elementary excitations of heisenberg ferrimagnetic spin chains

PHYSICAL REVIEW B 1 JUNE 1998-IVOLUME 57, NUMBER 21

Elementary excitations of Heisenberg ferrimagnetic spin chains

Shoji YamamotoDepartment of Physics, Faculty of Science, Okayama University, Tsushima, Okayama 700, Japan

S. Brehmer and H.-J. MikeskaInstitut fur Theoretische Physik, Universita¨t Hannover, 30167 Hannover, Germany

~Received 30 October 1997!

We numerically investigate elementary excitations of the Heisenberg alternating-spin chains with two kindsof spins, 1 and 1/2, antiferromagnetically coupled to each other. Employing a recently developed efficientMonte Carlo technique as well as an exact-diagonalization method, we verify the spin-wave argument that themodel exhibits two distinct excitations from the ground state which are gapless and gapped. The gapless branchshows a quadratic dispersion in the small-momentum region, which is of the ferromagnetic type. With theintention of elucidating the physical mechanism of both excitations, we make a perturbation approach from thedecoupled-dimer limit. The gapless branch is directly related to spin 1’s, while the gapped branch originatesfrom cooperation of the two kinds of spins.@S0163-1829~98!00921-7#

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I. INTRODUCTION

Extensive efforts have so far been devoted to verifyHaldane’s conjecture1 that the one-dimensional spin-SHeisenberg antiferromagnet exhibits qualitatively differeproperties according to whetherS is integer or half odd in-teger. The nontrivial energy gap immediately aboveground state was precisely estimated using a number ofmerical tools not only in the spin-1 case2–4 but also in thespin-2 case,5,6 while the valence-bond-solid model7 intro-duced by Afflecket al.significantly contributed to the understanding of the physical mechanism of the so-called Haldmassive phase. On the other hand, developing theO(3)nonlinear-s-model quantum field theory,1 Affleck8 pointedout that even integer-spin chains should be critical if a ctain interaction is added to the pure Heisenberg HamiltonActually various numerical methods9–15 revealed that thespin quantum number is no longer the criterion for the crcal behavior in a wider Hamiltonian space. Recently seveauthors16,17 even suggested the appearance of the Haldagap phases in half-odd-integer-spin chains with a magnfield applied. Thus the low-temperature properties of odimensional quantum antiferromagnets with one kind of shave more and more been elucidated.

In such circumstances, there has appeared brand-attempts18–30to explore the quantum behavior of mixed-spchains with two kinds of spins. These studies are furtclassified according to their main interests. Seveauthors18–21have been devoting their efforts to finding quatum integrable Hamiltonians and clarifying their critical bhavior. Although the models considered are generally coplicated, the generic description of a certain familyHamiltonians is interesting in itself and even allows usguess the essential consequences of mixed-spin chaindistinct attention is directed to mixed-spin chains with tsimplest interaction between the two kinds of spinAlternating-spin Heisenberg antiferromagnets with a singground state26,29 again present us the nontrivial ga

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problem.1,8 Recently Fukui and Kawakami26 have made anonlinear-s-model approach for a few models of this kinand have discussed a generic criterion for the critical mixspin chains. Their finding may stimulate many theoreticiato numerically investigate a variety of mixed-spin Hamiltnians and even lead to synthesis of novel mixed-spin-chmaterials. On the other hand, considering that all the mixspin-chain compounds synthesized so far exhibit a finground-state magnetization,31 we take a great interest inalternating-spin Heisenberg antiferromagnets with ferrimnetic ground states. This is the subject we discuss inpresent article.

