Exact ground state and elementary excitations of the spin tetrahedron chain

Transcription

1 Exact ground state and elementary exctatons of the spn tetrahedron chan Shu Chen, 1,2 Yupeng Wang, 2 W. Q. Nng, 1,3 Congjun Wu, 4 and H. Q. Ln 1 1 Department of Physcs and Insttute of Theoretcal Physcs, The Chnese Unversty of Hongkong, Hongkong, People s Republc of Chna 2 Bejng Natonal Laboratory for Condensed Matter Physcs, Insttute of Physcs, Chnese Academy of Scences, Bejng , People s Republc of Chna 3 Department of Physcs, Fudan Unversty, Shangha , People s Republc of Chna 4 Kavl Insttute for Theoretcal Physcs, Unversty of Calforna, Santa Barbara, Calforna 93106, USA Receved 14 February 2006; revsed manuscrpt receved 24 August 2006; publshed 21 November 2006 We study the antferromagnetc spn exchange models wth S=1/2 and S=1 on a one-dmensonal tetrahedron chan by both analytcal and numercal approaches. The system s shown to be effectvely mapped to a decoupled spn chan n the regme of strong rung couplng, and a spn sawtooth lattce n the regme of weak rung couplng wth spn 2S on the top row and spn S on the lower row. The ground state for the homogeneous tetrahedron chan s found to fall nto the regme of strong rung couplng. As a result, the elementary exctaton for the spn-1/2 system s gapless whereas the exctaton for the spn-1 system has a fnte spn gap. Wth the ad of the exact dagonalzaton method, we determne the phase dagram numercally and fnd the exstence of an addtonal phase n the ntermedate regme. Ths phase s doubly degenerate and s characterzed by an alternatng dstrbuton of rung snglet and rung spn 2S. We also show that the SU 3 exchange model on the same lattce has completely dfferent knd of ground state from that of ts SU 2 correspondence and calculate ts ground state and elementary exctaton analytcally. DOI: /PhysRevB PACS number s : Jm I. INTRODUCTION The study of quantum antferromagnetc spn models wth strong frustraton has attracted great attenton over the past decades. Early nvestgatons of the frustrated quantum magnets were partly motvated by the work of Anderson to search the resonatng-valence-bond RVB ground state n such systems. 1 In the frustrated quantum magnets, the magnetc orderng s generally suppressed by the frustraton. Some well-studed frustrated magnetc systems nclude, for example, the kagome lattce and pyrochlore lattce, n whch the nterplay between frustraton and quantum fluctuaton leads to rch varetes of phenomena. Recently, the frustrated magnets are beleved to be promnent canddates of realzng spn lqud states wth exotc ground state and deconfned fractonal exctatons. 2 4 Whle the mechansm of deconfnment n two-dmensonal 2D magnetc systems s less clear, the deconfnement of spnons n quas-one-dmensonal magnetc system, whch s closely related to the phenomenon of spn charge separaton, s well nvestgated. 5 8 Snce the 1980 s, a large number of low-dmensonal frustrated magnets have been syntheszed expermentally. 9 Generally speakng, strongly geometrcal frustraton n these systems allows the smple dmerzed state to be the ground state of the low-dmensonal frustrated spn system and opens a spn gap. So far, a varety of quas-one-dmensonal frustrated models have been studed theoretcally Addtonally, mportant progress has been made n trappng cold atoms under a hghly controllable way very recently, and thus t stmulates ntensve nvestgaton on how to smulate the magnetc systems usng cold atoms. A number of schemes have been proposed to mplement a varety of quantum spn models n optcal lattces. 