Dynamic Structure Factor of the Two-Dimensional Shastry-Sutherland Model Christian Knetter and G¨otz S. Uhrig Institut fu¨r Theoretische Physik, Universit¨at zu K¨oln, Zu¨lpicher Str. 77, D-50937 K¨oln, Germany (February 2, 2008) 4 for the dynamic structure factor which is measured by 0 We calculate the 2-triplon contribution to the dynamic inelastic neutron scattering (INS). 0 structure factor of the 2-dimensional Shastry-Sutherland 2 model, realized in SrCu2(BO3)2, by means of perturbative an cwoentfiinnduofluastubnoituanrdy2tr-tarnipsfloornmbaatinodnss.. TFohresreeablaisntdicspsahroawmleatregres 5.120 A 22 J wthuiegmihg.hptSroiencoitsuhiorenfisntinrdueilncatgsustripceefnraemcutitotrroadnqespucaeanntdttieitnraigntigsvteeroxunpngedlryeimrsotenannmtsd.oinmgeno-f J2 J1 ~b 33 2.905 A ] PACS numbers: 75.40.Gb, 75.50.Ee, 75.10.Jm b l e δ h - r t Quantum antiferromagnets which do not have a long ~ s range ordered ground state, so-called spin liquids, con- a a . t tinue to attract considerable interest. While there δ a v m are many 1-dimensional examples there are only a few 2-dimensional systems. Of strong recent interest is - FIG. 1. Shastry-Sutherland model with spin 1/2 on the d SrCu (BO ) [1], a realization of the 2-dimensional spin 2 3 2 dots; dimers are solid gray lines. The microscopic angles n 1/2 Shastry-Sutherland model [2] and distances apply to SrCu2(BO3)2 according to Ref. [1]. o [c H =J1 SiSj +J2 SiSj =J1Uˆ +J2Vˆ , Tdihmeeprsrimtaikteivneavsecetqoursala) awnhdileba˜spaanndthb˜esdpimanert-hlaettliactetiΓceeffΓ(AaBll <Xi,j> X[i,j] (1) distinghuishing horizontal and vertical dimers. 2 v where J andJ arethe intra-andinter-dimercouplings 1 2 8 as in Fig. 1. Two spins coupled by J are referred to as We briefly review the derivation of perturbative ef- 0 1 dimers. We focus on J ,J >0. fective operators (Hamiltonian and observable) in gen- 4 1 2 The state 0 with singlets on all dimers is an exact eral and derive the appropriate INS observable for the 9 | i 0 eigen-stateofH forallvaluesofJ1 andJ2 [2]. We found Shastry-Sutherland model. The corresponding spectral 3 0 to be the groundstate (singlet-dimer phase)of H for density, i.e., the dynamic structure factor, is calculated | i 0 x := J /J below 0.63 [3] while other results indicate via the T =0 Green function. We focus on the 2-triplon 2 1 t/ an instability at sl≈ightly higher values of x, for a review part above the flat, featureless 1-triplon band [7–9]. a see Ref. [4]. Thelimitofisolateddimers,i.e.,x=0,servesasstart- m In many papers it has been shown that the magnetic ing point of the perturbative analysis. The basic excita- - properties of SrCu (BO ) can be understood well by tion is given by promoting one singlet to a triplet. The d 2 3 2 H in (1) in the singlet-dimer phase [4]. Thus the bo- next higher excitation are two triplets and so on. Upon n o rate constitutes a particularly transparent case of a 2- switchingontheinter-dimercoupling(x>0)thetriplets c dimensional spin liquid. In view of the extensive spec- acquireadispersionandbecome dressedparticles,which : troscopicdataonthis systemquantitativetheoreticalre- we call triplons [10]. v i sults for spectral densities are highly desirable. But so Up to a constant, Uˆ in (1) counts the number of X faronlythenumericalexactdiagonalization(ED)forsys- triplonsandwedefine the particle-numberoperatorQ= r tems of 20 or 24 spins was possible [4]. This approach is Uˆ + 3N/4 (N: number of dimers). The perturbative a hampered by the finite size in two ways. First, the ener- part Vˆ in (1) decomposes into ladder operators Vˆ = gies of the excited states display strong