Bosonic representation of one-dimensional Heisenberg ferrimagnets

0.52 MB·English

by  Shoji Yamamoto

Bosonic representation of one-dimensional Heisenberg ferrimagnets
Shoji Yamamoto
Division of Physics, Hokkaido University, Sapporo 060-0810, Japan
(2 September2003)
4
Theenergy structureandthethermodynamicsofferrimagnetic Heisenbergchainsof alternating
0
spins S and s are described in terms of the Schwinger bosons and modified spin waves. In the
0
Schwingerrepresentation,weaveragethelocalconstraintsonthebosonsanddiagonalizetheHamil-
2
tonian at the Hartree-Fock level. In the Holstein-Primakoff representation, we optimize the free
n
energy in two different ways introducing an additional constraint on the staggered magnetization.
a
A new modified spin-wave scheme, which employs a Lagrange multiplier keeping the native energy
J
structurefreefromtemperatureandthusdiffersfromtheoriginalTakahashiScheme,isparticularly
4
stressedasanexcellentlanguagetointerpretone-dimensionalquantumferrimagnetism. Othertypes
2
of one-dimensional ferrimagnets and theantiferromagnetic limit S =s are also mentioned.

]
h PACS numbers: 75.10.Jm, 75.50.Gg, 75.40.Cx
c
e
m I. INTRODUCTION group[22,23]andquantumMonteCarlo[24,25]methods
in an attempt to illuminate dual features of ferrimag-
-
t netic excitations. More general mixed-spin chains were
a Significant efforts have been devoted to syn-
analyzed via the nonlinear σ model [26] with particu-
t thesizing low-dimensional ferrimagnets and under-
s lar emphasis on the competition between massive and
. standing their quantum behavior in recent years.
t massless phases. Quasi-one-dimensional mixed-spin sys-
a The first example of one-dimensional ferrimagnets,
m MnCu(S C O ) (H O) 4.5H O, was synthesized by tems[27,28]werealsoinvestigatedinordertoexplainthe
2 2 2 2 2 3· 2 inelastic-neutron-scattering findings [29,30] for the rare-
- Gleizes and Verdaguer [1] and followed by a series of or-
d dered bimetallic chain compounds [2] in an attempt to earth nickelates R2BaNiO5.
n
design molecule-based ferromagnets [3]. Caneschi et al. Inordertocomplementnumericaltoolsandtoachieve
o
c [4] demonstrated another approach to alternating-spin further understanding of the magnetic double structure
[ chains hybridizing manganese complexes and nitronyl of ferrimagnetism, several authors have recently begun

nitroxide radicals. The inorganic-organic hybrid strat- to construct bosonic theories of low-dimensional quan-
1
egyrealizedmorecomplicatedalignmentsofmixedspins tum ferrimagnets. The conventional spin-wave descrip-
v
6 [5]. There also exists an attempt at stacking novel tri- tion of the ground-state properties [22,31–33], a modi-
7 radicals into a purely organic ferrimagnet [6]. Mono- fiedspin-waveschemeforthelow-temperatureproperties
4 spin chains can be ferrimagnetic with polymerized ex- [34], and the Schwinger-bosonrepresentationof the low-
1 change interactions. An example of such ferrimagnets energy structure [35] and the thermodynamics [36], they
0 is the ferromagnetic-antiferromagnetic bond-alternating all reveal the potential of bosonic languages for various
4
copper tetramer chain compound Cu(C H NCl) (N ) ferrimagnetic systems. However, considering the global
0 5 4 2 3 2
[7]. The trimeric intertwining double-chain compound argument and total understanding over the bosonic the-
/
t Ca Cu (PO ) [8] is another solution to homometallic ory of ferromagnets and antiferromagnets [37–47], ferri-
a 3 3 4 4
m one-dimensional ferrimagnets, where the noncompensa- magnets are still undeveloped in this context especially
tion of sublattice magnetizations is of topological ori- in one dimension. In such circumstances, we represent
-
d gin. Besides one-dimensional ferrimagnets, metal-ion one-dimensionalHeisenberg ferrimagnets in terms of the
n magneticclusterssuchas[Mn O (CH COO) (H O) ] SchwingerbosonsandtheHolstein-Primakoffspinwaves.