13 In ths paper, we nvestgate both the ground- and exctedstate propertes of the spn models on a one-dmensonal 1D tetrahedron chan as shown n Fg. 1, wth ste spns S resdng n four corners of each tetrahedron. The basc unt,.e., a tetrahedron s composed of four spns wth equal antferromagnetc exchanges between each par of spns. The tetrahedron chan can be also vewed as a 1D pyrochlore strp n a 2D pyrochlore lattce, n whch only two of four corners of each tetrahedron are shared by a neghborng tetrahedron. Generally, a three-dmensonal 3D pyrochlore lattce s a network of corner-sharng tetrahedra and a 2D pyrochlore model, named also as a checkerboard-lattce model, s obtaned by a projecton of the 3D lattce on a plane. Dfferent from the 1D pyrochlore strp consdered n ths artcle, for both the 3D and 2D pyrochlore lattces, each corner of the tetrahedron s shared by a neghborng tetrahedron. As one of the most frustrated antferromagnets, the model of spn pyrochlore lattce has been nvestgated by a varety of technques ncludng the semclasscal large-s FIG. 1. Color onlne Schematc pctures of a the homogeneous spn tetrahedron chan, b the spn tetrahedron chan correspondng to model /2006/74 17 / The Amercan Physcal Socety

2 CHEN et al. lmt, large-n expanson of the SU N model, the contractor renormalzaton method based on the cluster expanson, and the bosonzaton method on the ansotropc lmt In spte of the ntensve research, even the ground state propertes of the 3D pyrochlore lattce are not well understood. For the 2D pyrochlore lattce, the numercal results based on exact dagonzatons have shown that the ground state has plaquette order. 18 However, for the 1D pyrochlore strp, we can determne ts ground state and elementary exctaton n an exact manner. Wth the ad of numercal dagonalzaton of the correspondng spn lattce systems wth small szes, we also nvestgate the quantum phase transtons of the ground state due to the change of exchange strengths along perpendcular rungs. The spn model on a 1D pyrochlore strp as shown n Fg. 1 s descrbed by the Hamltonan: H = J Ŝ Ŝ j, j where j denotes sum over all the nearest neghbors along the tetrahedron chan and Ŝ represents the spn operator resdng n ste. In ths work, we study both the spn-1/2 and spn-1 models on the pyrochlore strp. Our results show that the exctaton spectrum for the spn-1/2 system s gapless and the elementary exctaton of the spn-1 system has a fnte spn gap. Ths model can be extended to the cases where the strengths of bonds among the tetrahedra are not homogeneous. Here we only consder the nhomogeneous case as shown n Fg. 1 b, where we use J to represent the exchange strength along the vertcal rung of each tetrahedron. For convenence, we rewrte the Hamltonan of the spn tetrahedron chan correspondng to Fg. 1 b as follows: H A = J Ŝ,1 Ŝ,2 + Ŝ,3 + Ŝ,1 + Ŝ,2 + Ŝ,3 Ŝ +1,1 + J Ŝ,2 Ŝ,3. It s obvous that model 2 reduces to model 1 when J =J,.e., the homogeneous tetrahedron chan 1 s a specal case of model 2. Snce the model can be represented as a sum of local Hamltonan on each tetrahedron, a classcal ground state s obtaned whenever the total spn n the tetrahedron s zero for a homogeneous model. It s straghtforward that the classcal ground states have a contnuous local degeneracy. II. SPIN-1/ 2 LATTICE Frst, we consder the spn 1/2 case. As we have mentoned n the Introducton, there s a fundamental dfferent property between the 1D pyrochlore strp and ts hghdmensonal analoges: not all spns on a tetrahedron are equvalent. The spns on the vertcal rung of the tetrahedron are not coupled to the neghborng tetrahedron. We defne the total spn on each vertcal rung as Tˆ =Ŝ,2 +Ŝ,3. It s obvous that Tˆ 2 s conserved. Ths property enables us to smplfy the 1D model greatly. 