Let us introduce a Hamiltonian of alternatively alignetwo kinds of spinsS and s which are antiferromagneticallycoupled to each other:

H5J(j 51

N

~Sj•sj1dsj•Sj 11!, ~1.1!

whereSj25S(S11), sj

25s(s11), andN is the number ofunit cells. The bond alternationd has been introduced fordiscussion presented afterwards. We assumeS.s in the fol-lowing without losing generality. Because of the noncopensating sublattice magnetizations, this system exhibitsferrimagnetism instead of the antiferromagnetism. Applyithe Lieb-Mattis theorem32 to the Hamiltonian~1.1!, we im-mediately find (S2s) N-fold degenerate ground states. ThGoldstone theorem33 further allows us to expect a gaplesexcitation from the ferrimagnetic ground state. Thereforehere take little interest in the simple problem whethersystem is gapped or gapless. Alcaraz and Malvezzi22 inves-tigated the two cases of (S,s)5(1,1/2) and (S,s)5(3/2,1/2) and actually showed that in both cases the chis described in terms of thec51 Gaussian conformal fieldtheory under the existence of exchange anisotropy. Sugging that this should be the generic scenario for arbitrary frimagnetic Heisenberg chains, they further predicted thepearance of quadratic dispersion relations at the Heisen

13 610 © 1998 The American Physical Society

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57 13 611ELEMENTARY EXCITATIONS OF HEISENBERG . . .

points, which means that the model possesses a ferromnetic character. On the other hand, applying the spin-wtheory to the model, two groups24,25 have recently predictedthat there exists a gapped branch of elementary excitationwell as a gapless ferromagnetic branch. Their predictionquite interesting because it implies the coexistence offerromagnetism and the antiferromagnetism in the ferrimnets. This is the motivation for the present study. Employa quantum Monte Carlo technique and an exadiagonalization method, we here calculate energy eigenues of the elementary excitations with the total magnettion ( j 51

N (Sjz1sj

z)[M5(S2s)N71, which correspond tothe ferromagnetic and the antiferromagnetic branches,spectively. The numerical results are compared not only wthe spin-wave calculation but also with a perturbationproach from the decoupled-dimer limit (d50).

II. THE SPIN-WAVE APPROACH

First, we briefly review the spin-wave-theory result,24,25

which allows us to have a qualitative view of the low-enerstructure. We start from a Ne´el state withM5(S2s)N,namely, we define the bosonic operators for the spin detion in each sublattice as

Sj15A2Saj , Sj

25A2Saj† , Sj

z5S2aj†aj ,

~2.1!

sj15A2sbj

† , sj25A2sbj , sj

z52s1bj†bj .

In order to obtain the dispersion relations of the spin-waexcitations, we handle the boson Hamiltonian up to quadrorder. We define the momentum representation ofbosonic operators as

ak†5

1

AN(j 51

N

e22ia jkaj† ,

~2.2!

bk†5

1

AN(j 51

N

e2ia jkbj† ,

where k5p l /Na ( l 52N/211,2N/212, ,N/2) with abeing the distance between two neighboring spins. We nthat here the unit cell is of length 2a. Carrying out a Bogo-liubov transformation

ak5e2 ialk/2 coshuk ak1eialk/2 sinhuk bk† ,

~2.3!

bk5eialk/2 sinhuk ak†1e2 ialk/2 coshuk bk ,

with

~11d!tan@~k2lk!a#2~12d!tan~ak!50, ~2.4!

tan~2uk!52ASs

~11d!~S1s!

3A~11d!2 cos2~ak!1~12d!2 sin2~ak!,

~2.5!

we reach the diagonal Hamiltonian,

g-e

asise-

gt-l--

e-h-

a-

eice

te

H5E01(k

~vk2ak

†ak1vk1bk

†bk!, ~2.6!

where

E052J~11d!SsNJ

1J

2 (k

@A~11d!2~S2s!2116dSssin2~ak!

2~11d!~S1s!#, ~2.7!

vk75

J

2@A~11d!2~S2s!2116dSssin2~ak!

7~11d!~S2s!#. ~2.8!

