Characterization of ferrimagnetic Heisenberg chains according to the constituent spins

9 Pages·0.35 MB·English

by  Shoji Yamamoto

Characterization of ferrimagnetic Heisenberg chains according to the constituent spins
Shoji Yamamoto
Department of Physics, Okayama University, Tsushima, Okayama 700-8530, Japan
Takahiro Fukui
0
Institut fu¨r Theoretische Physik, Universit¨at zu K¨oln, Zu¨lpicher Strasse 77, 50937 K¨oln, Germany
0
0
2 Tˆoru Sakai
Department of Physics, Himeji Institute of Technology, Ako, Hyogo 678-1297, Japan
n
(Received 30 August 1999)
a
J
Thelow-energy structureand thethermodynamicproperties of ferrimagnetic Heisenbergchains

1 of alternating spins S and s are investigated by the use of numerical tools as well as the spin-
wave theory. The elementary excitations are calculated through an efficient quantum Monte Carlo
]
techniquefeaturingimaginary-timecorrelation functionsandarecharacterizedintermsofinteract-
h
c ing spin waves. The thermal behavior is analyzed with particular emphasis on its ferromagnetic
e and antiferromagnetic dual aspect. The extensive numerical and analytic calculations lead to the
m classification of the one-dimensional ferrimagnetic behavior according to the constituent spins: the
- ferromagnetic (S >2s), antiferromagnetic (S <2s), and balanced (S =2s) ferrimagnetism.
t
a
PACS numbers: 75.10.Jm, 65.50.+m, 75.40Mg, 75.30Ds
t
s
.
t
a I. INTRODUCTION ferromagnetic aspect, reducing the ground-state magne-
m tization,formagaplessdispersionrelation,whereasthose
- of antiferromagnetic aspect, enhancing the ground-state
d One-dimensional quantum ferrimagnets are one of the magnetization, are gapped from the ground state. As
n hot topics and recent progress [1–15] in the theoretical
a result of the low-energy structure of dual aspect, the
o understanding of them deserves special mention. Such
specific heat shows a Shottky-like peak in spite of the
c a vigorous argument a great deal originates in the pio-
[ ferromagnetic low-temperature behavior and the mag-
neeringeffortstosynthesizebinuclearmagneticmaterials
netic susceptibility times temperature exhibits a round
1 including one-dimensional systems. The first ferrimag-
minimum [3,10,18]. When the exchange coupling turns
v netic chain compound [16], MnCu(dto) (H O) 4.5H O
4 2 2 3· 2 anisotropic, the dispersion of the ferromagnetic excita-
(dto = dithiooxalato = S C O ), was synthesized by
0 2 2 2 tions is no more quadratic [1] and the plateau in the
GleizesandVerdaguerandstimulatedthepublicinterest
0 ground-statemagnetizationcurve[12]due tothe gapped
in this potential subject. The following example of an
1 antiferromagneticexcitationsvanishesviatheKosterlitz-
0 ordered bimetallic chain [17], MnCu(pba)(H2O)3·2H2O Thouless transition [15].
0 (pba = 1,3-propylenebis(oxamato) = C7H6N2O6), ex-
The quantum behavior of the model with higher spins
0 hibitingmorepronouncedonedimensionality,furtherac-
at/ tvievsattigedattiohnes.phTyhsiecraela[1ls8o,1a9p],paesarwedellaansicdheeam[2ic1a]lt[h2a0t],tihne- TishnooutghyeDtrsilolocnleeatraal.s[1th8a]tmiandethtehe(Sfi,rss)t a=tte(m1,p12t)tcoarsee-.