1 2 For the tetrahedron chan model 2, t s convenent to reformulate the Hamltonan as follows: H A = J + J 2 Ŝ,1 Ŝ +1,1 + J Ŝ,1 + Ŝ +1,1 Tˆ Tˆ 2 NJ S S +1, 3 where Tˆ 2 =T T +1 wth T =0,...,2S. In the strong couplng lmt J, a par of spns on each rung would form a snglet spn dmer wth T =0. Ths mples that the spns n the horzontal drecton along the chan s effectvely decoupled wth the spns on the vertcal rungs. Therefore, the ground state of H A s a product of the ground state of spn chan and rung snglets. Explctly, t s represented as follows: GS = BA S,2,S,3, 4 where BA denotes the Bethe-ansatz ground state wave functons of the 1D Hesenberg chan H chan =J Ŝ,1 Ŝ +1,1 and S,2,S,3 =,2,3,2,3 / 2 s the dmer snglet across the th vertcal rung. The correspondng ground state energy s E g A = E g BA N 3 4 NJ, where E g BA N s the ground state energy of the N-ste Hesenberg spn-1/ 2 chan. From the Bethe-ansatz soluton of Hesenberg model, we know the exact ground state energy E g BA N /N= J at the nfnte length lmt. In fact, utlzng the Ralegh-Rtz varatonal prncple, 11,19 21 we can exactly prove the state gven by Eq. 4 s the ground state of Hamltonan 2 as long as J 2J. To see t clearly, we can rewrte the Hamltonan 2 wth J =2J as the sum of a Hesenberg chan and 2N projecton operators, whch reads where 3J H A = J Ŝ,1 Ŝ +1,1 + 2 P3/2 Ŝ,1,Ŝ,2,Ŝ, J 2 P3/2 Ŝ,2,Ŝ,3,Ŝ +1,1 1, 6 P 3/2 Ŝ,1,Ŝ,2,Ŝ,3 = 1 3 Ŝ,1 + Ŝ,2 + Ŝ, s a projecton operator whch projects a three-spn state composed of S,1, S,2, and S,3 nto a subspace wth total spn 3/2. Now t s clear that the state gven by Eq. 4 s the ground state of the global Hamltonan because t s smultaneously the ground state of each local sub- Hamltonan. 11,20,21 Wth the same reasonng, the state 4 s of course the ground state of Hamltonan Eq. 2 for J 2J. It s not hard to check that the state 4 s an egenstate of the global Hamltonan H A by utlzng the denttes Ŝ,2 +Ŝ,3 S,2,S,3 =0, however, such an egenstate s not

3 EXACT GROUND STATE AND ELEMENTARY EXCITATIONS E N,0 = 3 4 NJ + JE 1/2 N, 9 where E 1/2 N =E 1/2,1 N,0 s the energy of the N-ste Hesenberg spn-1/ 2 chan and the Bethe-ansatz ground state energy E g 1/2 N /N= as N. In the second case, the tetrahedron chan can be effectvely descrbed by a -chan model conssted of N spns wth S=1/2 on the stes of the lower row and N spns wth T=1 on the stes of top row as shown n Fg. 2 c and the egenenergy can be represented as follows: E N,N = 1 4 NJ + JE 1/2,1 N,N, 10 FIG. 2. Three types of ground state for the spn-1/2 and spn-1 pyrochlore strps. The open and close crcles denote the spns wth S=1 and S=1/ 2, respectvely, for the spn-1/ 2 pyrochlore strp. For spn-1 pyrochlore strp, the open crcles represent the spns wth S =2 and the close crcles represent the spns wth S=1. necessary the ground state for arbtrary J. We note that the condton J 2J for the exstence of the ground state gven by 4 s just a suffcent condton whch s a very strong restrcton. In fact, t can be released to a wder parameter c regme. We expect that there s a crtcal value J and the system evolves nto another quantum ground state when J s smaller than J c. For the homogeneous pont J =J whch we are partcularly nterested n, although the above proof s no longer applcable, we can stll argue that Eq. 4 remans the ground state, and prove ths result by usng the numerc exact dagonalzaton method. Wth the ad of numercal dagonalzaton, we may determne the phase boundary of the model 2 precsely. Snce the total spn n every rung s conserved, the vertcal rungs are ether n snglets or trplets for a spn-1/2 pyrochlore strp. Therefore, the relevant egenstate of the pyrochlore strp can be classfed by the values of the total spns on the vertcal rungs. 