II. TWO COUPLED SPIN- The best studied system is the simple spin- ladder, i.e. two parallel spin- H R = L X? (S R S R ) () L I X = H (S R S R S R S R

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1 Heisenberg S = M. Reigrotzki, H. Tsunetsugu ;y, and T.M. Rice Theoretische Physik, ETH-Honggerberg, 89 Zurich, Switzerland y Interdisziplinares Projektzentrum fur Supercomputing, ETH-Zentrum, 89 Zurich, The properties of antiferromagnetic Heisenberg S = The study of antiferromagnetic Heisenberg spin- currently being investigated are (VO) P O 7 sum along Heisenberg spin- = X H S l S m X l $ I. INTRODUCTION Strong Coupling Expansions for Antiferromagnetic ladders has recently become Ladders of great interest because they oer the possibility of realizing spin liquid states with short range RVB (resonance valence bond) character [{5]. Experimental systems [6,7] and the homologous series of compounds Sr n Cu n O n [8,9]. The main evidence to date comes from numerical Switzerland investigations using both Lanczos diagonalization of nite ladders [,,], and (uly 994) density-matrix renormalization group [5], and quantum transfer-matrix methods []. These studies have shown that the low energy properties of ladders are Abstract governed by xed point behavior determined by the limit of strong interchain and weak intrachain coupling. In this paper we present analytic expansions about this strong coupling limit for ladders with up to four legs (or chains). We compare these with ladders expansions with the numerical results and nd that they hold qualitatively but not always quantitatively down to the isotropic limit (equal inter- and intrachain,, and 4 chains are expanded in the ratio of the intra- and interchain coupling constants. A simple mapping procedure is introduced couplings). Further we introduce a mapping scheme to relate the 4-chain ladder to the simple (i.e. -chain) ladder and the -chain ladder to the single chain. to relate the 4 and -chain ladders which holds down to moderate values of the expansion parameters. A second order calculation of the spin gap to the lowest triplet excitation in the - and 4-chain ladders The Hamiltonian of such a ladder takes the form is found to be quite accurate even at the isotropic point where the couplings are equal. Similar expansions and mapping procedures are? S l S m : () presented for the -chain ladders which are in the same universality The intrachain coupling is and the interchain coupling across the rungs is?. class as single chains. The summation over the length of the chains which will eventually tend to PACS numbers: innity is denoted by $ and the summation over the number of chains running parallel is denoted by l. In the strong coupling limit?, the part of the Hamiltonian with the the chains is treated as a perturbation of the system of uncoupled rungs. For the simplest spin ladder (-chains) the strong coupling perturbation expansion up to third order in =? is performed in Section. Ladders with three chains coupled together will be discussed in Section and a mapping to the single with an eective coupling determined by the splitting chain Typeset using REVTEX between the ground state and the rst excited state of a spin cluster. Lastly