m veal the general behavior of the model, they fixed the
alternatingmagneticcentersdonotneedtobemetalions
smaller spin to 1 in their argument with particular em-
- but may be organic radicals. 2
d phasis on the problem of the crystal engineering of a
Apracticalmodelofaferrimagneticchainistwokinds
n
molecule-based ferromagnet the assembly of the highly
o of spins S and s (S >s) alternating on a ring with anti- −
magnetic molecular entities within the crystal lattice in
c ferromagnetic exchange coupling between nearest neigh-
: bors, as described by the Hamiltonian, a ferromagnetic fashion. There also exists an extensive
v
numerical study [3], but the leading attention was not
i
X N necessarily directed to the consequences of the variation
r =J (Sj sj +sj Sj+1) . (1) of the constituent spins. So far the generic behavior,
a H · · rather than individual features, of ferrimagnetic mixed-
Xj=1
spin chains has been accentuatedor predicted, but there
are few attempts to characterize or classify the typical
Here N is the number of unit cells and we set the unit-
one-dimensional ferrimagnetic behavior as a function of
cell length equal to 2a in the following. The simplest
case,(S,s)=(1,1),has sofarbeen discussedfairly well. (S,s). In such circumstances, we aim in this article at
2 elucidating which property of the model (1) is universal
Thereliebothferromagneticandantiferromagneticlong-
and which one, if any, is variable with (S,s).
rangeordersinthegroundstate[2–4]. Thegroundstate,
which is a multiplet of spin (S s)N, shows elementary It is true that numerical tools are quite useful in this
−
excitations of two distinct types [8]. The excitations of context, but they are not almighty. Although the low-
1
temperature ferromagnetic behavior is quite interesting S(q,τ) 1;k Sz l;k +q 2e−τ[El(k0+q)−E1(k0)],
≃ h 0| q| 0 i
from the experimental point of view, it is hardly feasi- Xl (cid:12) (cid:12)
(cid:12) (cid:12)
ble to numerically take grand-canonical averages at low
(3)
enough temperatures. It is also unfortunate that with
the increase of (S,s), the available information is more
at a sufficiently low temperature, where k is the mo-
and more reduced in both quality and quantity. Then 0
mentum at which the lowest-energy state in the sub-
we have an idea of describing the model in terms of the
space is located. Therefore we can reasonably approx-
spin-wavetheory. The conventionalspin-wavetreatment
imate E (k +q) E (k ) by the slope ∂ln[S(q,τ)]/∂τ
oflow-dimensionalmagnetsmaydiscourageus. TheHal- 1 0 − 1 0 −
in the large-τ region. When we take interest in the
dane conjecture [22] revealeda qualitative breakdownof
lower edge of the excitation spectrum, such a treatment
the spin-wave description of one-dimensionalHeisenberg
is rather straightforward. The elementary excitations of
antiferromagnets. Neither quantum corrections [23] nor
the Haldane antiferromagnets were indeed revealed thus
additional constraints [24] to spin fluctuations end up
[26]. Here, due to the two kinds of spins in a chain and
with an overall scenario for the low-energy physics ap-
theresultantdualaspectofthelow-energystructure,the
plicable to general spins. However, Ivanov [7] has re-
relevant subspace and operator O are not uniquely de-
cently reported that the spin-wave description of one- j
fined. Since the total magnetization, M = (Sz+sz),
dimensional Heisenberg ferrimagnets is quite successful. j j j
Though his calculations were restricted to the ground- is a conserved quantity in the present sysPtem, we con-
sider calculating S(q,τ) independently in each subspace
state energy and magnetization, the highly accurate es-
with a given M [27]. The elementary excitation energies
timates obtained there are surprising enough, consid-
for the ferromagnetic branch are obtained from a single
ering the diverging ground-state magnetization in the
calculation,S(q,τ)inthesubspaceofM =0,whilethose
one-dimensionalantiferromagneticspin-wavetheory. For
for the antiferromagnetic branch from a couple of calcu-
antiferromagnets, quantum fluctuations of domain-wall
lations, S(q,τ) and the lowest energy in the subspace of
type, connecting the two degenerate N´eel states, are im-
M = (S s)N +1. We have taken Sz sz for O and
portant, whereas for ferrimagnets, domain-wall excita- found tha−t O = Sz sz extracts thejei±genjvalues ojf the
tions lead to magnetization fluctuations and are thus of j j − j
bonding (lower-energy) states in both subspaces. The
less significance. Thereforeitis likelythat the spin-wave
choice of the scattering matrices is a profound problem
approach is highly efficient for ferrimagnets. In an at-
in itself and is fully discussed elsewhere [8,28].
