Arikawa Mitsuhiro

Kuramoto Yoshio

Saiga Yasuhiro

Yamamoto Takashi

Physical Review Letters

Yamamoto, Takashi

Saiga, Yasuhiro

Arikawa, Mitsuhiro

Kuramoto, Yoshio

Institutional Repositories DataBase (IRDB)

Exact Dynamical Structure Factor of the Degenerate Haldane-Shastry Model

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Exact Dynamical Structure Factor of theDegenerate Haldane-Shastry Model著者 Yamamoto Takashi, Saiga Yasuhiro, ArikawaMitsuhiro, Kuramoto Yoshiojournal orpublication titlePhysical review lettersvolume 84number 6page range 1308-1311year 2000-02URL http://hdl.handle.net/10097/34960VOLUME 84, NUMBER 6 P H Y S I C A L R E V I E W L E T T E R S 7 FEBRUARY 20001308Exact Dynamical Structure Factor of the Degenerate Haldane-Shastry ModelTakashi Yamamoto,1 Yasuhiro Saiga,2 Mitsuhiro Arikawa,3 and Yoshio Kuramoto31Max-Planck-Institut für Physik komplexer Systeme, Nöthnizer Strasse 38, D-01187 Dresden, Germany2Institute for Solid State Physics, University of Tokyo, Roppongi 7-22-1, Tokyo 106-8666, Japan3Department of Physics, Tohoku University, Sendai 980-8578, Japan(Received 10 August 1999)The dynamical structure factor Sq, v of the K-component (K  2, 3, 4) spin chain with a 1r2interaction is derived exactly at zero temperature for the arbitrary size of the system. The result isinterpreted in terms of a free quasiparticle picture which is a generalization of the spinon picture in theSU2 case. The excited states consist of K quasiparticles each of which is characterized by a set ofK 2 1 quantum numbers. Divergent singularities of Sq, v at the spectral edges are derived analytically.The analytic result is checked numerically for finite systems.PACS numbers: 75.10.Jm, 05.30.Pr, 67.40.DbRecently much interest has been focused on magneticsystems with orbital degeneracy [1–6]. In the case oftwofold orbital degeneracy, the total degeneracy per sitebecomes 4 2 3 2, and the simplest model to realizethis situation in one dimension is the spin chain with SU4symmetry. The static property of the SU4 spin chain hasbeen studied mainly by numerical methods. It has beenreported that the spin correlation has a period of four unitcells, and that the asymptotic decay has a power-law expo-nent different from unity [3,4]. Such an exponent has alsobeen derived by use of conformal field theory [7]. In viewof this situation, one can naturally ask how the dynami-cal property depends on the number of internal degrees offreedom. Experimental investigations of orbitally degen-erate quasi-one-dimensional magnetic compounds such asNaV2O5 [8] are being performed with increasing accuracy.Hence it is useful to clarify the difference from systemswithout orbital degeneracy not only for static propertiesbut also for dynamic ones.In this Letter, we derive exact an analytic formula forthe dynamical structure factor Sq, v of the SUK spinchain at zero temperature, and provide intuitive interpreta-tion of the result in terms of quasiparticles obeying frac-tional statistics. We take the exchange interaction Jijdecaying as an inverse square of the distance: Jij JNp sinpi 2 jN22, where N is the number oflattice sites with unit spacing and J . 0. The model isgiven by [9,10]HHS 12X1#i,j#NJijPij , (1)where Pij is the exchange (or permutation) operator. It canbe written in the formPij KXd,g1Xdgi Xgdj , (2)where Xgdi changes the spin state d to g at site i. Inthe particular case of SU2, Pij is reduced to the spin0031-90070084(6)1308(4)$15.00exchange 2 Si ? Sj 1 12. This model is a generalizationof the Haldane-Shastry (HS) model [11,12] for the SU2chain, and hence is called the SUK HS model in thefollowing.In the original HS model, the spinons form an ideal spin12 “semion” gas [13], obeying the fractional exclusionstatistics [14]. The dynamical structure factor Sq, v ofthe SU2 HS model has a remarkably simple structure interms of the spinon picture: only two spinons contributeto Sq, v [15]. Since the semionic statistics is applica-ble only to the case of SU2, one has to take more gen-eral fractional statistics in order to apply a quasiparticledescription.To derive the exact formula for the dynamical structurefactor, we use the UK spin Calogero-Sutherland (CS)model [10] as an auxiliary. The Hamiltonian of the UKspin CS model is given byHspinCS  212NXi1≠2≠x2i1µpL∂2 X1#i,j#Nbb 1 Pijsin2 pL xi 2 xj, (3)where b . 