12 12 3 16 2 4
o [9] and [Fe (N C H ) O (OH) ]8+ [10], for which res- Basedonamean-fieldansatz,thelocalconstraintsonthe
8 3 6 15 6 2 12
c
onantmagnetizationtunneling [11–14]wasobserved,are Schwinger bosons are relaxed and imposed only on the
:
v also worth mentioning as zero-dimensional ferrimagnets. average. The conventional antiferromagnetic spin-wave
i The discovery of ordered bimetallic chain compounds formalism [48,49] is modified, on the one hand following
X
stimulated extensive theoretical interest in (quasi-)one- the Takahashi scheme [37,38] which was originally pro-
r
a dimensional quantum ferrimagnets. Early efforts [15] posedforferromagnets,whileontheotherhandintroduc-
were devoted to numerically diagonalizing alternating- ing a slightly different new strategy [50]. The Schwinger
spin Heisenberg chains. Numerical diagonalization, bosons and the modified spin waves both interpret the
combined with the Lanczos algorithm [16,17] and a low-energypropertiesfairlywellidentifyingtheferrimag-
scaling technique [18], further contributed to study- netic long-range order with a Bose condensation, while
ing modern topics such as phase transitions of the the two languages are qualitatively distinguished in de-
Kosterlitz-Thouless type [17,19] and quantized magne- scribing the thermodynamics. We demonstrate that the
tization plateaux [20,21]. Alternating-spin chains were new modified spin-wave scheme of all others depict one-
further investigated by density-matrix renormalization- dimensional ferrimagnetic features very well.
1
II. FORMALISM where k is defined as nπ/Na (n = 0,1, ,N 1) with
··· −
a being the distance between neighboring spins, and λ
A practical model for one-dimensional ferrimagnets is andµaretheLagrangemultipliersduetotheconstraints
two kinds of spins, S and s (S > s), alternating on a (2.6). Via the Bogoliubov transformation
ring with antiferromagnetic exchange coupling between
nearest neighbors, as described by the Hamiltonian, ak↑ = αk↑coshθk+βk†↓sinhθk,
a = α coshθ β† sinhθ ,
N k↓ k↓ k− k↑ k (2.8)
H=JnX=1(Sn·sn−1+sn·Sn) , (2.1) bbkk↑↓ == ββkk↑↓ccoosshhθθkk−+αα†k†k↓↑ssiinnhhθθkk,,
where N is the number of unit cells. The simplest case,
with
(S,s)=(1,1), has so far been discussedfairly well using
2
the matrix-productformalism [51], a modified spin-wave 2Ωcosak
scheme[50],theSchwinger-bosonrepresentation[36],and tanh2θk = , (2.9)
λ+µ
modern numerical techniques [22–25]. We make further
explorationsintohigher-spinsystemsanddevelopthean- the Hamiltonian (2.7) is diagonalized as
alytic argument in more detail.
=2NJSs+4NJΩ2
MF
H
2NJλ(2S+1) 2NJµ(2s+1)+2J ω
A. Schwinger-boson mean-field theory − − k
k
X
Letusdescribeeachspinvariableintermsoftwokinds +J ωk−σα†kσαkσ +ωk+σβk†σβkσ , (2.10)
of bosons as Xk σX=↑,↓(cid:16) (cid:17)
Sn+ =a†n↑an↓, Snz = 21 a†n↑an↑−a†n↓an↓ , where
s+ =b† b , sz = 1 b† b b† b , (2.2)
n n↑ n↓ n 2(cid:0)n↑ n↑− n↓ n↓ (cid:1) ω± ω± =ω (µ λ);
(cid:0) (cid:1) kσ ≡ k k± −
ω = (λ+µ)2 4Ω2cos2ak. (2.11)
where the constraints k −
p
a† a =2S, b† b =2s, (2.3) λ, µ, and Ω are determined through a set of equations
nσ nσ nσ nσ
σ=↑,↓ σ=↑,↓
X X n¯− cosh2θ +n¯+ sinh2θ +sinh2θ =NS, (2.12)
are imposed on the bosons. Then the Hamiltonian can kσ k kσ k k
k
be written as X(cid:0) (cid:1)
n¯− sinh2θ +n¯+ cosh2θ +sinh2θ =Ns, (2.13)
N kσ k kσ k k
H=2NJSs−2J Ω†n+Ωn++Ω†n−Ωn− , (2.4) Xk (cid:0)n¯− +n¯+ +1 coshθ sinhθ =NΩ(cid:1), (2.14)
nX=1(cid:16) (cid:17) kσ kσ k k
k
where Ω = (a b a b )/2 and Ω = X(cid:0) (cid:1)
n+ n↑ n↓ n↓ n↑ n−
−
(an↑bn−1↓ an−1↓bn↑)/2. The Hartree-Fock treatment where the thermal distribution functions n¯−
assumes th−e thermal average of the short-range antifer- α† α and n¯+ β† β are required tokσmin≡i-
romagnetic order to be uniform and static as h kσ kσiT kσ ≡ h kσ kσiT
mize the free energy and given by
Ω† = Ω =Ω. (2.5)
h n±iT h n±iT n¯± = 1 . (2.15)
The constraints (2.3) are correspondingly relaxed as kσ eωk±σ/kBT 1
−
N N The magnetic susceptibility is expressed as
a† a =2NS, b† b =2Ns. (2.6)
nσ nσ nσ nσ
nX=1σX=↑,↓ nX=1σX=↑,↓ χ= (gµB)2 n¯τ (n¯τ +1), (2.16)
4k T kσ kσ
In the momentum space the mean-field Hamiltonian B k τ=±σ=↑,↓
XX X
reads
where we have set the g-factors of spins S and s both
=2NJSs+4NJΩ2 4NJ(λS+µs)
MF equal to g. The internal energy should be given by
H −
2JΩ cosak(a b a b +H.c.)