22,23 For convenence, we use E N,M to represent the egenenergy of the state wth N-M spn snglets and M trplets on the vertcal rungs. It follows that the egenenergy s gven by E N,M = 3 4 NJ + MJ + JE 1/2,1 N,M, where E 1/2,1 N,M represents the energy of the lattce composed of N spns wth S=1/2 on the stes of the lower row and M spns wth T=1 on the top row. For each class of state wth N-M spn snglets and M trplets on the vertcal rungs, there are altogether C N M dfferent confguratons. It s nstructve to frst consder the followng two knds of confguraton whch correspond to two opposte lmts of the vertcal exchange: all the states on the vertcal rungs are spn snglets and all the states on the vertcal rungs are spn trplets. In the frst case, the orgnal model can be mapped to a spn chan model as dsplayed n Fg. 2 a and the correspondng egenenergy s gven by 8 where E 1/2,1 N,N denotes the egenenergy of the correspondng -chan model. The mxed spn chan can be vewed as an alternatng spn-1/ 2-spn-1 chan wth an addtonal next-nearest neghbor nteracton between the spns wth S =1/2. It s well known that there s a ferrmagnetc long-range order n the ground state of quantum ferrmagnetc Hesenberg chan The addtonal nteracton between the spns wth S=1/2 s a knd of frustraton whch makes t harder to compensate the spn wth S=1, therefore the long-range order stll exsts. Although no exact analytcal results for the mxed spn chan are known, ts ground state energy may be determned by numercal exact dagonalzaton method. For a mxed spn chan wth sze of 8+8, we g get ts ground state energy gven by E 1/2,1 8,8 /16 = We note that the ground state of a spn chan or a spn sawtooth model wth S =1/2 s exactly known. 19,20,27 It s clear that the state wth all snglets or trplets on the rungs are the ground state of the tetrahedron chan n the lmt of J and J, respectvely. Flppng a rung snglet nto trplet costs an energy of J, therefore the effect of the antferromagnetc couplng J s to prevent the spns on the rung formng trplet. On the other hand, a trplet n the rung acts effectvely as a spn wth S=1 whch lowers the total energy by nteractng wth ts neghborng spns wth S =1/2 on the lower rung. The competton between the two processes gves rse to the complexty of the phase dagram for the tetrahedron chan. An nterestng queston arsng here s whether some ntermedate phases exst between the phases wth fully pared snglets and trplets on the rungs. To determne the phase boundary of 2 numercally, n prncple, we need consder all the dfferent confguratons of T on the rungs. Among a gven class, we fnd that the confguraton wth the spns on the top row repellng each other has lower energy. For example, as shown n Fg. 2 b, the confguraton wth alternatng spn 1 and spn 0 on the top row has the lowest energy among the C N/2 N confguratons. After consderng all the rung confguratons, we get a phase dagram as shown n the Fg. 3. As expected, there s an ntermedate phase whch s effectvely descrbed by the ground state of ts equvalent model as shown n Fg. 2 b. Correspondng to Fg. 2 b, there s another equvalent confguraton whch s obtaned by totally shftng the spns on the top

4 CHEN et al. FIG. 3. The phase dagram for spn-1/ 2 tetrahedron chan wth varable vertcal exchange J n the parameter space of c=j /J. For c c 1, the system s n a decoupled phase whose ground state s a product of rung snglets and the crtcal spn lqud phase on the horzontal spn-1/2 chan; For c 2 c c 1, the ground state s a double degenerate mxed state wth the alternatng spn snglet and spn trplet on the rungs; For c c 2, the ground state s a ferrmagnetc state wth spn trplet on the rungs. FIG. 4. Color onlne The lowest energes of E N,M /J as a functon of c=j /J for N=4,6,8. The sold, dashed, and dotted lnes correspondng to the case wth M =N, M =N/2, and M =0 or c, b, a as dsplayed n Fg. 2. row a lattce space. In ths phase, the trplet and snglet on the rungs dstrbute n an alternatng way and the ground state s doubly degenerate. Next we wll gve a descrpton on how to determne the phase dagram as shown n Fg. 3. Snce the orgnal pyrochlore strp can be classfed by the values of the total spns on the vertcal rungs or equvalently by M, labellng the numbers of the rung trplets, the ground-state energy of our tetrahedral spn chan, for a gven value of J and J, wll be gven by the