0 is the coupling parameter and L is the sizeof the system. This continuous model is more tractablethan the SUK HS model, because the eigenfunctions ofthe model have been explicitly constructed [16,17]. Wetake the strong coupling limit b ! ` of the UK spin CSmodel. Then particles crystallize with the lattice parameterLN which is taken as the unit of length. We are then leftwith the center-of-mass motion, the lattice vibration andthe dynamics of the internal degrees of freedom which iscalled the “color.” The color dynamics is equivalent to thedynamics of the SUK HS model. The freezing trick de-scribed above was first introduced by Polychronakos [18],and has been applied to thermodynamics of lattice models[19–21]. This Letter is the first application of the freezingtrick to dynamical quantities.In the U2 spin CS model, Uglov has derived the exactformula of the dynamical spin-density correlation function© 2000 The American Physical SocietyVOLUME 84, NUMBER 6 P H Y S I C A L R E V I E W L E T T E R S 7 FEBRUARY 2000with a finite number of particles [17]. We shall first ex-tend his result to the case of K $ 3, and then take thestrong coupling limit. In doing so we have to make corre-spondence between physical quantities defined in the con-tinuum and discrete spaces. Let us define the followingoperator in the continuum space:Xgdq 1pLNXj1Xgdj e2iqxj , (4)where the momentum q takes values 2pnL with n anarbitrary integer. We first derive the dynamical structurefactor in the continuum model defined bySgdq, v; b XajajXgdq j0j2dv 2 Ea 1 E0 ,(5)where ja	 is the normalized complete basis of the systemwith eigenvalues Ea	, and j0 is the ground state. Weassume that N is an integer multiple of K so that the groundstate is a nondegenerate singlet.In the strong coupling limit the coordinate xj in Eq. (4)is written as xj  Rj 1 uj where Rj  j is a lattice pointand uj describes the lattice vibration. Except for the uni-form motion of the lattice described by uj  const, wemay regard uj as a small quantity. In fact, the density re-sponse can be shown to be smaller than the spin responseby O b21. Then the dynamical structure factor of theSUK HS model is given simply by the strong couplinglimit of Eq. (5) provided that one restricts q in the rangeof the first Brillouin zone: jqj # p .The dynamical structure factor (5) can be derived in amanner analogous to the case of K  2 [17]. However, thefollowing observations are necessary for generalization.First, each excited state relevant to Eq. (5) transforms asone of the weight vectors for the adjoint representation ofSUK. This observation allows us to find the selectionrule for the SUK spin. In the SU2 case, this selectionrule is reduced to the simple fact that excited states relevantto Eq. (5) are spin-triplet states. Second, in order to derivethe matrix element in Eq. (5) we find a convenient set ofoperators given byJa 1pLXq√aXb1Xb,K2a1bq 1K2aXb1Xa1b,bq!(6)with a  1, . . . , K 2 1. More details of the calculationwill be presented elsewhere.In order to give the formula of Sgdq, v, we fixsome notations for partitions [17]. Let LKN be theset of all partitions whose lengths are less than N 1 1and the largest entry is less than K 1 1. Namely wehave LKN  l  l1, l2, . . . , lN  j K $ l1 $ l2 $. . . $ lN $ 0	. For a partition l, we define subsets byCK l  i, j [ l j j 2 i  0 modK	 and HK l i, j [ l j li 1 l0j 2 i 2 j 1 1  0 modK	. Nextwe define the concept type of a partition. The type ofa given partition l contains sufficient information fordetermining the sets CK l and HK l explicitly. Weintroduce a reductive transformation t on the set of allpartitions as follows [22]: (i) If there exist K rows orK columns which have the same number of boxes in apartition, remove those rows or columns. (ii) Apply thereduction (i) repeatedly until the newly generated partitionis no longer reducible.We then determine a subset AKN of LKN as the imageof t, i.e., AKN  tLKN . For any partition l [ LKN ,we say that n is of the type l if n  tl. The totalnumber of types increases from 3 in the case of SU2 to25 in SU3, and to 252 in SU4. For a box s  i, j [l, the numbers lls  l0j 2 i and l0ls  i 2 1 arecalled the leg length and coleg length, respectively. Forany subset n , l, the order jnj is defined as the numberof boxes in n.Now we give the exact formula for the dynamical struc-ture factor Sgdq,v for a finite size of the system withg fi d. We