− k↑ k↓− k↓ k↑ 1
Xk E = 2(EMF+2NJSs)−2NJSs, (2.17)
+2J λa† a +µb† b , (2.7)
kσ kσ kσ kσ
where
Xk σX=↑,↓(cid:16) (cid:17)
2
E =2NJSs+4NJΩ2 2NJ(2λS+2µs+λ+µ) ω±(k)=(S+s)cosh2θ 2√Sscosaksinh2θ
MF − 1 k− k
+2J ωk+J n¯τkσωkτσ. (2.18) ±(S−s)≡ωk±(S−s), (2.26a)
Xk Xk τX=±σX=↑,↓ ω±(k)= S + s Γ 2Λ cosh2θ
Arovas and Auerbach [44] pointed out that relaxing the 0 "(cid:16)rs rS (cid:17) − # k
original constraints (2.3) into Eq. (2.6) leads to dou- S s
blecountingthenumberofindependentbosondegreesof 2Γ + Λ cosaksinh2θk
freedom. Therefore,in Eq. (2.17),we havecorrectedthe −" −(cid:16)rs rS (cid:17) #
mean-field artifact reducing the overestimated quantum S s
, (2.26b)
fluctuation.
± s − S
r r
(cid:16) (cid:17)
B. Modified spin-wave theory: Takahashi scheme γ (k)=2√Sscosakcosh2θ (S+s)sinh2θ , (2.27a)
1 k k
−
S s
γ (k)= 2Γ + Λ cosakcosh2θ
Nextweconsiderasingle-componentbosonicrepresen- 0 " − rs rS # k
tation of each spin variable at the cost of the rotational (cid:16) (cid:17)
symmetry. We start from the Holstein-Primakoff trans- S s
+ Γ 2Λ sinh2θ . (2.27b)
formation −" rs rS − # k
(cid:16) (cid:17)
S+ = 2S a†a a , Sz =S a†a , The conventional spin-wave scheme naively diagonal-
n − n n n n − n n (2.19) ize the Hamiltonian (2.20) and ends up with the num-
s+n =bq†n 2s−b†nbn, szn =−s+b†nbn. ber of bosons diverging with increasing temperature. In
q order to suppress this thermal divergence, Takahashi
Treating S and s as O(S) = O(s), we can expand the [38]considered optimizing the bosonic distribution func-
Hamiltonian with respect to 1/S as tions under zero magnetization and obtained an excel-
lent description of the low-temperature thermodynam-
= 2NJSs+E1+E0+ 1+ 0+O(S−1), (2.20) ics for low-dimensional Heisenberg ferromagnets. For
H − H H
ferrimagnets, this idea is still useful [34,50] but never
where Ei and i give the O(Si) quantum corrections applies away from the low-temperature region as it is.
H
to the ground-state energy and the dispersion relations, The zero-magnetization constraint plays a role of keep-
respectively. Via the Bogoliubov transformation ing the number of bosons finite under ferromagnetic in-
teractions but does not work so under antiferromagnetic
a = α coshθ β†sinhθ , interactions. Takahashi [38] and Hirsch et al. [39] pro-
k k k− k k (2.21)
b = β coshθ α†sinhθ , posed constraining the staggered magnetization, instead
k k k− k k of the uniform magnetization, to be zero as the anti-
they are written as ferromagnetic version of the modified spin-wave theory.