2 II. TWO COUPLED SPIN- The best studied system is the simple spin- ladder, i.e. two parallel spin- H R = L X? (S R S R ) () L I X = H (S R S R S R S R ): (4) R The intrachain (interchain) coupling is given by (respectively? ), while S i R I ji = H X R s : : : (t t 4 8? # (6) : ki L X p j e ; ikx l js : : : t l : : : si; (7) = L!(k) =?? 8 =?? k ( cos k) (8) cos? 4 cos k cos k cos k ): (? : (9)? 4? R the case of 4-chain ladders is treated in Section 4 using two dierent approaches. t t t t ) : : : s : (5) First the four chains are considered as a system of two simple ladders running This excitation leads to a correction of the ground state energy. Up to third order parallel, with a small coupling? between the two adjacent chains. Second, the four chains are treated in the same way as the simple ladder by starting in the limit perturbation in the strong coupling limit, the ground-state energy per rung is " E g? = L? 6 of non-interacting rungs, each with four spins. In the second case in analogy to the -chain system a mapping to the simple ladder is performed. The rst excited state of the unperturbed ladder is obtained by promoting one rung to a triplet state. The L-fold degeneracy of this state is lifted in rst order THE SIMPLE SPIN LADDER CHAINS: perturbation in the strong-coupling limit. To rst order in =? the eigenstates are given by Bloch states chains strongly coupled together. The Hamiltonian has the form where the l-th rung is excited to a triplet with S z =. The excitation energy!(k) H = H H I ; () for magnon excitations up to third order =? in the strong-coupling limit is where H and H I are given by H = H I = To second order this result is equal to that obtained by Barnes et. al. in [], except The energy has a minimum at for the additional second order term / cos k. denotes the spin operator on the R-th rung on chain i. For chains of length L k =. Therefore the spin gap =!() is periodic boundary conditions are introduced by dening S i L Si. The eigenstates of the rung Hamiltonian H R are given by a singlet state jsi with energy E s = 4? and three triplet states jt i with spin z-component = ; ; In Fig. the dispersion relation in second and third order is plotted for a ratio of =? = :5. The third order correction improves the agreement to numerical and energy E t = 4?. The eigenstates of H are direct products of rung states. data from []. The deviation near k = is due to magnon processes. Fig. shows the spin gap in second and third order compared to numerical data for a A. Spin Gap and Dispersion relation of magnon excitations 8 Heisenberg ladder []. At the isotropic point =? = results from Barnes et. al. [] for 8 and ladders are included. In the isotropic region, the third When applying perturbation theory in =? for the strong-coupling limit, the rst order correction to the ground state energy vanishes, since H I pairwise excites order correction unfortunately leads to worse agreement than the second order correction, which is surprisingly good in the innite-length limit for the simple two adjacent singlets of the unperturbed ground state ji = js : : : si to a linear combination of triplets with total spin S = : Heisenberg spin ladder. 4

3 LADDERS WITH THREE SPIN- CHAINS III. A system of three coupled spin- realized by mapping the system to the single Heisenberg spin- in Fig. a. The two spin- H H R=) and their splitting into spin-subspaces upon switching on the perturbation H I = H I? S R S R S R S R X j= S j R S j R: () IV. LADDERS WITH FOUR COUPLED SPIN- To study the ladders with 4 parallel coupled Heisenberg spin-? S R S R S R S4 R 4X in the chains and the splitting to the lowest triplet is equal to. The eective coupling in rst order perturbation theory in the strong coupling limit then is has a degenerate ground state in the chains strong-coupling limit which leads to an additional complication compared to the e : (4) study of the simple ladder (or any other system of an even number of chains coupled The results of the exact diagonalization of a cluster are shown in Fig. together). White et. al. [5] give an explanation of the fundamental dierence of 4. The separation of the second from the rst excited state is greater than the even and odd number of spin chains coupled together as due to the behavior of splitting e of the singlet and the triplet, reaching its minimum of = e at the topological spin defects. Since the rung states are already degenerate, perturbation isotropic point =? =. Thus the mapping should give reasonable results for This can be theory for degenerate systems must be used from the beginning. temperatures k B T e. At isotropy, from Fig. 5 we obtain a value e = :8. chain which has A more complete density matrix renormalization group (DMRG) study of isotropic been studied extensively using an eective coupling in rst order perturbation Heisenberg coupled chains by White et. al. [5] leads to qualitatively comparable in =?. Later we use the exact diagonalization of a spin cluster to determine results. They consider nite ladders with open boundary conditions so that there is the eective coupling. a nite excitation energy determined by the velocity of the des Cloizeaux-Pearson The Hamiltonian of the system is given in analogy to the one of the ladder by mode. In a single chain this is proportional to the coupling constant. The ratio H = H H I ; () of these velocities in the -chain and -chain systems gives a direct measure of renormalized coupling e :68. with CHAINS H = H R = () H I R = H I = strong coupling limit, two dierent approaches are taken. First the 4 chains will The energy levels of a rung system together with their degeneracy are depicted be treated as two simple ladders coupled together and expanded around the strong states are denoted by jd i (doublet), corresponding rung coupling limit. Secondly, we map the 4-chain system to a simple double-chain ladder with renormalized coupling constants. The Hamiltonian of this system is to S z =, the spin- state by q ~ E (quartet) with ~ corresponding to S z = ; ; ;. the sum of three terms H = H H H with The energy levels of an uncoupled pair of rungs (with Hamiltonian H = (5) H = are shown in Fig. b. S j R S j R (6) H = j= In rst order perturbation theory the ground state of a pair of rungs is H = E g =? 4 ()? S R S R: (7) 6 5