obtainSgdq, v X0ljFKl j2dv 2 El , (7)where the primed summation is restricted so as to sat-isfy the momentum conservation q  2pjCK ljNand the color selection rule for l [ LKN . The lat-ter is conveniently implemented by introducing asubset AK;gdN of AKN , and decomposes the sum-mation over l by each type n  tl such thatn [ AK;gdN . For example, in the case of K  3,we have A3;21N  2, 1, 1, 3, 3, 1, 3, 3, 2, 2	 andA3;13N  1, 2, 2, 3, 2, 1, 1	.In Eq. (7) the excitation energy is given byEl J4µ2pN∂2 ∑N 2 1 jCK lj 2 2Xs[CK ll0ls∏,(8)and the squared form factor byjFKl j2 1NQs[CK l1,1	 l0ls2Qs[HK l lls lls 1 13Ys[CK lN 2 l0lsN 2 l0ls 2 1. (9)Since we consider the case of zero external magneticfields, the SUK symmetry demands that Sgdq,v isactually independent of g, d as long as g fi d. We canprove this fact using the expressions (7)–(9). In the par-ticular case of K  2, our formulas (7)–(9) generalize theknown one [15] to the arbitrary size of the system. Wehave checked the validity of Eqs. (7)–(9) with K  2 and3 by comparing with the numerical result for N # 24 andN # 15, respectively. The numerical result is obtainedvia exact diagonalization and the recursion method. Theagreement is excellent in both cases of K  2 and 3. InFig. 1, we present the result for K  3 and N  15.1309VOLUME 84, NUMBER 6 P H Y S I C A L R E V I E W L E T T E R S 7 FEBRUARY 2000FIG. 1. Numerical result of the dynamical structure factorSq, v in the cases of K  3 and N  15. The vertical andhorizontal axes represent the rescaled energy and momentum,respectively. The intensity is proportional to the area of thecircle. The solid lines are the dispersion lines of the elementaryexcitations in the thermodynamic limit. The analytic resultsare in excellent agreement with numerical ones, and are notdistinguishable from the latter.We now consider the quasiparticle interpretation ofthe color selection rule. For labeling the excited statesrelevant to Sgdq, v, it is more convenient to use theconjugate partition l0  l01, . . . , l0K  [ LNK insteadof l [ LKN . Each l0i has the information on the mo-mentum and SUK spin of a quasiparticle. We call thisquasiparticle a spinon following the SU2 case. Thespinon is considered to be an object possessing the SUKspin. Here the SUK spin means the K 2 1 eigenvaluess1, . . . , sK21 of a set of operators h1, . . . , hK21 wherehg is defined by hg PNi1 Xggi 2 Xg11g11i 2 forg  1, . . . , K 2 1. For the SU2 case, this definitiongives the z component of spin. The SUK spin of thespinon with l0i is specified by a certain condition on thepair i, l0i. Since the condition for general K is rathercomplicated, we give an example in the case of K  3.The SU3 spin for l0i is assigned as follows: It is0, 12 if i, l0i  0, 0, 1, 1, 2, 2 mod3; 12, 212if i, l0i  0, 1, 1, 2, 2, 0 mod3; and 212, 0 ifi, l0i  0, 2, 1, 0, 2, 1 mod3. It is important tonote that a spinon transforms as a weight vector of thefundamental representation K̄ of SUK.From the formulas (7)–(9), we can conclude that therelevant excited states for the SUK HS model consistof K spinons. Moreover the conditions on the type ofexcited states lead to an important consequence: K-spinonexcited states have K 2 1 different SUK spins. That is,the excited states contain K 2 1 species of quasiparticles.For instance, in the case of K  3, excited states relevantto S13q, v consist of three spinons with SU3 spins0, 12, 0, 12, and 12, 212. For the SU2 case,1310we recover the well-known fact that only two spinons withthe same spin contribute to Sq, v [15].The K-spinon excitation belongs to the tensor represen-tation K̄≠K of SUK. This representation contains theadjoint representation as an irreducible component. Fromthe condition stated above for the K-spinon excitation, wesee that the K-spinon excitation transforms as one of theweight vectors for the adjoint representation. This is con-sistent with the condition for the SUK spins of the excitedstates which are relevant to Sgdq, v.Now we present the thermodynamic limit of the for-mulas (7)–(9). Performing a procedure similar to that inRefs. [23,24], for K  2, 3, and 4, we obtain the final re-sult as follows:Sq, v  AKX1#a,b#KKYi1Z 121dki jFKab kj23 dq 2 p 2 pkdv 2 ek , (10)where ek  pys2KPKi1 1 2 k2i  with ys Jp2, and pk  pKPKi1 ki . In the above for-mula, AK is a normalization constant given byAK 2KpK3K 2 1KYj1GK 2 1KG jK2, (11)and the form