finite size effects T +T + T , where the index i denotes the number −1 0 1 since these states are spatially extended, in particular of triplons created (destroyed) by T . i the bound states built from two elementary excitations. TheoriginalHamiltonianismappedbyaperturbative Second, the ED provides only isolated spikes instead of CUT[3,9,11]toaneffectiveHamiltonian(inunitsofJ ) 1 continuousdistributions. Inthisarticle,weremedythese drawbacks by making use of recent conceptual progress ∞ ′ in the method of perturbative continuous unitary trans- Heff(x)=Uˆ + xk C(m)T(m) (2) formations (CUTs) [5,6]. We provide high order results kX=1 Xm 1 where m = (m ,m ,... ,m ),m 0, 1 . This ef- randthe otheroneonthedimer atr+d. Theprimitive 1 2 k i ∈ { ± } fective Hamiltonian conserves the number of triplons: vectorsa and bspan the dimer-lattice Γ (Fig. 1). The eff [H ,Q] = 0. In each order k, H is a sum of vir- vectorK+σQ lies within the (first) Brillouinzone (BZ) eff eff tual processes T(m) = T T weighted by rational ofthe duallatticeΓ∗ spannedbythe vectorsa∗ andb∗; coefficients C. The summ1′··is· rmesktricted by the triplon- as usual Q = (π,π)effin units of the inverse dimer-lattice conservation condition mP1 + +mk = 0. The effec- constant. The additional quantum number σ 0,1 ··· ∈ { } tive Hamiltonian can be decomposed into irreducible n- is chosen such that K lies within the magnetic Brillouin particle operators H [5] H =H +H +H +.... zone(MBZ)whichisthe(first)Brillouinzoneofthedual n eff 0 1 2 The matrix elements of the irreducible H for the lattice Γ∗ . The exchange parity of the two triplons is n AB infinite system can be computed perturbatively on fi- fixedby σ,K,d S =( 1)S σ,K, d S,hencewerestrict | i − | − i nite clusters due to the linked cluster theorem. For the to d=(d ,d )>0: [d >0or(d =0andd >0)]. 1 2 1 1 2 Shastry-SutherlandmodelH isconvenientlysettozero; Forfixedtotalmom⇔entumK,H (15thorder)isasemi- 0 1 H and H were determined previously [3,9] to obtain infinite matrix in d and σ while H (14th order) is rep- 1 2 2 the 1- and 2-triplon energies. resented by a 84 84 matrix. The matrix elements are × Applying the same transformation as for H other ob- polynomials in x calculated previously [3,12]. servables are also mapped onto their effective counter- We turn to analyzing the appropriate observable for parts [5] the INS experiment on the SrCu (BO ) . It reads 2 3 2 (q) = Sz(x )eiq·xi , where Sz(x ) is the z- eff(x) = ∞ xkk+1 C˜(m;i) (m;i) (3a) FcTohmepxonemnutsPotfinthotesbpeiincoantftuhseedpowsiitthionthxeiv(edcoittosrisnrFig.Γ1). O O i eff Xk=0 Xi=1|mX|=k which denote the positions of the dimer centers. ∈ (m;i):=T T T T , (3b) The momentum transfer q measured in experiment O m1··· mi−1O mi··· mk is any vector in the dual space whereas the excitations where istheinitialobservable. Ausefuldecomposition of the Shastry-Sutherland model are labeled best by O of eff reads the momenta K MBZ. The usual backfolding im- O ∈ plies K(q) = q mod (Γ∗ ) and K(q) + σ(q)Q = q ∞ AB mod (Γ∗ ) whence σ(q) = 0 for q MBZ mod (Γ∗ ) eff = d,n , (4) eff ∈ eff O O and σ(q)=1 otherwise. nX=0dX≥−n Weconstructanoperator definedforKandrsuch, N where d indicates how many particles are created (d that (K;r) and (q;x) have the same action on the ≥ N F 0) or destroyed (d < 0) by whereas n 0 is the ground state 0 . The operator will then be used to Od,n ≥ | i N minimum number of particles that must be present for obtain the effective operator. It is a crucial feature par- tohaveanonzeroaction. ForT =0measurements ticular to the Shastry-Sutherland model that the triplon d,n O only the operators