Thus the spin-wave approach suggests the coexistence oferromagnetism and the antiferromagnetism in the pressystem. In the ferromagnetic branch (vk

2), the spin wavereduces the total magnetization and exhibits a quadraticpersion relation in the small-momentum region:

vk→02 5

JdSs~2ak!2

~11d!~S2s!, ~2.9!

while in the antiferromagnetic branch (vk1), the spin wave

enhances the total magnetization and have the gappedtation spectrum:

vk→01 5J~11d!~S2s!1

JdSs~2ak!2

~11d!~S2s!. ~2.10!

As long asS ands are different from each other, the modkeeps the two aspects of low-lying excitations. We furthnote that the qualitative character of the model remainschanged over the whole region ofd. It is the purpose in thisarticle to confirm this scenario employing efficient numericmethods.

III. A PERTURBATION APPROACH

Second, prior to the numerical approach, let us carry operturbation calculation with the intention of elucidating tnature of the elementary excitations. In the decoupled-dimlimit, we can easily find the low-lying eigenstates. Figur1~a!, 1~b!, and 1~c! represent, respectively, the ground stauC&, the ferromagnetic excitation at an arbitrary unit cellj ,uC j↓&, and the antiferromagnetic excitation at an arbitra

unit cell j , uC j1&, of the Hamiltonian ~1.1! with (S,s)

5(1,1/2) atd50. When we turn on the exchange interactibetween the dimers, the localized excitations can hopneighboring unit cells with an amplitude proportional tod.We take account of this effect using the degenerate perbation. We introduce a representation of the matrix-prodtype as

uC&5g1↑s

^ h1s

^¯^ gN↑s

^ hNs , ~3.1!

uC j↓&5g1

↑s^ h1

s^¯^ gj

↓s^ hj

s^¯^ gN

↑s^ hN

s , ~3.2!

uC j1&5g1

↑s^ h1

s^¯^ gj

↑1^ hj

1^¯^ gN

↑s^ hN

s , ~3.3!

where

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13 612 57SHOJI YAMAMOTO, S. BREHMER, AND H.-J. MIKESKA

gj↑s5@ u0& j &u1& j ], gj

↓s5@&u2& j u0& j ],

gj↑15@2&u1& j 0], ~3.4!

hjs5F2u↑& j

u↓& jG , hj

↑15F2u↑& j

0 G , ~3.5!

with u6& j , u0& j being theSjz eigenstates andu↑& j , u↓& j thesj

z

eigenstates. Now the dispersion relations of the eigenstwith M5N/271, vk

7 , are calculated as

vk25

^Ck↓uHuCk

↓&

^Ck↓uCk

↓&2EG5

4

9Jd@12cos~2ak!#1O~d2!,

~3.6!

vk15

^Ck1uHuCk

1&

^Ck1uCk

1&2EG5

3

2J1

1

18Jd@7212 cos~2ak!#

1O~d2!, ~3.7!

where uCk↓&5N21/2( j 51

N e22ia jkuC j↓& and uCk

1&5N21/2( j 51

N e22ia jkuC j1& are the Fourier transforms ofuC j

↓&and uC j

1&, respectively, and EG5^CuHuC&/^CuC&52J(11d/9)N is the ground-state energy within the first oder of d. Although the Heisenberg point (d51) is far fromthe decoupled-dimer limit, it is interesting enough that tqualitative characters of both branches remain unchanged increases: The ferromagnetic branchvk

2 is gapless andproportional tok2 in the small-k region, while the antiferro-magnetic branchvk

1 is gapped.For an arbitrary combination of (S,s), the similar argu-

ment can be developed and qualitatively the same resuobtained. For example, in the case of (S,s)5(3/2,1/2), wefind the dispersion relations

vk25

5

8Jd@12cos~2ak!#1O~d2!, ~3.8!

vk152J1

1

8Jd@726 cos~2ak!#1O~d2!, ~3.9!

with EG52(5/4)J(11d/4)N, considering the elementarexcitations shown in Fig. 2. Thus we are more and m

FIG. 1. Schematic representations of the ground state~a!, theelementary excitation in the subspace ofM5N/221 ~b!, and theelementary excitation in the subspace ofM5N/211 ~c! of theferrimagnetic chain with spin 1 and spin 1/2 in the decoupled-dimlimit. The arrow~the bullet symbol! denotes a spin 1/2 with its fixed~unfixed! projection value. The solid~broken! segment is a single~triplet! pair. The circle represents an operation of constructinspin 1 by symmetrizing the two spin 1/2’s inside.