lowest value of Eq. 8 wth M =0,1,...,N. In Fg. 4, we show as a functon of c=j /J all the lowest egenenerges for the lattce szes wth 4, 6, and 8 tetrahedra. The energes correspondng to the pont of c=0 ntercept ponts wth the vertcal axs are the lowest energes of E 1/2,1 N,M, from above to below, wth M =0,1,...,N. The slope values of the straght lnes depend upon the number M. Our numercal analyss shows that the ground state n the whole parameter space c s determned by three knds of confguratons wth M =0,N/2,N, correspondng to the confguratons of a, b, and c dsplayed n Fg. 2 and ther crossng ponts determne the phase transton ponts. For smplcty, we omt all the excted energy levels for the case of N=8 n the Fg. 4. From our numercal results, we obtan the up crtcal value c 1 = and the down crtcal value c 2 = for the orgnal pyrochlore strp wth a sze of 24 stes 8 tetrahedra or N=8. Smlarly, we get c 1 = and c 2 = for N=4, c 1 = and c 2 = for N=6. Here c 1 corresponds to the crossng ponts of lnes a and b, whereas c 2 corresponds to the crossng ponts of lnes b and c. Snce we need much smaller memory sze to dagonalze systems of a and b than system of c, therefore we can calculate even larger system to determne c 1. For example, we get c 1 = for N=10 and c 1 = for N=12. In Fg. 5, we analyss the fnte sze scalng of c 1 and c 2. The lnear ft of c 1 and c 2 gves c 1 =0.9767± and c 2 = ± for N. We can also determne the phase boundares c 1 and c 2 n an alternatve way. By extrapolatng the ground state energy of a and b to nfnte sze, we then determne c 1 and c 2 by the crossng of the energy levels of a, b and b, c, respectvely. In Fg. 6, we dsplay the fnte sze scalng of the ground state energes of a, b, and c. The ground state energes per ste, obtaned by lnear ft to the nfnte lmt, correspondng to the confguratons of a, b, and c are ± , ± and ± , respectvely. By ths way and usng Eq. 8, wegetc 1 = and c 2 = n the lmt of nfnte sze. If we use the ground state energy obtaned by Bethe ansatz for a, wegetc 1 = Before endng the dscusson of the spn-1/ 2 pyrochlore strp, we would lke to remark on a generalzed spn pyrochlore strp where the horzontal exchange s varable. Wth the same reasonng as that of the model 2, we can easly get the suffcent condton for the exstence of the fully dmerzed state on all the vertcal rungs, whch reads J 2J and s rrelevant wth the horzontal exchange. When the horzontal exchanges are zero, the model has totally dfferent ground sates and t falls nto the class of 1D damond mode, 28,29 whch has also fully dmerzed state on the vertcal rungs as the ground state but the ground state s hghly degenerate wth a degeneracy of 2 N because the unpared N spns are completely free

5 EXACT GROUND STATE AND ELEMENTARY EXCITATIONS FIG. 7. The phase dagram for spn-1 tetrahedron chan. For c c 1, the system s n a decoupled phase whose ground state s a product of rung snglets and Haldane phase on the horzontal chan; For c 2 c c 1, the ground state s a double degenerate mxed state wth the alternatng spn snglet and spn quntet on the rungs; For c c 2, the ground state s a ferrmagnetc state wth spn quntet on the rungs. FIG. 5. Color onlne The phase boundares c 1 and c 2 versus the szes of the system. III. SPIN-1 LATTICE WITH SU(2) SYMMETRY The above method can be drectly extended to deal wth the spn-1 case. For the spn tetrahedron chan 2 wth S =1, we can prove that the ground state of 2 s a product of fully dmerzed snglets on the rungs and the ground state of horzontal spn-1 chan as long as J 4J. Explctly, the ground state can be represented as follows: GS = Haldane S,2,S,3 11 where Haldane denotes the ground state of the 1D spn-1 chan and S,2,S,3 = m= 1 1 m+1 m,2 m,3 s a spn snglet across the th vertcal rung. The proof s rather smlar to ts spn-1/ 2 correspondence and s straghtforward when we