matter. vacuum 0 is not changed by the CUT since it is an ex- Od,0 | i Theenergyandmomentumresolvedn-particlespectral acteigen-state. Withasuitableconventionforthesinglet density for the operator is given by orientation the action of on 0 is O F | i (n)(ω,K)= π−1Im (n)(ω,K) , (5) (q)0 = (8) S − G F | i isin(q δ ) eiq·rv rv +isin(q δ ) eiq·rh rh . where (n) is the retarded n-particle Green function · v | i · h | i G Xrv Xrh 1 (n)(ω,K)= 0 † 0 . The sums run over all vertical dimers rv and horizontal G (cid:28) (cid:12)(cid:12)On,0ω− ni=1Hn+i0+On,0(cid:12)(cid:12) (cid:29)(6) dimers rh. A state r is defined by one triplon with (cid:12)(cid:12) P (cid:12)(cid:12) Sz = 0 on the dimer|ait r and singlets elsewhere. The Since expectation values do not change under unitary vectors δ are defined in Fig. 1. v/h transformations the Green function (6) is not altered if The appropriate local operator (r) using the dis- the the effective operators are substituted for the initial tances r Γ reads (r) = Sz(r)N Sz(r), where the ones. Using the decompositionsofH andof aswellas subscripts∈0 aeffnd 1 distNinguish th0e tw−o sp1ins on dimer r O theconservationoftriplonsbyHeff theindividualsectors such that (r)0 = r . The momentum space repre- of different triplon numbers can be analyzed separately. sentation isNgive|niby (|σ¯i=1 σ) We focus on the 2-triplon sector and introduce the 2- − triplon momentum states [3,12] (q)=a(q) (σ,K)+b(q) (σ¯,K) (9a) N N N (cid:12)σ(q),K(q) 1 (cid:12) σ,K,d S = ei(K+σQ)·(r+d/2) r,r+d S , (σ,K)= ei(K+σQ)·r (r) (cid:12) (9b) | i √N Xr | i (7) N Xr N a(q)=i[sin(q δ )+sin(q δ )]/2 (9c) v h where S 0,1,2 is the total spin of the two triplons · · ∈ { } b(q)=i[sin(q δ ) sin(q δ )]/2 , (9d) and r,r+d is the state of one triplon on the dimer at · v − · h | i 2 which ensures (q)0 = (q)0 for all q. the contrary,ED data for the specific heat [15] indicates N | i F | i Including the microscopic details for SrCu (BO ) thatlowervaluesofxfitbettertoexperimentthanlarger 2 3 2 (Fig.1)anddenotingqbyq=ha˜∗+kb˜∗ fixesthescalar ones, cf. the data for 16 spins. productstoq δ =0.717(h k)andq δ =0.717(h+k). A possible weakness in our analysis are the necessary v h · − · This completes the derivation of the excitation operator extrapolations. InRef.[3],wedidnotuseOPTbutDlog (q) for the INS with momentum transfer q. Pad´e approximants which allow for power-law singular- N The action of the effective local operator (r) from ities. This led in a very robust way to the instability eff N Eq. (3) on 0 is implemented on a computer. Although at x 0.63. OPT does not allow for power-law sin- | i ≈ (r) exclusively produces 1-triplon states when acting gularities. Hence it leads to a smoother dependence on N on 0 (r)leadstostatescontaininganarbitrarynum- x. No instability occurs below 0.7 so that the precise eff | iN ber of triplons. Focusing here on the 2-triplon channel position of the instability is still an open issue. We em- we have to deal with . The calculations of the am- phasize thatin spite of the smootherOPT extrapolation 2,0 N plitudes of 2,0 can be performed on finite clusters, see the parameters x = 0.603,J1 = 6.16 meV still yield a N Refs. [6,12]. better agreement with experiment than the parameters By substituting 2,0(r) for (r) in Eq. (9b) we ob- x=0.635,J1=7.33 meV. N N tain (q) which excites the same type of 2-triplon 2,0 N momentum states σ,K(q),d that are used for the ef- | i fective Hamiltonian. The 2-triplon amplitudes A = σ,K,d σ,K,d 2,0(q)0 with K MBZ define a vector in the 300 h |N | i ∈ quantum numbers σ and d. Each component is calcu- q=(2.0,0.0), x=0.603, J1=6.160meV