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eas

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convinced that the scenario of the low-energy structshould be valid for an arbitrary Heisenberg ferrimagnet. Atually, recent numerical studies22,24 on the present Hamil-tonian reported that the low-temperature properties aresentially the same regardless of the values ofS ands as longas they differ from each other. Alcaraz and Malvezzi22 foundthat the model with exchange anisotropy is describedterms of the Gaussian critical theory in both cases of (S,s)5(1,1/2) and (S,s)5(3/2,1/2). In such circumstances, wrestrict our numerical investigation to the case of (S,s)5(1,1/2).

IV. NUMERICAL PROCEDURE

In order to calculate the low-lying eigenvalues of thmodel, we here use two numerical tools, which possessspectively, both advantageous and weak points of their oand are complementary to each other. One is the exdiagonalization method employing the Lanczos algorithCalculation of the energy levels reduces to the diagonaltion of the 6N36N matrix representing the Hamiltonian. Iconstructing basic states, we use direct products ofsingle-spin states indicated by the projection values. Tpresent Hamiltonian commutes with the total magnetizatM and therefore splits into 3N11 blocks labeled byM ~onthe assumption thatN is even!. Since we perform the calculation under the periodic boundary condition, each eigvalue is further classified by its total wave numberk.

We treat the chains ofN58,10,12, where we restrict thcalculation to the lowest energy level in each subspaSince our main interest is to reveal the nature of the elemtary excitations, we investigate the subspaces ofM5N/2 andM5N/271 for all the chain lengths we treat. We calculathe subspaces ofM<N/222 as well forN58,10 in order tofurther elucidate the ferromagnetic nature of the model. Dto the two kinds of spins, the construction of the basic stais somewhat complicated. The base dimension reachesmillion in the case ofN512 andM5N/221.

The chain length we can reach with the diagonalizatmethod is much smaller than one with a Monte Carlo tenique. However, having in mind the small correlation lengof the present system,24,25 the diagonalization result is fruitful enough to obtain a general view of the low-energy stru

r

a

FIG. 2. Schematic representations of the ground state~a!, theelementary excitation in the subspace ofM5N/221 ~b!, and theelementary excitation in the subspace ofM5N/211 ~c! of theferrimagnetic chain with spin 3/2 and spin 1/2 in the decoupldimer limit. The notation is the same as one in Fig. 1 except forcircle representing an operation of constructing a spin 3/2 by smetrizing the three spin 1/2’s inside.

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57 13 613ELEMENTARY EXCITATIONS OF HEISENBERG . . .

ture. Actually, the ground-state energy forN512 coincideswith theN→` extrapolated value within the first five digitsWe further note that even the ground-state energy forN58is so close to the thermodynamic-limit value as to showcoincidence within the first three digits. This fact is quhelpful in estimating the dispersion relations in the lonchain limit, although the eigenvalues for different chalengths should be distinguished.

The other approach is based on a quantum Monte Ctechnique12,34–36which one of the authors has recently dveloped. The idea is summarized as extracting the loedge of the excitation spectrum from imaginary-time qutum Monte Carlo data at a low enough temperature. Timaginary-time correlation functionS(q,t) is generally de-fined as

S~q,t!5^eHtOqze2HtO2q

z &, ~4.1!

whereOq5N21( j 51N Oje

iq ja0 is the Fourier transform of anarbitrary local operatorOj with a0 being the length of theunit cell, and ^A&[Tr@e2bHA#/Tr@e2bH# denotes the canonical average at a given temperatureb215kBT. WhileS(q,t) as a function oft generally exhibits a complicatemultiexponential decay, it may efficiently be evaluated asufficiently low temperature as34