tempt to demonstrate such an idea, we try to describe
Althoughthechainlengthwecanreachwiththeexact-
not only the low-energy structure but also the thermal
diagonalization method is strongly limited, the diago-
behavior of the Heisenberg ferrimagnetic spin chains (1)
nalization results are still helpful in the present system
withinthe frameworkofthe spin-wavetheory. Extensive
whose correlation length is generally so small as to be
numerical calculations, supplemented by the spin-wave
comparable to the unit-cell length (see Fig. 5 bellow).
analysis, fully reveal the one-dimensional ferrimagnetic
Actually, the ground-state energies for N = 10 coincide
behavior as a function of the constituent spins.
with their thermodynamic-limit values within the first
severaldigits. TheLanczosalgorithmgivesthemostpre-
II. ELEMENTARY EXCITATIONS ciseestimatefortheground-stateenergyandtheantifer-
romagnetic excitation gap, whereas the quantum Monte
Carlotechniqueisnecessaryfortheevaluationofthecur-
Inordertoinvestigatethelow-energystructure,weem-
vature of the small-momentum dispersion.
ployanewquantumMonteCarlotechnique[25]aswellas
On the other hand, we consider a spin-wave descrip-
theconventionalLanczosdiagonalizationalgorithm. The
tion of the elementary excitations as well. We introduce
idea is in a word expressed as extracting the low-lying
the bosonic operators for the spin deviation in eachsub-
excitations from imaginary-time quantum Monte Carlo
lattice via
data at a low enough temperature. The imaginary-time
correlation function S(q,τ) is generally defined as
S+ = 2S a†a a , Sz =S a†a ,
S(q,τ)= eHτO e−HτO , (2) j q − j j j j − j j (4)
q −q s+ =b† 2s b†b , sz = s+b†b ,
(cid:10) (cid:11) j jq − j j j − j j
where O = N−1 N O e2iaqj is the Fourier trans-
q j=1 j
form of an arbitrarPy local operator Oj, which may be whereweregardS ands asquantitiesofthe sameorder.
an effective combination of the spins S and s, and the TheHamiltonian(1)isexpressedintermsofthe bosonic
thermal average at a given temperature β−1 = k T, operators as
B
A Tr[e−βHA]/Tr[e−βH], is taken in a certain sub-
hspaice≡. S(q,τ) can be represented in terms of the eigen- =Eclass+ 0+ 1+O(S−1), (5)
H H H
vectors and eigenvalues of the Hamiltonian, l;k (l =
| i
1,2, ) and E (k) (E (k) E (k) ), and behaves where E = 2sSJN is the classicalground-stateen-
l 1 2 class
··· ≤ ≤ ··· −
like ergy, and and are the one-body and two-body
0 1
H H
2
terms of the order O(S1) and O(S0), respectively. We is the O(S1) quantum correctionto the ground-stateen-
mayconsiderthesimultaneousdiagonalizationofH0 and ergy, and α†k and βk† are the creation operators of the
1 in the naivestattempt to go beyond the noninteract- ferromagnetic and antiferromagnetic spin waves of mo-
H
ing spin-wave theory. Indeed, the higher-order terms we mentum k whose dispersion relations are given by
take into account, the better description of the ground-
statepropertiesweobtain[7]. However,suchanidea,asa ω± =ω (S s), (8)
k k± −
whole, ends in failure, bringing a gap to the lowest-lying
ferromagnetic excitation branch and thus qualitatively with
misreading the low-energy physics. Therefore, we take
))))
an alternative approachat the idea of first diagonalizing ωk = (S s)2+4Sssin2(ak). (9) acbd
0 andnextextractingrelevantcorrectionsfrom 1. 0 q − ((((
H H H
is diagonalized as [3,4] Using the Wick theorem, is rewritten as
1
H
H0 =E0+JXk (cid:16)ωk−α†kαk+ωk+βk†βk(cid:17) , (6) H1 =E1−JXk (cid:16)δωk−α†kαk+δωk+βk†βk(cid:17) 8888
....