lated to 8th order in x. 250 q=(2.5,0.5), x=0.603, J1=6.160meV The 2-triplon energy and momentum resolved spec- q=(2.0,0.0), x=0.635, J1=7.325meV q=(2.5,0.5), x=0.635, J=7.325meV traldensityof isobtainedbyevaluatingthe 2-particle 1 N 200 Green function (6) via tridiagonalization [13,14] ) u (2)(ω,K;x)= σ,d|Aσ,K,d(x)|2 . S(a. 150 G P b2 ω a 1 (10) − 0− b2 100 ω a 2 1 − − ω −··· 50 For fixed K, the continued fraction coefficients a and b i i are obtained by repeated application of H +H (ma- 1 2 0 trixindandσ)ontheinitial2-particlemomentumstate 4.0 4.5 5.0 5.5 6.0 6.5 7.0 Energy ω in meV f = (q)0 (vector in d and σ). Prior to the eval- 0 2,0 | i N | i uation we extrapolate the matrix elements of H and 1 FIG.2. Two-triplon contribution to the spectral densities H andtheamplitudesof f byoptimizedperturbation 2 | 0i oftheobservableN(q)representingtheINSonSrCu2(BO3)2 theory (OPT) introduced in Ref. [6]. The values used fortwosetsofparameters. Forclarity,abroadeningof0.02J1 for the OPT parameterα are 0.20for the elements OPT − is used. The inaccuracy of the peak positions due to the of H , 0.80 for the elements of H , and 0.25 for the 1 2 − extrapolation is about 2.5%. amplitudes of [12]. N In Fig. 2, the spectral densities of are plotted for 2,0 N the two momenta K = (0,0) and (0,π), which translate Tounderstandthe2-triplonstatesthroughoutthe BZ, toq=(h,k)asindicated. Resultsareshownfortwosets we calculate the spectral density for 150 different mo- of parameters. The set x = 0.635,J = 7.33 meV was 1 menta. Color-codingtheintensitiesleadstoFig.3where proposed from the analysis of the magnetic susceptibil- we follow the experimentally traced path in dual space. ity χ(T) [15]. We proposed the set x = 0.603,J = 6.16 1 The black lines are the most relevant eigen-energies of meV previously [3] based on the analysis of excitations H +H extracted from the 84 84 matrix representing energies at T = 0. Clearly, the differences between the 1 2 × the full matrix of H plus a part of H at fixed q and two sets matter. Both the positions and the weights of 2 1 x. Enlarging this matrix from 84 84 to 112 112 does thecurvesdifferfromonesettotheother. Comparingto × × not lead to visible changes. The dashed lines mark the high resolution INS data [16] we come to the conclusion lower and upper bound of the 2-particle continuum de- that the parameters x=0.603,J =6.16 meV fit signifi- 1 rivedfromthe 1-triplondispersion[9]. The energyrange cantly better, both concerningthe positions andandthe depicted is chosen according to a recent high resolution weights of the peaks. So we favor this set of parameters. INS measurement of SrCu (BO ) [16]. The 1-triplon It is objected that χ(T) is not well described [4]. But 2 3 2 contributionwouldappearasasharp,flatandhighlyin- χ(T) is also strongly influenced by the presence of inter- tensive (red) band at about 3 meV from which no new layer coupling [3,15] which is not known. So a definite insight is gained. conclusiononthebasisofχ(T)aloneisverydifficult. On 3 Ourresultscompareexcellentlywith the experimental levelrepulsion. (ii) The quantitative agreementwith the data. The main conclusion is that the rather flat bands high resolutionINS [16]is very good. Not only the over- of the 2-triplon states can indeed be understood. Pre- all shape of the structure factor but also prominent de- viously, experiment [17] and theory [3,18,19] found evi- tailslikethe intensiveflatmodesatabout5meVandthe dence for significant correlated hopping of two triplons. modes just below the continuum at 5.75meV are repro- So it came asa surprise thathigh resolutionINS showed duced. This observation supports our choice for the pa- very flat features only. Previous results were limited in rametersx=0.603,J =6.16meV.