S~q,t!5(l

u^1;k0uSqzu l ;k01q&u2e2t@El ~k01q!2E1~k0!#,

~4.2!

whereu l ;k& ( l 51,2, ) andEl(k) (E1(k)<E2(k)<¯) arethe l th eigenvector and eigenvalue of the Hamiltonian ink-momentum space, andk0 is the momentum at which thlowest-energy state in the subspace is located. Now it issonable to approximateE1(k01q)2E1(k0) by the slope2] ln@S(q,t)#/]t in the large-t region satisfying

t@E2~k01q!2E1~k01q!#@ lnu^1;k0uSq

zun;k01q&u2

u^1;k0uSqzu1;k01q&u2 ,

~4.3!

for an arbitraryn.In case the lower edge of the spectrum is separated f

the upper bands or continuum by a finite gap, or in case ospectral weightsu^1;k0uOq

zu1;k01q&u2 being relatively large,the inequality~4.3! is well justified even at smallt’s and alogarithmic plot ofS(q,t) is expected to exhibit fine linearity in a wide region oft. Actually, for single-spin Heisenberg chains with an arbitrary spin quantum number or a ctain bond alternation, it was shown12,34–36 that the tdependence ofS(q,t) is essentially approximated bysingle exponent at each momentumq at a sufficiently lowtemperature.

Here, due to the two kinds of spins in a chain,Oj is notuniquely defined. We show in Fig. 3 logarithmic plotsS(q,t) as a function oft setting several operators forOj .We have set (bJ)21 andn equal to 0.02 and 200, which arerespectively, low and large enough36 to remove the finite-temperature effect and the finite-n effect. In order to estimateeachS(q,t), we have carried out a few million Monte Carlsteps spending several days on a supercomputer or aweeks on a fast workstation. Energy difference between

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ground state and the lowest state with an arbitraryq is ob-tained throughS(q,t) calculated in the subspace ofM50.

In Fig. 4 we plot the excitation energies as a function oqobtained by estimating the slope2] ln@S(q,t)#/]t in thelargest-t region available. In the case ofOj5sj

z , the multi-exponential behavior ofS(q,t) is remarkable even in thelarge-t region and thus prevents us from precisely estimatthe energy eigenvalues. In all the other cases, we obtainful data within the numerical precision. With the presedata, we can at least conclude that spin-1/2’s do not hmuch effect on the lowest-lying excitations, which suggethat the elementary excitations of ferromagnetic nature mqualitatively be described by the simple picture shownFig. 1~b! even at the Heisenberg pointd51. While Oj5Sj

z

1sjz brings somewhat higher energies thanOj5Sj

z and Oj

5Sjz2sj

z , we find no difference beyond the numerical accracy between the cases ofOj5Sj

z andOj5Sjz2sj

z . It will be

FIG. 3. Logarithmic plots ofS(q,t) versus the imaginary timetat several choices ofOj taking a few values of momentumq for theHeisenberg ferrimagnetic chain ofN532 with (S,s)5(1,1/2) andd51: ( (2aq/p516/64), 1 (2aq/p548/64) with Oj5Sj

z ; ,

(2aq/p516/64), 3 (2aq/p548/64) with Oj5sjz ; L (2aq/p

516/64), h (2aq/p548/64) with Oj5Sjz1sj

z ; n (2aq/p516/64),s (2aq/p548/64) withOj5Sj

z2sjz . The numerical un-

certainty is all within the size of the symbols.

FIG. 4. Quantum Monte Carlo estimates of excitation energas a function ofq for the chain ofN532. Heres, h, L, and3

have, respectively, been obtained fromS(q,t)’s with Oj5Sjz , Oj

5sjz , Oj5Sj

z1sjz , andOj5Sj

z2sjz at the subspace ofM50. The

error bars are attached to the data obtained withOj5sjz and Oj

5Sjz1sj

z . The numerical uncertainty of the rest of the data iswithin the size of the symbols. GS denotes the ground state.