+ irrel+ quart, (10) 0000
H H
where
where the O(S0) correction to the ground-state energy,
E0 =J [ωk (S+s)] , (7) E1, and those to the dispersions, δωk±, are, respectively,
Xk − given by
6666
....
0000
pppp

////

kkkk
aaaa
2222
4444
....
0000
) 2222
2 ....
))/ 0000
211
/,,)
1221
,//,
1332
((((
0000
....
0000
0000 000 00 00 0 0 000 0000
.... ... .. .. . . ... ....
5443 433 23 22 2 1 111 0000
JJJJ //// ))))kkkk((((EEEE
E = 2JN Γ2+Γ2+ S/s+ s/S Γ Γ , (11) III. THERMODYNAMIC PROPERTIES
1 − h 1 2 (cid:16)p p (cid:17) 1 2i
sin2(ak) Γ
δω± =2(S+s)Γ + 2 [ω (S s)] , (12) The dual structure of the excitations leads to unique
k 1 ωk √Ss k± − thermal behavior. In order to complement quantum
MonteCarlothermalcalculations,especiallyatlowtem-
with peratures, we consider describing the thermodynamics
in terms of the spin-wave theory. Introducing the addi-
1 S+s tional constraint of the total magnetization being zero
Γ = 1 ,
1
2N1Xk (cid:18)√Sωskcos−2(a(cid:19)k) (13) isnutcocetehdeedcoinnveonbttiaoinnainlgspainn-wexavceelltehnetordye,scTraipktaihonashoif [t2h9e]
Γ2 = , low-temperature thermal behavior of one-dimensional
−N ω
Xk k Heisenberg ferromagnets. The present authors have re-
cently applied the idea to the spin-(1,1) ferrimagnetic
2
while the irrelevant one-body terms Heisenberg chain [10]. Here we develop the method for
general spin cases and make a detailed analysis of its
(S s)2 cos(ak) validity as a function of (S,s). The core idea of the so-
= J − Γ (α β +α†β†), (14)
Hirrel − √Ss 1Xk ωk k k k k criazleleddamsocdoinfiterdolslipnign-wthaevebothsoeonrynu[3m0b,3e1rs] cbaynibmepsousminmgaa-
certainconstraintonthemagnetization. Fromthispoint
and the residual two-body interactions are both
Hquart ofview,thezero-magnetizationconstraint,whichisquite
neglectedsoastokeeptheferromagneticbranchgapless.
reasonable for isotropic magnets, plays a relevant role in
The resultant Hamiltonian is compactly represented as
ferromagnets. Theresultantlow-temperatureexpansions
[29] of the specific heat and susceptibility of the spin-s
E +J ω−α†α +ω+β†β , (15) Heisenberg ferromagnetic chain,
H≃ g Xk (cid:16) k k k k k k(cid:17)
e e C 3 ζ(3) 1
= 2 t12 t+O(t32), (18)
with NkB 8s21 √π − 2s2
χJ 2s4 ζ(1)
ω± =ω± δω±, (16) = t−2 s25 2 t−32
k k − k N(gµB)2 3 − √π
E =E +E +E . (17)
eg class 0 1 s ζ(1) 2
+ 2 t−1+O(t−12), (19)
2(cid:20) √π (cid:21)
Now we compare all the calculations in Fig. 1. The
diagonalization results fully demonstrate the steady ap-
with t = k T/J and Riemann’s zeta function ζ(z), in-
plicability and good precision of our Monte Carlo treat- B
deedcoincidewiththethermodynamicBethe-ansatzcal-
ment. The spin-wave approach generally gives a good
culations [32] for s= 1 within the leading few terms.
description of the low-energy structure. Even the free 2
In the present system, the zero-magnetization con-
spin waves allow us to have a qualitative view of the
straint is explicitly represented as
elementary excitations. The relatively poor description
of the antiferromagnetic branch by the free spin waves
Sz+sz =(S s)N σn−σ =0, (20)
implies that the quantum effect is more relevant in the h j ji − − k
Xj Xk σX=±
antiferromagnetic branch. Here we may be reminded of
e
the spin-wave treatment of mono-spin Heisenberg mag- where n± = n±P (n−,n+) with P (n−,n+) be-
nets. Fortheferromagneticchains,thespin-wavedisper- k n−,n+ k k
ingtheprobaPbilityofn− ferromagneticandn+ antiferro-
sions are nothing but the exact picture of the elemen-
magneticspinwavesappearinginthek-momentumstate.
