Exact expressions are derived for some excited states in a linear quantum spin system for which the exact ground state has been studied in the last decade

Caspers, W.J.

Magnus, W.

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Some exact excited states in a linear antiferromagnetic spin system

University of Twente Research Information

Volume88A, number2 PHYSICSLETTERS 22 February1982SOME EXACT EXCITED STATES IN A LINEAR ANTIFERROMAGNETIC SPINSYSTEMW.J. CASPERS andW. MAGNUS’Twente Universityof Technology,Enschede,TheNetherlandsReceived20 October1981Revisedmanuscriptreceived12 November1981Exactexpressionsarederivedfor someexcited statesin a linear quantumspin systemfor which the exactgroundstatehasbeenstudiedin the last decade.1. Hamiltonianandnotation. Weconsidera linear describesthe couplingbetweenthekth and the(k + 1)-stantiferromagneticlatticewhich containsa next near- cell,which contains I nn interactionand2 nnn inter-est neighbourinteraction(NNN) with halfthe strength actions.of the nearestneighbourinteraction(NN). The hamil-tonian is of the Heisenbergtype: 2. Eigenstatesandeige,ivaluesofHOk. The eigen-statesof Hok are chosento besimultaneouseigen-— ~,. ~ statesof (S2k 1 +S2k)2and~~kl +S~k,and can~= i ‘ ~= i I thusbelabeledby the totalspinSk and its z-compo-nentMk. In thetwo-spin representationof thekth cell,2N is the numberof spins(S1 = 1/2). thespectrumof Hok splits up into a singletand a trip-Assumingperiodicboundaryconditions,we put let, asis shownin table 1.~ = Si for every lattice site i. For convenience,the chainis divided into N cells of 2 spins,andH can 3. ThegroundstateofH. Considerthe productbesplit up into two parts[1—41: NN N V~~o>0>1 ® 0>2 ®~® ‘°>N = “10>k. (5)HOIiHOk and H’=~H,~,k+l , (2) k1turns outto be notonly thegroundstateof H0,where butalso an eigenstateof H H0 +H’ with the lowest— — 2 3 possibleeigenvalueE0 = —3N. Thereare two eigen-Hok — ‘~2k—152k —2[(S2k1 +S2k) — ~] (3) .statesof H, correspondingwith t is lowestpossibledescribestheinternalinteractionin the kth cell, con- eigenvalue.The otheroneis found by translationovertamingthespinpair (52k_1 S2k)~and onelattice spacing[1—41.= ~2k S2k+l 4. Shorthandnotations.Let I~k(M)>be a many-+ 21 S +S •~ ~ ~ ~ cell statein which only thekth cell is in a triplet state~S2k—1 2k+ 1 2k 2k+2) 4j (with qua tumnumberM). In thesamew y I ~kl(M,M’)>= 2 ~ + ~ ~ denotesa statewith only two cells (k, 1) in a triplet( 2k—i 2k) 2k+1 + 2k~ 2k+1 +S2k+2) state(MrespM’). In otherwords,if a particularcellOn leaveof absencefrom Instituutvoor TheoretischeFysika, / doesnotappearin thenotationof a many-cellstate,UniversiteitLeuven,B-3030 Leuven,Belgium. it is assumedto be in the cell groundstate,i.e. in the0 03l-9163/82/0000—0000/$02.75© 1982 North-Holland 103Volume88A, number2 PHYSICSLEJT[RS 22 lebruary 1982Table I~1kIO)k=2~2(+ ~)k_I_+)k)a) 0 0 “okI°~k= ~3~°~kIM)k = I I 11ok I lM>k ==2~2(i+~~)k+~ 1 0= I 1 1a) +_-)k denotesa cell statewith spin up anddownon sites2k — 1. 2k respectively,andso on.singletstateI0)~.Finally, is the Kroneckerdeltasymbol.With thehelp of theseequations,the following setISM>kl = ~ (lM~1M2 I 1 1SM>I~kl~Ml, M2)>, of eigenstates has been constructed for the full harnil-tonian:S = 0,1,2, M= --S, ~S + I S — l,S, (6) (1) A triplet (S = 1) ofeigenstateswith energyE=—3N + 4, containing only states with one excited cell:describestheadditionof the cell spinsof the cellskand!,when both cells arein a triplet state.K1M1 1M21 NX 1 ISM> is a Clebsch—Gordancoefficient [5] I ~~) = ~ (~)k I~k(M)>, N