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13 614 57SHOJI YAMAMOTO, S. BREHMER, AND H.-J. MIKESKA

shown in the next section that the thus-obtained lowest-lyenergy eigenvalues are in good agreement with the exdiagonalization result. This method is applicable to ratlong chains but is not successful in obtaining the highlying eigenvalues except for the special fortunate case34

Thus the diagonalization technique is necessary and usfor the antiferromagnetic branch withM.N/2 even thoughit is inferior to the Monte Carlo method in treating lonchains.

V. NUMERICAL RESULTS AND DISCUSSION

We plot in Fig. 5 the quantum Monte Carlo and the exadiagonalization calculations of the excitation energies afunction of momentumq for the chain without bond alternation, where the results of the spin-wave theory and the fiorder perturbation from the decoupled-dimer limit are ashown. The lower band is the lower edge of the excitatspectrum and consists of the lowest-lying eigenvalues wM5N/221. It exhibits a quadratic dispersion at smallq’s aswas expected. The upper band consists of the lowest-lyeigenvalues withM5N/211 and is separated from thground state by a finite gap. It is the scenario predictedthe analytic approaches that we here observe. The quanMonte Carlo finding is in good agreement with the diagonization result. The diagonalization calculation indicates tthe chain-length dependence of the dispersion relationquite weak even in the vicinity of the zone boundary andzone center, which is consistent with the extremely smcorrelation length.24,25

In order to perform a quantitative comparison betweennumerical findings and the analytic results, let us consithe curvature of the dispersion,v, which is defined by

vk25v~2ak!2. ~5.1!

The quantum Monte Carlo method, the spin-wave calcution, and the perturbation approach, respectively, gvQMC/J50.37(1), vSW/J51/2, andvpert/J52/9. The spin-wave theory overestimates the true value, while the per

FIG. 5. Quantum Monte Carlo and exact-diagonalization callations of the lowest energies as a function of momentum insubspaces ofM5N/271 for the Heisenberg ferrimagnetic chainN532 with spin 1 and spin 1/2. The numerical uncertainty iswithin the size of the symbols. The results of the spin-wave theand the first-order perturbation from the decoupled-dimer limitalso shown by solid and broken lines, respectively. GS denotesground state.

gct-r-.ful

-a

t-onh

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yum-tisell

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-e

r-

bation approach underestimates that. We have made atempt to obtain another estimate ofv using the exact-diagonalization result. Although the Lanczos algorithresults in much more precise raw data than the Monte Ctechnique, yet the attempt was not as successful as one uthe Monte Carlo data because of the lack of data poiHowever, we have confirmed that the diagonalization emate ofv possibly coincides withvQMC within the numericalaccuracy. We note that the spin-wave estimatevSW accordswith one for the Heisenberg ferromagnet of spin 1/2 in tunit of the unit-cell length being equal to unity. Furthermothe calculation of the lowest levels in the subspace ofM5N/222 results in energy eigenvalues below the twmagnon continuum. The obtained state should be a tmagnon bound state.37 All these facts again emphasize thferromagnetic aspect of the present model. However, in ctrast with the ferromagnet, the true valuevQMC is reducedfrom vSW due to the quantum effect. Thus the elementaexcitations in the sector ofM,N/2 can be regarded as spwaves modified by quantum fluctuations.

For the antiferromagnetic branch, on the other hand, b

FIG. 6. The lowest energies as a function of momentum insubspaces ofM5N/221 ~a! and M5N/211 ~b! for the bond-alternating Heisenberg ferrimagnetic chains ofN532 with spin 1and spin 1/2. Heres ~u!, h ~2!, L ~1!, andn ~3! represent theexact-diagonalization~quantum Monte Carlo! estimates atd50.8,d50.6, d50.4, andd50.2, respectively. The numerical uncetainty is all within the size of the symbols. The results of the spwave theory and the first-order perturbation from the decoupdimer limit are also shown by solid and broken lines, respectivewhere the values ofd are 0.8, 0.6, 0.4, and 0.2 from top to bottoexcept for the perturbation result for the antiferromagnetic brain the small-q region with the values ofd being 0.2, 0.4, 0.6, and0.8 from top to bottom. GS denotes the ground state.