Their scheme was applied to extensive antiferromagnets
E = 2NJ 2√SsΓ (S+s)Λ , (2.22a) in both two [38,39,52–55] and one [40,41] dimensions.
1
− − Theconventionalspin-waveprocedureassumesthatspins
h S i s ononesublatticepointpredominantlyup,whilethoseon
E0 = 2NJ Γ2+Λ2 + ΓΛ , (2.22b) the other predominantly down. The modified spin-wave
− " −(cid:16)rs rS (cid:17) # treatmentrestoresthesublatticesymmetry. Weconsider
the naivest generalizationof the antiferromagnetic mod-
ified spin-wave scheme to ferrimagnets.
Hi =J ωi−(k)α†kαk+ωi+(k)βk†βk The constraint of zero staggered magnetization reads
Xk h
+γ (k) α β +α†β† , (2.23) a†nan+b†nbn =N(S+s). (2.28)
i k k k k
n
(cid:0) (cid:1)i X(cid:0) (cid:1)
where In order to enforce this condition, we first introduce a
Lagrange multiplier and diagonalize the effective Hamil-
1 tonian
Γ= cosaksinh2θ , (2.24)
k
2N
1 Xk H=H+2Jν a†nan+b†nbn . (2.29)
Λ= 2N (cosh2θk−1), (2.25) e Xn (cid:0) (cid:1)
k Then the ground-state energy and the dispersion rela-
X
tions are obtained as
3
E = 2NJSs+E ; E =E +4NJΛν, (2.30) Theground-stateenergyandthedispersionrelationsare
g 1 1 1
−
ω± =ω±(k); ω±(k)=ω±(k)+2νcosh2θ , (2.31) simply given by
k 1 1 1 k
e e
keeping oenly the beilinear terms and as Eg =−2NJSs+E1; ωk± =ω1±(k), (2.40)
E = 2NJSs+E +E , (2.32) within the up-to-O(S1) treatment and by
g 1 0
−
ω± =ω±(k)+ω±(k), (2.33)
k 1 0 e Eg =−2NJSs+E1+E0; ωk± =ω1±(k)+ω0±(k),
considering the O(S0) interactions as well. In terms of
e (2.41)
the spin-wave distribution functions
in the up-to-O(S0) treatment. They are nothing but the
1
n¯± = , (2.34) T =0 findings in the Takahashi scheme.
k eωk±/kBT −1 At finite temperatures we replace α†kαk and βk†βk in
the internal energy and the magnetic susceptibility are the spin-wave Hamiltonian (2.23) by
expressed as [56]
∞
n¯∓ n∓P (n−,n+), (2.42)
E =Eg+ n¯τkωkτ, (2.35) k ≡ k
k τ=± n−X,n+=0
XX
χ= (gµB)2 n¯τ(n¯τ +1). (2.36) where Pk(n−,n+) is the probability of n− ferromagnetic
3kBT k τ=± k k and n+ antiferromagnetic spin waves appearing in the
XX k-momentum state and satisfies
θ , defining the Bogoliubov transformation (2.21), is de-
k
termined through P (n−,n+)=1, (2.43)
k
nX−,n+
γ (k) 2νsinh2θ γ (k)=0, (2.37)
1 k 1
− ≡
for all k’s. Then the free energy is written as
provided we treat as a perturbation to .