tary excitations, while for the antiferromagnetic chains,
Equation (20) straightforwardly proposes that the ther-
thosearegenerallynomorethanaqualitativeview. The
malspindeviationshouldcancelthe N´eel-statemagneti-
point is that in the presentsystem the spin-wavepicture
zation. By minimizing the free energy
is efficient for both elementary excitation branches and
the spin-wave series potentially lead to an accurate de-
F =E + (n−ω−+n+ω+)
scription. More specifically, the divergence of the boson g k k k k
numbers, which plagues the antiferromagneticspin-wave Xk
e e e e
treatmentinonedimension,doesnotoccurinthepresent +k T P (n−,n+)lnP (n−,n+), (21)
B k k
system. This viewpoint is further discussed in the final Xk nX−,n+
section. We conclude this section by listing in Table I
the spin-waveestimates ofafew interestingquantities in with respect to P (n−,n+) at each k under the
k
comparisonwiththenumericalsolutions,wherewedefine condition (20) as well as the trivial constraints
the curvature v as ω− =v(2ak)2. P (n−,n+)=1, we obtain
k→0 n−,n+ k
P
e
4
C = 3 S−s 21 ζ(32)t12 1 t+O(t32), (22) mferargonmetaigcnaetsipcecatspaetctSf/osr=S/s1, while. aAhnuonthderredcopnesricdeenrt-
NkB 4(cid:18) Ss (cid:19) √2π − Ss → ∞
ation also leads us to such a picture. Since the pertur-
χJ Ss(S s)2 e e e ζ(1)
= − t−2 (Ss)21(S s)32 2 t−32 vation from the decoupled dimers [8] qualitatively well
N(gµB)2 3 − − √2π describesthelow-lyingexcitationsofthesystem,wepro-
eζ(1) 2 e pose in Fig. 2 an idea of decomposing ferrimagnets
+(S s) 2 t−1+O(t−21), (23) intoferromagnetsandantiferromagnets,whereweletthe
− (cid:20)√2π(cid:21)
decoupled dimers and the Affleck-Kennedy-Lieb-Tasaki
e e
valence-bond-solidstates[34]symbolizeferrimagnetsand
wheret=k T/Jγ withγ =1 Γ (S+s)/Ss Γ /√Ss.
B − 1 − 2 integer-spin gapped antiferromagnets, respectively. Now
The specific heat C has been obtained by differentiating
we expect spin-(S,s) ferrimagnets to behave like combi-
e
the free energy F, whereas the susceptibility χ by calcu-
nations of spin-(2s) antiferromagnets and spin-(S s)
lating( M2 M 2)/3T,wherewehavesetthegfactors −
of the shpinsiS−handis both equal to g, because the differ- ferromagnets. Since the antiferromagnetic excitations
of the ferrimagnetic ground state are gapped, the low-
ence between them amounts to at most several percent
temperature behavior of ferrimagnets should be only of
of themselves in practice [33].