even. (8)~k=15. Someexcitedstates.Excitedstatescanbe con- (2) A singlet (S = 0), with energy E = —3N ÷4,structed exactly, by taking appropriate linear combi- containing only states with two excited cells:nationsof somebasicstates1CIk(M)> and ISM)kl. InNorder to prove that thoseareexacteigenstatesofH, I ~ (—l)”~i00>k,k+l , N even. (9)one hasto figure outhow H’ actson the basicstates. ~Using conventional Clebsch—Gordan technology, andapplying the Wigner—Eckart theorem [5] for irredu- (3) A triplet (S = 1), with energy E = —3N+ 8,cible tensoroperators,one finds: containingonly stateswith oneor two excitedcells:Hkk+lkl)o>=0,~/Ikk+1I~k~Ml)>_~llM1>kk+I ‘ N L k=1~ (_l)k+1IlM>kl] , Nodd. (10)H~k+lI~k+1(M2)>=~2I1M2>k,k+1‘ l~k<l~NAll states are normalized to 1.k+1 V1k k+1(Ml ,M2)>6. Concludingremarks. The presentedexcitedstates= _4~ ~ (lF 1 —~ 1100> l~1M1 Ill 1M~) correspondto exacteigenstatesof thehamiltonianHpM~M~(7) given in (1), andare characterizedby a fixed maximumX K 1—u 1M2 Ill 1M~)l~k, k÷1 (M~, Mi)> numberof cell excitations.Recently,ShastryandSutherland[3] investigated,for thesamesystem,ex-cited statesof a differentnature.Theyconsideredthe+ ~ KiM1 1M2Ill lM’>(I ~k(M)> +1~k+1(M’))), two phas of thesyst mrepresent dby singl tpairsM’(2k — 1,2k) and (2!, 21 + 1) respectively,separatedhlk,k+1ISM>k,k+1 = [S(S + 1) 4IISM>kk+1 by defects,i.e. single spins.To meetperiodicboundaryconditionsthenumberof defectsshould be even.In+ V’2öSl II~k(M)>+ I ~‘~k+1 (RI)] . particular they analyzedstateswith two defectsand104Volume88A, number2 PHYSICSLETTERS 22 February1982solved for thesestatesanapproximatesecularprob- Referenceslem,resultingin elementaryexcitationswith thesamewave number,energyandspinasthoserepresentedby [1] P.M. vandenBroek,Phys.Lett. 77A (1980)261.thestatesI ~w> and I ~) of ourformulas(8)and(9) [2j P.M. vandenBroek,W.J. Caspers,M.W.M. Willemse,respectively.This agreementmeritsfurtherdetailed Physica104A (1980)298.[31 B.S. Shastryand B. Sutherland,Phys.Rev. Lett. 47analysis. (1981)964.For further exactresultsin linear quantumspin [41C.K. MajumdarandD.K. Ghosh,J.Phys.C3 (1970)911;systemswe may referto refs. [6,71. J.Math.Phys.10 (1969)1388,1399.[51AR.Edmonds,Angularmomentumin quantummechan-7. Two-dimensionalsystems.Usingthesamemeth- ics (PrincetonU.P.,1960).[6j J.C. BonnerandM.E. Fisher,Phys.Rev. 135A(1964)od, it is also possibleto constructexacteigenstates 640.for sometwo-dimensionalversionsof the presented [7] C.J. Thompson,in: Phasetransitionsandcritical phenom-hamiltonian.Detailswifi be publishedin a separate ena,Vol. 1, eds.C. DombandandM.S. Green(Academicpaper. Press,New York, 1976)p. 425.The authorsare very muchindebtedto Dr.F.W. Wiegel for critical readingof themanuscript.105

51AR.Edmonds, Angular momentum in quantummechan7. Two-dimensionalsystems. Using the same meth- ics (Princeton U.P.,

For further exact results in linear quantum spin [41C.K. Majumdar

for some two-dimensional versions of the presented [7] C.J. Thompson, in: Phase transitions and critical phenomhamiltonian. Details wifi be published in a separate ena,

od, it is also possible to construct exact eigenstates 640.

resulting in elementary excitations with the same wave number, energy and spin as those represented by [1] P.M. van den Broek,

the states I~w> and I ~) of our formulas (8)and (9) [2j P.M. van den Broek,W.J. Caspers, M.W.M.Willemse, respectively. This agreement merits further detailed Physica 104A

Some exact excited states in a linear antiferromagnetic spin system

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