4 Periodic boundary conditions again are introduced by S i L The indices u and l denote simple ladder Hamiltonians () on the upper resp. lower! (k) =?? 4 4!? 8? 5 : (9) 5!? ()? E s = 4 (??) q 4???? i js E t = 4? q?? i jt E t E t the coupling? t 4? q?? = E 6 p 4 # () ;? The dispersion relation for the two branches is shown in Fig. 6 for =? =?. As in the double chain system, the interchain coupling (coupling between chains one and two and between three and four) is? and the intrachain coupling is. Fig. 7 shows the spin gap for the two branches. S i. The coupling between the two ladders is?. B. Four coupled spin- chains The Hamiltonian is again of the form (5){(7). The interaction Hamiltonian A. Two coupled spin- ladders H between ladders will not be treated as perturbation of the one-rung eigenstates The part H H of H is the sum of two simple ladder Hamiltonians: of the double ladder but instead exact eigenstates of the sum H H are the basisstates for a perturbative treatment of the intrachain coupling. Additionally the H H H u H l : (8) problem is mapped to the simple ladder by exact diagonalization of the 4 cluster. simple ladder.. Exact one-rung eigenstates To determine the ground state energy and the energy of the low-lying excitations, perturbation in and? in the strong coupling limit?? up to The 6 one-rung eigenstates of H H are denoted by second order is applied. The rst excited state is L-fold degenerate, the factor arising since promoting a singlet to a triplet on one rung can be done on either of the two coupled ladders. The L-fold degeneracy is lifted as in the simple ladder system by the perturbation H (transforming to Bloch states); the remaining -fold degeneracy is a spin degeneracy. The second perturbing Hamiltonian H will lift () E t = 4 (?? ) the -fold degeneracy into even and odd parity states, the parity-transformation P chain being dened by reversing the chain order (P chain : i! 5 i). ~E E q = 4 (?? ) q For the ground state the energy per rung to second order in and? is E g L =? 4? 4!?? There are two singlets js i, three triplets jt i, t E and t E with Sz components = ; ; and a quintet q ~E with S z components ~ = ; ; ; ;. With the corrections obtained for the rst excited state with odd ( ) and even parity (), the dispersion relation for the 4-chain system as two coupled ladders For? =?, the ground state energy per spin of the 4-chain system is " p g E? = 4L up to second order perturbation in the strong coupling limit is k ( cos k) cos? 4 4 up to the second order in =?. By the same reasoning as for the double ladder,!??? 4? leads to a splitting of the rst excitation from the ground state cos k into an even and odd parity state. For the odd parity state the dispersion relation 8 7

