Block Spin Density Matrix of the Inhomogeneous AKL

arXiv:0804.1741v1[quant-ph]10Apr2008NonamemanuscriptNo.(willbeinsertedbytheeditor)BlockSpinDensityMatrixoftheInhomogeneousAKLTModelYingXu·HoshoKatsura·TakaakiHirano·VladimirE.KorepinReceived:date/Accepted:dateAbstractWestudytheinhomogeneousgeneralizationofa1-dimensionalAKLTspinchainmodel.Spinsateachlatticesitecouldbedifferent.Undercertainconditions,thegroundstateofthisAKLTmodelisuniqueandisdescribedbytheValence-Bond-Solid(VBS)state.Wecalculatethedensitymatrixofacontiguousblockofbulkspinsinthisgroundstate.Thedensitymatrixisindependentofspinsoutsidetheblock.Itisdiagonalizedandshowntobeaprojectorontoasubspace.Weprovethatforlargeblockthedensitymatrixbehavesastheidentityinthesubspace.ThevonNeumannentropycoincideswithRenyientropyandisequaltothesaturatedvalue.KeywordsAKLT·DensityMatrix·Entanglement·ValenceBondSolidPACS75.10.Pq·03.65.Ud·03.67.Mn·03.67.-a1IntroductionQuantumentanglementisafundamentalmeasureofhowmuchquantumeffectswecanobserveandusetocontrolonequantumsystembyanother,anditistheprimaryNSFGrantDMS-0503712YingXuC.N.YangInstituteforTheoreticalPhysicsStateUniversityofNewYorkatStonyBrook,StonyBrook,NY11794-3840,USAE-mail:[e-mail]tofAppliedPhysicsTheUniversityofTokyo,7-3-1Hongo,Bunkyo-ku,Tokyo113-8656,JapanE-mail:[e-mail]tofAppliedPhysicsTheUniversityofTokyo,7-3-1Hongo,Bunkyo-ku,Tokyo113-8656,JapanE-mail:[e-mail]calPhysicsStateUniversityofNewYorkatStonyBrook,StonyBrook,NY11794-3840,USAE-mail:[e-mail]2resourceinquantumcomputationandquantuminformationprocessing[9],[49].Theentanglementofquantumstates,particularlyrelatedwithspinsystemshasbeenat-tractingagreatdealofinterest,seeforexample[5],[15],[18],[24],[26],[28],[32],[37],[43],[46],[53],[58],[64],[65],[70].QuantumentanglementcanbequantifiedbythevonNeumannentropyandRenyientropyofasubsystem,asdiscussedin[18],[21],[22],[36],[38],[66],[71].AnarealawofthevonNeumannentropyinharmoniclatticesystemshasbeenproposedandstudiedin[13],[14],[58].Entanglementpropertiesalsoplayanimportantroleincondensedmatterphysics,suchasphasetransitions[55],[56]andmacroscopicpropertiesofsolids[25],[62].Extensiveresearchhasbeenundertakentounderstandquantumentanglementforcorrelatedelectrons,interactingbosonsaswellasvariousothersystems,see[3],[7],[10],[11],[12],[16],[17],[19],[27],[33],[34],[35],[39],[40],[42],[44],[45],[48],[51],[52],[54],[57],[59],[60],[61],[62],[67],[68],[69],[72]forreviewsandreferences.InthispaperwestudytheinhomogeneousgeneralizationoftheAKLTspinchainmodel.TheoriginalhomogeneousmodelwasintroducedbyAffleck,Kennedy,LiebandTasaki[1],[2]withallbulkspinsbeingthesame.Theinhomogeneousgeneralizationwasintroducedin[41]inwhichspinsatdifferentlatticesitesmaytakedifferentvalues.Theconditionsoftheuniquenessofthegroundstatewerediscussedandcorrelationfunctionsinthegroundstatewereobtained.TheuniquegroundstateisknownastheValence-Bond-Solid(VBS)state[6],[41],whichisveryimportantincondensedmatterphysics.TheVBSstateiscloselyrelatedtoLaughlinansatz[47]andfractionalquantumHalleffect[6].Itenablesustounderstandgroundstatepropertiesofanti-ferromagneticsystemwithaHaldanegap[30].UniversalquantumcomputationbasedonVBSstateshasalsobeenproposed[63].Thedensitymatrixofacontiguousblockofbulkspins(wecallitthedensitymatrixlaterforshort)ofthehomogeneousAKLTmodelhasbeenstudiedextensivelyin[18],[23],[38],[41],[64],[71].Itcontainsinformationofallcorrelationfunctions[6],[37],[38],[71].Moreover,thedensitymatrixwasshown[18],[38]tobeindependentofthesizeofthechainandthelocationoftheblockrelativetotheends.Itwasdiagonalizedin[71]andprovedtobeaprojectorontoasubspace.Itwasconjecturedin[71]thatthestructureandpropertiesofthedensitymatrixisgeneralizableto:(i)theinhomogeneouschain;(ii)higherdimensionallattices;(iii)generalgraphs.Inthispaper,weshallprovethefirstpartoftheconjecture,i.e.thatforthe1-dimensionalinhomogeneousAKLTmodel.Thegeneralversionoftheinho-mogeneousmodelwasfirststudiedin[41].InordertowritedowntheHamiltonianandconditionsfortheuniquenessofthegroundstate,wefirstintroducenotations.ConsiderasystemofalinearchainofNbulkspinsandtwoendingspins.BySjwedenotethevectorspinatsitejwithspinvalueSj.ThenweassociateapositiveintegernumbertoeachbondofthelatticeanddenotebyMij(Mij=Mji)thebondnumberbetweensitesiandj.Theymustberelatedtobulkspinsbythefollowingrelation2Sj=Mj−1,j+Mj,j+1(1)with2S0=M01and2SN+1=MN,N+1forendingspins.Theconditionforsolvabilityofrelation(1)isN+1Xj=0(−1)jSj=0.(2)3Solutiontorelation(1)undercondition(2)isMj,j+1=2jXl=0(−1)j−lSl≥1.(3)Moredetailscanbefoundin[41].NowwedefinedtheHamiltonianoftheinhomogeneousAKLTmodelasH=NXj=0Sj+Sj+1XJ=Sj+Sj+1+1−Mj,j+1CJ(j,j+1)PJ(j,j+1).(4)HeretheprojectorPJ(j,j+1)describesinteractionsbetweenneighboringspinsjandj+1,whichprojectsthebondspinJj,j+1≡Sj+Sj+1ontothesubspacewithtotalspinJ(J=Sj+Sj+1+1−Mj,j+1,...,Sj+Sj+1).AnexplicitexpressionofPJ(j,j+1)wasgivenin[41].CoefficientCJ(j,j+1)cantakearbitrarypositivevalue.ThisHamiltonian(4)hasauniquegroun