(iii)Finally,wehave 1 resolution (in momentum and in energy [17]), analysed demonstrated that perturbative CUTs are capable and only two points of the BZ [3] or were restricted to low well-suitedtoquantitativelycalculatecomplexquantities values of x [18,19]. Fig. 3 shows that there are many like spectral densities also for two-dimensional models. bound states distributed over a fairly large energy range We are indebted to N. Aso, K. Kakuraiand coworkers of about 1.5meV. This range corresponds to the previ- for making the INS data availableto us priorto publica- ousexpectationofenhancedcorrelatedhopping. Butthe tion. Financial support by the DFG in SFB 608 and in smoothly connected eigen-energies do not display a sig- SP 1073 is gratefully acknowledged. nificant dependence on momentum. We interpret this finding as evidence for level repulsion. Due to the neg- ligible 1-triplon kinetic energy there is a relatively large number of individual states involved. Their energetic re- pulsion renders each individual band very flat. [1] R.W.SmithandD.A.Keszler,J.SolidStateChem.93, 430 (1991). [2] B. S. Shastry and B. Sutherland, Physica 108B, 1069 7.0 (1981). [3] C. Knetter, A. Bu¨hler, E. Mu¨ller-Hartmann, and G. S. Uhrig, Phys. Rev.Lett. 85, 3958 (2000). 6.5 [4] S.MiyaharaandK.Ueda,J.Phys.: Condens.Matter15, R327 (2003). 6.0 [5] C. Knetter,K. P.Schmidt, and G. S.Uhrig, J. Phys.A: ) V e Math. Gen. 36, 7889 (2003). m [6] C. Knetter, K. P. Schmidt, and G. S. Uhrig, Eur. Phys. gy ( 5.5 J. B 36, 525 (2004). er [7] S. Miyahara and K. Ueda, Phys. Rev. Lett. 82, 3701 n E 5.0 (1999). [8] Z. Weihong, C. J. Hamer, and J. Oitmaa, Phys. Rev. B 60, 6608 (1999). 4.5 [9] C. Knetter, E. Mu¨ller-Hartmann, and G. S. Uhrig, J. Phys.: Condens. Matter 12, 9069 (2000). 4.0 [10] K. P. Schmidt and G. S. Uhrig, Phys. Rev. Lett. 90, q = (2,0) (2.5,0) (2.5,0.5) (2,0) 227204 (2003). K = (0,0) ( π /2, π /2) (0, π ) (0,0) [11] C. Knetter and G. S. Uhrig, Eur. Phys. J. B 13, 209 (2000). FIG. 3. Color-coded spectral density of N in the en- [12] C. Knetter, PhD Thesis, Universit¨at zu K¨oln, 2003, ergy-momentum plane. Intensities as indicated by the scale http://kups.ub.uni-koeln.de/volltexte/2003/942 on theright hand side. Parameters are thesame as in Fig. 2 [13] V. S. Viswanath and G. Mu¨ller, The Recursion Method; ApplicationtoMany-BodyDynamics,Vol.m23ofLecture Notes in Physics (Springer-Verlag, Berlin, 1994). We also computed the energy and momentum inte- [14] D. G. Pettifor and D. L. Weaire, The Recursion Method grated weights. At x = 0.603, we find that about and its Applications, Vol. 58 of Springer Series in Solid 50%+25% of the full weight, known from a straightfor- State Sciences (Springer, Berlin, 1985). ward sum rule, is covered by the 1- and 2-triplon ex- [15] S. Miyahara and K. Ueda, J. Phys. Soc. Jpn. 69, citations, respectively. The remaining 25% must be at- 72(Suppl.B) (2000). tributedtohighertriplon-excitations. Thisfindingagrees [16] N.Aso,H.Kageyama,M.Nishi,andK.Kakurai,unpub- nicely with experiment, see e.g. the constantmomentum lished. scan in Fig. 2(a) of Ref. [17]. [17] H. Kageyama et al.,Phys.Rev. Lett.84, 5876 (2000). [18] Y. Fukumoto,J. Phys.Soc. Jpn. 69, 2755 (2000). In conclusion, we like to stress three main results. (i) [19] K.Totsuka,S.Miyahara, andK.Ueda,Phys.Rev.Lett. Thescenarioofstronglydispersingmodesinthe2-triplon 86, 520 (2001). sector cannot be held up. We showed that these modes are also rather flat and we argue that this stems from 4