5 the low-lying excitations of four coupled Heisenberg spin- chains up to second for! (k) :659 :75 =???! (k) :66 :667 =??? k cos (:86 :5 cos k :469 cos k); () k cos (:86 :55 cos k :8 cos k) (4) where D (E ; E ; ) E ( q?)?? ; (6) = E p? > :7, where the condition D (E ; E ; ), which is = couplings, -chain(;? =? G? ; () ) = ( a)x ( a)x a G(x) a)x ( a)x ; () ( order perturbation theory in the strong coupling limit is E ; =? : (7) These three states are sucient to determine the coupling constants: = E ; E ; ; (8) q = D D ; (9)? and is the average energy separation between the ground state singlet and the two triplets. There is one other spin triplet consisting of two magnons, with the energy, E ;tt = (? ). Fig. shows the energy levels of the 4 cluster as a function of the intrachain for the even parity state. The dispersion relation for the two branches is shown coupling,. The ground state is a spin singlet and there are two triplets above it. The two triplets are split with a separation increasing with. This splitting in Fig. 8. For the odd parity branch, the spin gap minimum is always at k =. On the other hand, the minimum of the even parity branch jumps from k = to corresponds to the band width of propagating magnons along the chain direction. At around =? :64, another triplet goes down and becomes lower than E ;. k = at =? = :6. This explains the kink in the curve of the even parity spin This triplet is the two magnon state. Therefore, this crossing corresponds to the gap in Fig. 9. point where the bottom of the two-magnon continuum becomes lower than the top C. Mapping to an eective simple ladder of the one-magnon mode. However, these two triplet eigenstates, having dierent parities with respect to the mirror symmetry, P chain, do not mix to each other, and As in Section, let us map the 4-chain ladder into an eective simple ladder to this level crossing does not have signicant consequences. determine the spin gap. Throughout this section, we assume? =? again. To The energy spectrum Fig. has the same structure as the system, at this end, we calculate the eigenenergies of a 4 cluster, and determine the eective least for small 's. The corresponding eective couplings are determined by (8) coupling constants of the eective simple ladder, = (;? ),? =? (;?), and (9). The result is shown in Fig.. The mapping breaks down for large so as to reproduce the same low-energy spectrum. In this approximation, the spin gap of the 4-chain ladder is then given by that of the simple ladder as necessary for a real?, is no longer satised. 4-chain(;? ) = -chain( (;? );?(;? )): (5) We can now estimate the spin gap of the 4-chain ladder by using (5). Here we use a Pade approximation for the -chain determined by the strong coupling The energy spectrum of a cluster is easily calculated. The ground state is limit (9) and the weak-coupling asymptotic form, /? []: a spin singlet, and there are two magnons, bonding and antibonding combinations of the two states in which one rung is a spin singlet and the other is a triplet. The energies of these three lowest states are 9

6 by using () for (5) with the values of 's and?'s shown in Fig.. obtained magnon can be regarded as a bound state of two S = simple mapping procedure is found to work quite well down to the isotropic limit where a lim?! -chain=?. This Pade approximant has a correct asymptotic form in both limits,? =! and? =!. We use here a = determined by when compared to numerical results. The low energy behavior of the two systems will be similar and the spinons will now extend over the three chains. Lastly the -chain = :5 at? =, but this form does not agree so well with the numerical results [] at <? = <. success of the expansion around the strong coupling limit of the Heisenberg ladders encourages one to consider a corresponding expansion for the doped ladders Figure shows the result of the spin gap for the 4-chain ladder. The gap is described by a t- model which have shown interesting results in mean eld and numerical investigations [4,5]. The values determined by numerical diagonalization [5,] are included at? =. The result of the second-order perturbation () is also included. This mapping shows a correct tendency. V. CONCLUSIONS In this paper we presented the results of analytic expansions about the limit of strong interchain coupling. It is convenient to expand around this strong coupling limit since the qualitative behavior and therefore the universality class of the system remains unchanged. An analytic expansion allows one to examine the evolution of the system to the case of isotropic coupling. A magnon is an elementary S = excitation, but the spin density distribution which is localized on a single rung in the strong coupling limit evolves continuously into a distribution spread over several rungs at isotropy. With decreasing interchain coupling the magnon spectrum changes its dispersion relation from a simple cosine band as the size of the magnons increases and longer range hopping matrix elements enter. An S = spinons [] on individual chains, but since the interchain coupling changes the excitation spectrum completely this analogy is only qualitative at best. The 4-chain ladder also has a spin gap to S = magnon excitations and so belongs to the same universality class as the -chain ladder, and a mapping procedure between 4-chain and -chain ladders is possible. This works well down to moderate values of the intra- to interchain coupling but not down to the isotropic limit where they are equal. On the other hand a second order perturbation for the spin gap works surprisingly well down to the isotropic limit. The -chain ladder can be mapped onto the single chain and a