rewrte the Hamltonan 2 n the followng form: H A = h Ŝ,1,Ŝ,2,Ŝ,3 + h Ŝ,2,Ŝ,3,Ŝ +1,1 + J Ŝ,1 Ŝ +1,1 12 wth h Ŝ,1,Ŝ,2,Ŝ,3 =JŜ,1 Ŝ,2 +Ŝ,3 + J 2 Ŝ,2 Ŝ,3 and h Ŝ,2,Ŝ,3,Ŝ +1,1 =J Ŝ,2 +Ŝ,3 Ŝ +1,1 + J 2 Ŝ,2 Ŝ,3. Now t s easy to fnd that the ground state of h s a product of pared FIG. 6. Color onlne The ground state energes of systems a, b, and c versus the szes of the system. snglet on the rung and a free spn on the unpared ste as long as J 4J. By the varatonal prncple, we conclude that the state 11 s the ground state of the spn-1 model gven by Eq. 2 for J 4J. Certanly, the suffcent condton for the exstence of a fully dmerzed ground state on the rung can be released to a lower bound. In prncple, we can determne t numercally followng a smlar scheme of the spn-1/2 case. Snce the total spn T on each vertcal rung can be 0, 1 or 2, the spn-1 tetrahedron chan can be mapped to a mxed sawtoothlke model accordng to ts confguratons of the vertcal rungs. Several relevant confguratons are: Spn snglets on all the vertcal rungs wth the egenenergy gven by E T=0 = 2NJ + JE 1 N, 13 where E 1 N denotes the egenenergy of a spn-1 chan of N stes. From the known numercal results, 30 we get ground state energy per ste E 1 g N /N= n the large N lmt. Spn trplets on all the vertcal rungs wth the egenenergy gven by E T=1 = NJ + JE 1,1 N,N. 14 Here E 1,1 N,N represents the egenenergy of a spn-1 chan of 2N stes Spn quntet T=2 on all the vertcal rungs wth the egenenergy gven by E T=2 = NJ + JE 1,2 N,N. 15 Here E 1,2 N,N represents the egenenergy of a mxed spn-1 and spn-2 chan wth spn 1 on the ste of the lower row and spn 2 on the ste of top row. Just smlar to the spn-1/2 pyrochlore strp, we need to consder all the dfferent confguratons of the rungs. Numercally, we fnd the phase dagram s rather smlar to that of the spn-1/2 case. As dsplayed n Fg. 7, there s also an ntermedate phase between the fully pared snglet state wth T =0 and state wth all T = 2. The ntermedate state s twofold degenerate wth the confguraton of alternatng snglet and quntet on rungs. Such a state can be schematzed n terms of Fg. 2 b wth the open crcle denotng the state wth T =2. From our numercal results, we obtan c 1 = and c 2 = for the orgnal tetrahedron chan wth a sze of 18 stes N=6 as well as c 1 = and c 2 = for N=4. Wth smlar reasonng as the spn-1/2 case, we can get c 1 = for an even larger system wth N=8. Lnear fttng of datum of c 1 versus 1/N 2 and extrapolatng t to the lmt of the nfnte

6 CHEN et al. sze, we get c 1 = ± for N. As a natural generalzaton, t s straghtforward to extend the spn pyrochlore model wth SU 2 symmetry to the case wth arbtrary spn S. For the spn-s model, a suffcent condton for the exstence of rung-dmerzed ground state s J 4 3/ s+1 for a half-nteger spn and J 4 for nteger spn. In the dmerzed phase, the horzontal chan s decoupled wth the spns on the rungs and therefore the elementary exctaton of the tetrahedron chan s gapless for half-nteger-spn model or opens a gap for the nteger-spn model. IV. SPIN-1 LATTICE WITH SU(3) SYMMETRY For the spn-1 system, the most general Hamltonan has a bquadratc exchange term besdes the blnear term and t exhbts much rcher quantum phase structures than the blnear model. 32 When the bquadratc exchange has the same strength of the blnear exchange, the Hamltonan H =J j h, j wth h Ŝ,Ŝ j = Ŝ Ŝ j + Ŝ Ŝ j 2 16 owns the SU 3 symmetry. For spn-1 systems of transton metal compounds where two electrons are coupled ferromagnetcally by Hund s rule, the bquadratc exchange term orgnates from a fourth order perturbaton process. Its magntude s thus small compared to the blnear terms, and thus the SU 3 symmetry s not applcable. However, n the cold atomc physcs, most atoms have hgh hyperfne multplets, thus t s possble to acheve hgh symmetres. For example, the 6 L atom s wth nuclear spn 1 and electron spn 1/2. In a weak magnetc feld, electron spn s polarzed, whle nuclear spn remans free. Recent studes ndcated that the three nuclear spn components can be descrbed by an approxmate SU 3 symmetry. 