ferromagnetic aspect. At low temperatures there is in-
deed no effective contributionfrom the spin-(2s) antifer-
(1,1/2) (3/2,1/2) romagneticchainwithanexcitationgap[22]immediately
abovethegroundstate. Inthiscontext,wearesurprised
(a) (a)
but pleased to find that provided S = 2s, the expres-
sions(22) and(23)coincide with those for ferromagnets,
(18)and(19),exceptforthequantumrenormalizingfac-
(b) (b) tor γ. The low-temperature thermodynamics should be
dominatedbythesmall-momentumferromagneticexcita-
tions. Within the linearized spin-wavetheory,the small-
(c) (c)
momentum dispersions of Heisenberg ferrimagnets and
ferromagnets are characterized by the curvatures
(3/2,1) (2,1) v(S,s)-ferri = Ss J, (24)
2(S s)
(a) (a) −
v(S−s)-ferro =(S s)J, (25)
−
(b) (b) respectively, where we find that they coincide with each
other only when S = 2s. The criterion, S = 2s, is fur-
ther convincing when we consider the high-temperature
(c) (c) behavior. The paramagnetic behavior of the spin-(S,s)
ferrimagnet is given as
FIG. 2. Schematic representations of the M = (S −s)N F(S,s)-ferri
= ln[(2S+1)(2s+1)] , (26)
groundstatesofspin-(S,s)ferrimagneticchainsofN elemen- Nk T −
B
tarycellsinthedecoupled-dimerlimit(a),theAKLTground k Tχ(S,s)-ferri 1
states of spin-(2s) antiferromagnetic chains of 2N spins (b), B = [S(S+1)+s(s+1)] , (27)
the M = 2N(S−s) ground states of spin-(S−s) ferromag- Ng2µ2B 3
netic chains of 2N spins (c). The arrow (the bullet symbol)
whereas those of the spin-(S s) ferromagnet and the
denotes a spin 1/2 with its fixed (unfixed) projection value.
−
The solid segment is a singlet pair. The circle represents an spin-(2s) antiferromagnet are as follows:
operation of constructing spins S and s by symmetrizing the
spin 1/2’s inside. The relation (a) ≈ [(b)+(c)]/2 may be F(S−s)-ferro+F(2s)-antiferro
expected. Nk T
B
= ln[(2S 2s+1)(4s+1)] , (28)
− −
Theanalyticexpressions(22)and(23)giveusabird’s- kBT(χ(S−s)-ferro+χ(2s)-antiferro)
eye view of the one-dimensional ferrimagnetic behavior. Ng2µ2
B
The spin-(S,s) ferrimagnet turns into the spin-s antifer-
1
romagnet in the limit of S s, whereas it looks like = [(S s)(S s+1)+2s(2s+1)] . (29)
→ 3 − −
the spin-S ferromagnet in the limit of S/s . In
→ ∞
this sense, the subtraction S s may be regarded as Theseasymptoticvaluesagreewitheachotheronlywhen
−
the ferromagnetic contribution, while the residual spin S = 2s. This is simply the consequence of the degrees
amplitude 2s as the antiferromagnetic one. No ferro- offreedom. InferrimagnetsofS >2s,the ferromagnetic
5
spin degrees of freedom overbalance the antiferromag- numberofthe bosons nσ ratherthanthe subtrac-
σ=± k
netic ones, while in ferrimagnets of S < 2s, vice versa. tion σn−σ. InoPurnaivestattemptto improvethe
OnlythebalancedferrimagnetwithS =2siswellapprox- theorPy,σw=e±replkace the constraeint (20) by
imatedasthe simple combinationofthe spin-(S s)fer- e
−
romagnet and the spin-(2s) antiferromagnet. However, Sz sz =(S+s 2Γ )N
we note that even in the case of S = 2s, the similarity h j − ji − 1
Xj
between the ferrimagnetic behavior, (22) and (23), and
nσ
theferromagneticone,(18)and(19),doesnotgobeyond (S+s) k =0. (30)
− ω
theleadingfewtermsshownhere. Theferromagneticfea- Xk σX=± ek
tures of ferrimagnets are thermally smearedout. On the ))))
other hand, the ferrimagnetic behavior further deviates Hoacbdwever,the alternativecondition (30)changesthe low-
((((
from the purely ferromagnetic one due to the quantum temperature description, (22) and (23), which should be
effect characterized by γ. The low-temperature expan- keptunchangedunderanyartificialconstraint,aswellas
sions (22) and (23) imply that as temperature goes to considerablyunderestimatestheSchottky-likecharacter- 0000
....
zero, the quantum effect is reduced for the specific heat istic peak of the specific heat. In an attempt to remove 5555
C,whereasenhancedforthesusceptibilityχ. Inthelimit Γ1 from Eq. (30), we reach a phenomenological modifi-
of S/s , the quantum corrections Γ and Γ both cation
1 2
→ ∞
vanish.
nσ
:Sz sz : =(S+s)N (S+s) k =0,
Although the spin-wave theory combined with the ad- h j − j i − ω
ditional constraint (20) is so useful, it should further be Xj Xk σX=± ek 0000
modifiedathighertemperaturessoastocontrolthetotal (31) 4.4.4.4.