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57 13 615ELEMENTARY EXCITATIONS OF HEISENBERG . . .

spin-wave analysis and perturbation calculation are nosuccessful as ones for the ferromagnetic branch. Considethat quantum effects are, in general, much more remarkin antiferromagnets than in ferromagnets, the most naiveture of the elementary excitations illustrated in Figs. 1 anmay have to be significantly modified for the antiferromanetic branch.

Finally in this section, we show in Fig. 6 the calculatiofor the chains with bond alternation. Although the quantuMonte Carlo calculation is generally in good agreement wthe diagonalization result, the agreement seems to be sowhat poorer in the small-d region. This is convincing keeping in mind that the decrease ofd may cause the freezing othe spin configuration in Monte Carlo sampling and therefa huge number of Monte Carlo steps are needed to refinedata accuracy in the small-d region. It is needless to say thathe perturbation calculation is more justified in the smaldregion. As the model approaches the decoupled-dimer liboth ferromagnetic and antiferromagnetic bands becometer and approach 0 and (3/2)J, respectively. The spin-wavresult is fairly good in the ferromagnetic branch but retively poor in the antiferromagnetic branch. This is convining considering that the spin wave correctly describeslow-lying excitations of the ferromagnets, while it is validmost qualitatively for the antiferromagnets.

VI. SUMMARY

We have investigated the low-energy structure of therimagnetic alternating-spin chains with spin 1 and spin 1Motivated by the spin-wave analysis of the model, we hamainly calculated the eigenvalues with an arbitrary momtum of the one-magnon states, namely, the lowest-lyinggenvalues in the subspaces ofM5N/271. The chain-lengthdependence of the dispersion relations is extremely wewhich is consistent with the considerably small correlatlength of the system,j,2a.24,25The qualitative character othe model remains unchanged under the existence of b

v.

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it,t-

--e

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alternation. The ferromagnetic branch is gapless and shoquadratic dispersion in the small-momentum region, whthe antiferromagnetic branch is separated from the grostate by a finite gapD. D/J in the thermodynamic limit isestimated to be 1.759 14~1! at the Heisenberg pointd51.We made an attempt to understand the low-lying excitatithrough the first-order perturbation calculation from tdecoupled-dimer limit. The ferromagnetic excitations amore or less dominated by spin 1’s, whereas the mechanof the antiferromagnetic excitations was less revealed. Qutum Monte Carlo snapshots38 may help us to inquire furtheinto the antiferromagnetic fluctuations.

We have further carried out the diagonalization calcution in the subspace ofM5N/222 and found the two-magnon bound state below the continuum. This fact empsizes the ferromagnetic aspect of the sector ofM,N/2 in theferrimagnet. We expect that the Heisenberg ferrimagnethaves like a ferromagnet at low enough temperatures, wits antiferromagnetic aspect may appear atkBT*D.

ACKNOWLEDGMENTS

A part of this work was carried out while S.Y. was staing at the Hannover Institute for Theoretical Physics. Sgratefully acknowledges the hospitality of the institute at thtime. The authors would like to thank U. Neugebauer forgreat help in coding the Lanczos algorithm. They are furtgrateful to T. Tonegawa, T. Fukui, and S. K. Pati for theuseful comments. This work was supported by the GermFederal Ministry for Research and Technology~BMBF! un-der Contract No. 03-MI4HAN-8, by the Japanese MinistryEducation, Science, and Culture through Grant-in-Aid N09740286, and by a Grant-in-Aid from the Okayama Foudation for Science and Technology. Most of the numericomputation was done using the facility of the Zuse Coputing Center, Berlin and the Supercomputer Center, Intute for Solid State Physics, University of Tokyo.

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