    0

    0.0

    0

    BT

    N=7,,=1,J=O.3

    4

    Fig.2ThedistributionofconcurrenceoftheLMGmodelintheFMcaseforJ=-1,=-0.3.Here,N

    reDresentstotalparticlenumberofLMGmode1.TheconcurrencesofthesystemswithN=3,4

    ,5,and7are

    shownin(a),(b),(c),and(d),respectively. CommunicationsinTheoreticalPhysicsVo1.56 Theconcurrenceofthe

    issomewhatdifflentfrom

    FMcaseatfinitetemperature

    thatoftheAFMcase.Theen-

    tangledregionisconcentrated{ustinoneconnectedcon

    tinuousarea.Itisalsosymmetricaboutthemagneticfield h.AsshowninFig.2.jnsometemperature.theconcur

    rencehasatleasttwopeaksthatarepositedsymmetric abouth=0.OnlyifparticlenumberN>3.several

Report this document

For any questions or suggestions please email
cust-service@docsford.com