33 It s well known that the SU 3 exchange model on a chan has qute dfferent propertes from that of the SU 2 blnear model. Therefore we may expect that the SU 3 tetrahedron chan H = J h Ŝ,2,Ŝ,3 + J 3 =1 h Ŝ,,Ŝ +1,1 + J h Ŝ,1,Ŝ,2 + h Ŝ,1,Ŝ,3 17 also dsplays dfferent phase structure from that of ts SU 2 correspondence 2. The model wth J =J was ntally proposed by three of us wth Zhang n Ref. 34 as an example of the SU N generalzaton of the Majumdar-Gosh model, however, no analytcal results have been gven there. Observng that the Hamltonan can be wrtten as a sum of the Casmr of the total spn n each tetrahedron and the representatons wth the smallest Casmr made out of four stes n the fundamental representatons s three dmensonal, we concluded that the state of trmer products s the GS of the SU 3 tetrahedron chan. The ground state s twofold degenerate. Explctly, the ground state of the SU 3 spn tetrahedron chan can be represented as follows: or where GS 1 = T S,1,S,2,S,3 GS 2 = T S,2,S,3,S +1,1, T S,S j,s k = 1 6, j, k represents a trmer state whch s a snglet composed of three spns on ste, j, and k. Here denotes the spn on ste wth the value takng 1, 0, or 1 and s an antsymmetrc tensor. In the followng, we shall calculate the ground state energy and elementary exctaton of Eq. 17 analytcally. For convenence, we make a shft of constant to the Hamltonan 17 by replacng h Ŝ,Ŝ j wth Pˆ,j=h Ŝ,Ŝ j 1. We note that modfcaton of J does not lft the degeneracy of the left- and rght-trmer states. For J J, the state of trmer products s of course the ground state of the SU 3 tetrahedron chan and the correspondng ground state energy s E g = 2NJ NJ. Breakng a snglet of trmer wll cost a fnte energy, thus the elementary exctaton of the SU 3 tetrahedron chan has an energy gap. For a three-ste cluster, the trmer snglet s represented by a Young tableaux 1 3 and the frst excted state above the snglet are represented by the Young tableaux When a trmer snglet s broken, t decomposed nto a monomer and a pared dmer. For a system wth degenerate ground state, the monomer and dmer can propagate freely n the background of trmerzed ground state and lower the energy further. In prncple, two type of exctatons are avalable n a pyrochlore chan, ether a magnonlke exctaton produced by flppng a trmer state nto ts excted state or a par of deconfned objects composed of a dmer plus a monomer. For our system wth doubly degenerate ground state, the spnonlke exctatons have lower energy. The deconfned exctatons behave lke doman-wall soltons whch connects two spontaneously trmerzed ground states. Explctly, we represent an excted state wth a dmer at ste 2m 1 and a monomer at ste 2n s represented as m,n = T S m 1,1,S m 1,2,S m 1,3 m S m,1, T S m,2,s m,3,s m+1,1 T S n 1,2,S n 1,3,S n,1, d S n,2,s n,3 T S n+1,1,s n+1,2,s n+1,3..., where d S,S j = 1 2 j j wth represents a dmer. The correspondng momentum-space wave functon s k m,k d = 1 m n M e mk m +nk d m,n. The exctaton spectrum can be calculated drectly by usng the above varatonal wave functon. Because there exsts no ntrnsc mechancs responsble for bndng the dmer and