7 [4] M. Sigrist, T.M. Rice, and F.C. Zhang, Phys. Rev. B 49, 58. REFERENCES [5] H. Tsunetsugu, M. Troyer, and T.M. Rice, Phys. Rev. B 49, 678. [] E. Dagotto,. Riera, and D.. Scalapino, Phys. Rev. B 45, 5744 (99); [] T. Barnes, E. Dagotto,. Riera and E. S. Swanson, Phys. Rev. B 44 (99) 96. [] T. M. Rice, S. Gopalan, and M. Sigrist, Europhys. Lett., 445 (99). [4] S. Gopalan, T. M. Rice, and M. Sigrist, Phys. Rev. B 49, 89 (994). [5] S. R. White, R. M. Noack, and D.. Scalapino, \Resonating Valence Bond Theory of Coupled Heisenberg Chains", (unpublished). [6] D. C. ohnston et al., Phys. Rev. B 5, 9 (987). [7] R. S. Eccelston, T. Barnes,. Brody, and.w. ohnson, \Inelastic neutron scattering from the spin ladder compounds (VO) P O 7 ", (unpublished). [8] M. Takano, Z. Hiroi, M. Azuma, and Y. Takeda, AP Series 7, (99). [9] M. Azuma, Z. Hiroi, M. Takano, K. Ishida, and Y. Kitaoka, \Observation of a spin gap in SrCu O comprising spin / quasi-d two-leg ladders", (unpublished). [] D. Poilblanc, H. Tsunetsugu, T.M. Rice, \Spin Gaps in coupled t{ ladders", (unpublished). [] M. Troyer, H. Tsunetsugu, and D. Wurtz, \Thermodynamics and spin gap of the Heisenberg ladder calculated by the look-ahead Lanczos algorithm", (unpublished). [] S. P. Strong and A.. Millis, Phys. Rev. Lett. 69, 49 (99). [] In a single spin chain, elementary excitations are topological S = excitations usually called spinons now, and physical S = excitations are two spinon scattering states: L. D. Faddeev and L. A. Takhtajan, Phys. Lett. 85A, 75 (98). 4

8 FIG.. Dispersion relation of simple Heisenberg spin- 4 r ~ E q E r d jd r i? r 7 r 5 PP r 5 r PP r FIGURES (a) (b) nd order pert. rd order pert. exact diag.? [ω(k)- ] / - PP 5 D = D D D D D D ) = (D D ) (D D = D D D (D (D D ) = (D D ) D ) = (D D ) 5 - π/ π k - D D = D D - - ladder for =? = :5. The =? (b) These combine to give FIG.. (a) The rung energies for the -chain ladder. data of exact diagonalization are taken from Ref. []. the eigenenergies of a pair of rungs, and under the action of H I they split yielding the spectrum for a cluster..8 nd order pert. rd order pert. 8 cluster cluster Enenrgy / - S= S= - /.6.4 FIG.. Spin gap for a magnon excitation of simple Heisenberg spin- extrapolated value for is taken from Ref. [] The ladder. Enenrgy / - S= - S= FIG. 4. Diagonalization of cluster classied by total spin S. 5 6

9 FIG. 6. Dispersion relation for double Heisenberg spin- FIG. 7. Spin gap for a magnon of double Heisenberg spin- FIG. 8. Dispersion relation for four coupled Heisenberg spin- exact diag. st order pert. () eff / / ( ) = ladder. FIG. 5. Eective coupling obtained by mapping the -chain to the simple chain.. ω = =. / =.7 ω(k) / ω ω(k) / / =. π/ π k ω ω π/ π k chains. ladder. 8 7

10 FIG. 9. Spin gap for a magnon of four Heisenberg spin- = /. () ( ) (Effective coupling) / * chains. FIG.. Eective couplings when the 4-chain ladder is mapped to the simple ladder. Energy / () ( ) ground state magnon state magnon state / = mapping nd order pert. Poilblanc et al. White et al FIG.. Energies of 4 cluster. FIG.. Spin gap of the 4-chain ladder calculated by mapping to a simple ladder. 9