)
))2 JJJJ
21/)
/,11 0000////
12
,, ....TTTT
,/22 3333
13 BBBB
/(
((3 kkkk
(
0000
....
2222
0000
....
1111
0000
....
0000
8822 00 66 88 4466 44 22 22 0000
.... .. .. .. .... .. .. .. ....
0011 11 00 00 0000 00 00 00 0000
BBBB
kkkkNNNN //// CCCC
))))
acbd
((((
0000
....
5555
0000
....
4444
)
))2 JJJJ
21/)
/,11 0000////
12,, ....T T T T
,/22 3333
13/( BBBB
3 kkkk
((
(
0000
....
2222
0000
....
1111
0000
....
0000
5000 0 00 00 50 0 0000
.... . .. .. .. . ....
1642 3 14 21 02 1 0000
BBBB BBBB
ggggNNNN //// TTTT kkkk
mmmm cccc
2222 2222
antiferromagnetic coupling between nearest neighbors in
common? The surprising efficiency of the present spin-
wave treatment can be attributed to the orderedground
state [2] of ferrimagnets and thus to the nondiverging
sublattice magnetization. The key constant Γ is noth-
1
ing but the quantum spin reduction
1 1
a†a = b†b , (32)
N h j jig N h j jig
Xj Xj
where A denotes the ground-state average of A. Γ
g 1
h i
monotonically decreases as S/s increases, diverging at
S/s = 1 and vanishing for S/s . Since the spin re- ))
duction Γ can be a measure for→th∞e validity of the spin- 11
wave desc1ription, the spin-wave theory in general works 2,2,
better as S/s, as well as Ss, increases. ((
Inthis context,it isinteresting to observethe ground- 88
state spin correlations,
SzSz for r =2la,
f (r)= j j+l (33)
S (cid:26)Szsz for r =(2l+1)a, ))
j j+l 11
szsz for r =2la, ,,
f (r)= j j+l (34) 22
s (cid:26)szSz for r =(2l+1)a. //
j j+l+1 33
((
66
aa

)) //
22
// rr
11
,,
22
3/3/ 44
((
))
22
//
11
22
,,
11
((
))
ab
((
00
00 05 00 0 50 00 05
.. .. .. . .. .. ..
14 30 20 1 00 11 21
- -- --
Ss
))rr(( ff
[8] S.Yamamoto,S.Brehmer,andH.-J.Mikeska,Phys.Rev. nard, M. Drillon, and P. Legoll, Inorg. Chem. 29, 3325
B 57, 13610 (1998). (1990).
[9] S. Yamamoto and T. Sakai, J. Phys. Soc. Jpn. 67, 3711 [21] A.Caneschi,D.Gatteschi,P.Rey,andR.Sessoli, Inorg.
(1998). Chem. 27, 1756 (1988); A.Caneschi, D.Gatteschi, J.-P.
[10] S. Yamamoto and T. Fukui, Phys. Rev. B 57, 14008 Renard, P. Rey, and R. Sessoli, ibid. 28, 1976 (1989);
(1998); S. Yamamoto, T. Fukui, K. Maisinger, and U. 2940 (1989).
Schollw¨ock,J.Phys.: Condens.Matter10,11033(1998). [22] F. D. M. Haldane, Phys. Lett. 93A, 464 (1983); Phys.
[11] K.Maisinger, U.Schollw¨ock,S.Brehmer,H.-J.Mikeska, Rev. Lett.50, 1153 (1983).
and S. Yamamoto, Phys. Rev. B 58, 5908 (1998); [23] T. Oguchi, Phys.Rev. 117, 117 (1960).
A. K. Kolezhuk, H.-J. Mikeska, K. Maisinger, and U. [24] S. M. Rezende, Phys.Rev.B 42, 2589 (1990).
Schollw¨ock, Phys. Rev.B 59, 13565 (1999); [25] S. Yamamoto, Int.J. Mod. Phys.C 8, 609 (1997).