7 EXACT GROUND STATE AND ELEMENTARY EXCITATIONS monomer together to form a bound state n a spontaneously trmerzed system, t s reasonable to assume that the dmer and monomer are well separated and they could be treated separately. Smlar schemes have been used to evaluate the exctaton spectrum n the spn-sawtooth system. 19,20 Under such an approxmaton, the exctaton spectrum can be represented as a sum of monomer part and dmer part,.e., k m,k d k m + k d. The state n s not orthogonal wth the nner product gven by n n = 3 1 n n, thus k d has a nontrval norm k d k d =4/ 5 3 cos k d. Wth a smlar scheme as solvng the spectrum of exctatons of the spn-1/2 model, 11,19,20 we get the spectrum for the dmer k d = 5 4 d 3 cos k d d 4 20 wth d =2J. In the large J lmt, a monomer can only hop around n the left and rght phases along the horzontal drecton wthout breakng addtonal trmer snglet, therefore ts exctaton spectrum forms a flat band,.e., k m =0. When J s close to J, the monomer actually can move around several corners,.e., the monomer can be n the ste of m,2 and m,3 ether, therefore the wave functon for a monomer has resonatng structure at the mth trangle n the pyrochlore. The process of hoppng from ste m,1 to m,2 or m,3 accompanes wth an energy cost of J -J, and thus the exctaton spectrum s stll a flat band for the homogeneous pyrochlore wth J =J. For the nhomogeneous case, we take the monomer exctaton as a three-ste cluster consstng of three sngle monomers at stes m,1, m,2, and m,3,.e., m = 1 m S m,1 + m S m,2 3 + m S m,3. Smlarly, after consderable algebra, we get the spectrum for the monomer wth m =J -J. k m = 2 3 m 6 cos k m m 15 V. CONCLUSIONS 21 We have proposed and studed a class of frustrated lattce whch can be vewed as a 1D strp of the pyrochlore lattce or a tetrahedron chan. For the general Hesenberg exchange model, we gve an exact proof for the exstence of the ground state consstng of the rung-dmerzed state and the ground state of the decoupled chan. The phase dagrams of the spn-1/2 and spn-1 tetrahedron chan are gven and the phase boundares are precsely determned for the small-sze systems. For both the spn-1/ 2 and spn-1 systems, there exst three phases, say, the fully dmerzed snglets on the rungs plus a decoupled chan, a mxed phase wth alternatng spn snglet and state wth total rung spn 2S on the rungs, and a ferrmagnetc phase wth long-range order, as the strength of vertcal exchanges vares from nfnty to mnus nfnty. We also studed the SU 3 spn-exchange model on the 1D tetrahedron chan, for whch the ground sate s a double degenerate trmerzed state and the elementary exctatons are fractonalzed topologcal exctatons. Our results ndcate that the propertes of the ground state for the pyrochlore systems wth half-nteger and nteger spns or the systems wth the same spns but dfferent nternal symmetres SU 2 and SU 3 for spn-1 systems are qute dfferent. S. C. would lke to acknowledge hosptalty of the physcal department of the Chnese Unversty of Hong Kong durng hs vst and he also thanks the Chnese Academy of Scences for fnancal support. Ths work s supported n part by NSF of Chna under Grant No and Grant No , and RGC C. W. s supported by the NSF under the Grant No. Phy P. W. Anderson, Mater. Res. Bull. 8, ; P. Fazekas and P. W. Anderson, Phlos. Mag. 30, R. Moessner, K. S. Raman, and S. L. Sondh, cond-mat/ unpublshed. 3 F. Alet, A. M. Walczak, and M. P. A. Fsher, cond-mat/ unpublshed. 4 T. Senthl, A. Vshwanath, L. Balents, S. Sachdev, and M. P. A. Fsher, Scence 303, L. D. Faddeev and L. A. Takhtajan, Phys. Lett. 85A, I. Affleck, cond-mat/ unpublshed. 7 A. A. Nersesyan and A. M. Tsvelk, Phys. Rev. B 67, A. O. Gogoln, A. A. Nersesyan, and A. M. Tsvelk, Bosonzaton n Strongly Correlated Systems Cambrdge Unversty Press, Cambrdge, See, for example, Proceedngs of the Conference on Hghly Frustrated Magnetsm, August 2003, Grenoble, France specal ssue, J. Phys.: Condens. Matter 16, C. K. Majumdar and D. K. Ghosh, J. Math. Phys. 10, B. S. Shastry and B. Sutherland, Phys. Rev. Lett. 47, See, for example, the revew artcle of P. Lechemnant n, Frustrated spn systems, edted by H. T. Dep World Scentfc, Sngapore, 2003, and references theren. 13 L.-M. Duan, E. Demler, and M. D. Lukn, Phys. Rev. Lett. 91, ; J. J. Garca-Rpoll, M. A. Martn-Delgado, and

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