factor is given byFKab k jka 2 kbjgKQ1#i,j#K ,i,jfia,b jki 2 kjjg0KQKi1 1 2 k2i 12gK 2(12)with gK  K 2 1K and g0K  21K . Since the for-mula (10) for Sgdq, v does not depend on the pairg, d with g fi d, we have omitted the superscript. Un-fortunately our exact result is inconsistent with a conjec-ture proposed several years ago [25].For K  2, the formula reproduces the result ofHaldane-Zirnbauer [15] which was obtained by a com-pletely different method. Notice that the second productin the numerator of Eq. (12) is absent in the SU2 case.We can derive static structure factor Sq by integratingover v in Eq. (10). In the low energy limit we recover theresults of Ref. [9] which are obtained by conformal fieldtheory. By analyzing the form factor for q  2kF withkF  pK , we can show that it has the asymptotic formSq  a1jq 2 2kF ja121 (13)with the exponent a1  2 2 2K and a nonuniversal con-stant a1. In the real space the spin (or color) correlationdecays as b1 cos2kFxjxj2a1 with a certain coefficientb1. A similar analysis shows that there are also weakersingularities around q  2lkF for l  2, . . . , K 2 1 withexponents al  2l1 2 lK. These K 2 1 singularitiescorrespond to K 2 1 gapless bosonic modes [7,9].The spinon interpretation of the formula (10) goes as fol-lows: As in the case of finite systems, the excited states forVOLUME 84, NUMBER 6 P H Y S I C A L R E V I E W L E T T E R S 7 FEBRUARY 2000Sq, v in the thermodynamic limit consist of K spinonswith K 2 1 different SUK spins. This fact means, asin the low energy limit, the dynamics of the SUK HSmodel can be described by the K 2 1 species of quasi-particles. This simple structure reflects the Yangian sym-metry of the SUK HS model [26]. In the form factor(12), the factor jka 2 kbjgK represents the statistical inter-actions of spinons with the same SUK spin, while thefactor jki 2 kjjg0K represents those of spinons with differ-ent SUK spins. We refer to Ref. [27] for a more detailedexplanation of statistical interactions. It will be interest-ing to consider the relation between our results and theexclusion statistics in conformal field theory discussed inRefs. [28,29].The support of Sq, v represents the region in the mo-mentum-frequency plane where Sq, v takes the nonzerovalue. We see that the support of Sq, v as determinedfrom the formula (10) is compact, i.e., there is no intensityoutside of the finite area. In the SU3 case, for example,the support is determined asv # ys2pq2p 2 q  eUq for 0 # q # 2p ,v $ 3ys2pq2p3 2 q for 0 # q # 2p3 ,v $ 3ys2pq 2 2p34p3 2 qfor 2p3 # q # 4p3 ,v $ 3ys2p q 2 4p3 2p 2 qfor 4p3 # q # 2p .For a general value of K , there are K lower boundaries asgiven byeLj q  Kys2p q 2 2pjK 2p j 1 1K 2 q ,for 2pjK # q # 2p j 1 1K with j  0, 1, . . . ,K 2 1.The behavior of Sq, v near the boundaries of thesupport is derived for general K as follows: We canshow that there is a stepwise discontinuity at the upperboundary v  eUq. On the other hand, there are di-vergent singularities at the lower boundaries v  eL0 qand v  eLK21q. Here Sq, v diverges by the powerlaw with the exponent 21K . At the other lower bound-aries v  eLj q with j fi 0, K 2 1, Sq, v has thresh-old singularities but no divergence. As in the SU2 case[15,30], we expect that the divergences at two of the lowerboundaries occur also in the SUK Heisenberg model withthe nearest-neighbor exchange.In conclusion, we have derived the exact formulas (7)and (10) for Sq, v of the SUK HS model for the ar-bitrary size of the system at zero temperature. Our exactresult of Sq, v for K # 4 is likely to be valid for largerK as well. We have also clarified the quasiparticle pictureof the spin dynamics. The relevant excited states consistof K spinons with K 2 1 different SUK spins.T. Y. and Y. S. wish to acknowledge the support of theCREST from the Japan Science and Technology Corpora-tion. T. Y. wishes to acknowledge the support of the VisitorProgram of the MPI-PKS.[1] Y. Q. Li, M. Ma, D. N. Shi, and F. C. Zhang, Phys. Rev.Lett. 81, 3527 (1998).[2] S. K. Pati, R. R. P. Singh, and D. I. Khomskii, Phys. Rev.Lett. 81, 5406 (1998).[3] Y. Yamashita, N. Shibata, and K. Ueda, Phys. Rev. B 58,9114 (1998).[4] B. Frischmuth, F. Mila, and M. 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Exact Dynamical Structure Factor of the Degenerate Haldane-Shastry Model

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