[12] T. Kuramoto, J. Phys. Soc. Jpn. 67, 1762 (1998); 68, [26] S. Yamamoto, Phys.Rev.Lett. 75, 3348 (1995).
1813 (1999). [27] S. Yamamoto and S. Miyashita, Phys. Lett. A 235, 545
[13] S.Yamamoto, Phys.Rev. B59, 1024 (1999). (1997).
[14] C.Wu,B.Chen,X.Dai,Y.Yu,andZ.-B.Su,Phys.Rev. [28] S. Yamamoto, Phys.Rev.B 51, 16128 (1995).
B 60, 1057 (1999). [29] M. Takahashi, Phys. Rev.Lett. 58, 168 (1987).
[15] T.SakaiandS.Yamamoto,Phys.Rev.B60,4053(1999); [30] M. Takahashi, Phys. Rev.B 40, 2494 (1989).
S.YamamotoandT.Sakai,J.Phys.: Codens.Matter11, [31] J. E. Hirschand S.Tang, Phys.Rev.B 40, 4769 (1989);
5175 (1999). S.Tang,M.E.Lazzouni,andJ.E.Hirsch,Phys.Rev.B
[16] A. Gleizes and M. Verdaguer, J. Am. Chem. Soc. 103, 40, 5000 (1989).
7373 (1981); 106, 3727 (1984). [32] M. Takahashi and M. Yamada, J. Phys. Soc. Jpn. 54,
[17] Y.Pei,M.Verdaguer,O.Kahn,J.Sletten,andJ.-P.Re- 2808 (1985); M. Yamada and M. Takahashi, ibid. 55,
nard,Inorg. Chem. 26, 138 (1987). 2024 (1986).
[18] M.Drillon,J.C.Gianduzzo,andR.Georges,Phys.Lett. [33] M.Hagiwara, K.Minami, Y.Narumi,K.Tatani,andK.
96A, 413 (1983); M. Drillon, E. Coronado, R. Georges, Kindo, J. Phys.Soc. Jpn. 67, 2209 (1998).
J. C. Gianduzzo, and J. Curely, Phys. Rev.B 40, 10992 [34] I. Affleck,T. Kennedy,E. H.Lieb, and H.Tasaki, Phys.
(1989). Rev. Lett. 59, 799 (1987); Commun. Math. Phys. 115,
[19] M. Verdaguer, A. Gleizes, J.-P. Renard, and J. Seiden, 477 (1988).
Phys.Rev.B 29, 5144 (1984). [35] N. Fujiwara and M. Hagiwara, to be published in Solid
[20] O. Kahn, Y. Pei, M. Verdaguer, J.-P. Renard, and J. State Commun..
Sletten, J. Am. Chem. Soc. 110, 782 (1988); P. J. van [36] S. Yamamoto, to be published in Phys. Rev. B 61, No.
Koningsbruggen,O.Kahn,K.Nakatani,Y.Pei,J.-P.Re- 2 (2000).
TABLE I. Comparison between the linear-spin-wave (LSW), interacting-spin-wave (ISW), and numerical estimates of the
ground-state energy Eg, the antiferromagnetic excitation gap ∆, and the curvature of the ferromagnetic dispersion, v. The
most accurate numerical estimates of Eg and ∆ are obtained by the exact diagonalization (Exact), whereas those of v by the
quantumMonte Carlo (QMC).
(S,s) Eg/NJ ∆/J v
LSW ISW Exact LSW ISW Exact LSW ISW QMC
(1,1) −1.4365 −1.4608 −1.4541(1) 1 1.6756 1.759(1) 1 0.3804 0.37(1)
2 2
(3,1) −1.9580 −1.9698 −1.9672(1) 2 2.8025 2.842(1) 3 0.3390 0.31(1)
2 2 8
(3,1) −3.8281 −3.8676 −3.861(1) 1 1.5214 1.615(5) 3 1.1319 0.90(3)
2 2
(2,1) −4.8729 −4.8973 −4.893(1) 2 2.6756 2.730(5) 1 0.8804 0.76(2)
9