H0 e H1
F =E +J P (n−,n+) nτωτ
g k k
C. Modified spin-wave theory: A new scheme Xk nX−,n+ τX=±
+k T P (n−,n+)lnP (n−,n+). (2.44)
B k k
Although the Takahashi scheme overcomes the diffi- Xk nX−,n+
culty of sublattice magnetizations diverging thermally,
theobtainedthermodynamicsisstillfarfromsatisfactory We minimize the free energy with respect to Pk(n−,n+)
(see Fig. 2 later on). Within the conventionalspin-wave enforcing a condition
theory,thequantumspinreduction,thatis,thequantum
Sz sz +2δ :Sz sz :
fluctuation of the ground-state sublattice magnetization h n− niT ≡h n− n iT
per unit cell, reads S+s n¯τ
=S+s k =0, (2.45)
− N ω
ha†naniT=0 =hb†nbniT=0 ≡δ Xk τX=± k
π S+s dk 1
aswellasthetrivialconstraints(2.43). Inthesecond-side
= , (2.38)
Z0 (S s)2+4Sssin2(k/2)2π − 2 compactexpression,thenormalorderingistakenwithre-
− spect to both operatorsα and β. Equation(2.45)claims
and divergesqat S = s. The Takahashi scheme settles that the thermal fluctuation (S + s) k(n−k + n+k)/ωk
this quantum divergence as well as the thermal diver- should cancel the full, or classical, N´eel order (S +s)N
gence. However, the number of bosons does not diverge rather than the quantum mechanicallyPreduced one (S +
in the ferrimagnetic ground state. Without quantum di- s 2δ)N. Without consideration of the quantum fluc-
−
vergence, it is not necessary to modify the dispersion tuation 2δ, which is absent from ferromagnets but pe-
relations (2.26a) into the temperature-dependent form culiar to ferrimagnets, the present scheme breaks even
(2.31). While the thermodynamics should be modified, the conventional spin-wave achievement at low temper-
the quantum mechanics may be left as it is. atures. Numerically solving the thermodynamic Bethe-
Such an idea leads to the Bogoliubov transformation Ansatz equations,TakahashiandYamada [57]suggested
freefromtemperaturereplacingEq. (2.37)byγ (k)=0, that the conventional spin-wave theory correctly gives
1
that is, the low-temperature leading term of the specific heat.
Boththe Takahashischeme with Eq. (2.28)andthe new
2√Sscosak scheme with Eq. (2.45) indeed keep unchanged the con-
tanh2θ = . (2.39)
k ventional spin-wave findings
S+s
4
C 3 S sζ(3) considerablyunderestimatetheantiferromagneticexcita-
− 2 t1/2 (T 0), (2.46)
NkB ∼ 4r Ss √2π → tion energies. The quantum correlation has much effect
ontheantiferromagneticexcitationmodeandsuchanef-
where t = k T/J within the up-to-O(S1) treatment, fectiswellincludedintotheSchwinger-bosoncalculation
B
while t=k T/γJ with γ =1+Γ/√Ss (S+s)Λ/Ssin even at the mean-field level.
B
−
the up-to-O(S0) treatment. The conventional spin-wave Nextwecalculatethethermodynamicproperties. Fig-
approach gives no quantitative information on the mag- ure 2 shows the temperature dependence of the specific
netic susceptibility, whereas the modified theory reveals heat. The Schwinger-boson mean-field theory is still
highlysuccessfulatlowtemperatures,whilewithincreas-
χJ Ss(S−s)2 t−2 (T 0). (2.47) ing temperature, it rapidly breaks down failing to repro-
N(gµ )2 ∼ 3 → duce the Schottky-type peak. The mean-field order pa-
B
rameter Ω monotonically decreases with increasing tem-
In terms of the optimum distribution functions
perature and reaches zero at
1
n¯± = , (2.48) k T S+s+1
k e[Jωk±−ν(S+s)/ωk]/kBT −1 BJ = ln(1+1/S)+ln(1+1/s). (3.1)
the free energy at the thermal equilibrium is written as
Above this temperature, Ω sticks at zero suggesting no
F =E +ν(S+s)N k T ln(1+n¯τ), (2.49) antiferromagnetic correlation in the system. The onset
g − B k of the paramagnetic phase at a finite temperature is a
k τ=±
XX
mean-field artifact and the particular temperature (3.1)
where ν is the Lagrangemultiplier due to the constraint is an increasing function of S and s. The modified spin-
(2.45). wave theory based on the Takahashi scheme also fails to
describe the Schottky peak. Because of the Lagrange
multiplier ν, which turns out a monotonically increasing
III. RESULTS function of temperature, the dispersion relations (2.31)
lead to endlessly increasing energy and thus nonvanish-
First we calculate the ground-state energy E and the ingspecificheatathightemperatures. Only the modified
g
antiferromagneticexcitationgapω+ andcomparethem spin-wave theory based on the new scheme succeeds in
k=0
with numerical findings in Table I. At T = 0, the Taka- interpreting the Schottky peak. Since the antiferromag-
hashi scheme and the new scheme lead to the same re- netic excitation gap is significantly improved by the in-
sults both giving ν = 0. We are fully convinced that clusionoftheO(S0)correlation,theinteractingmodified
the spin-wave treatment better works for larger spins. spin waves reproduce the location of the Schottky peak
We further learn that the spin-wave approach is bet- fairly well. Mixed-spin trimeric chain ferrimagnets have
ter justified with increasing S/s as well as Ss, which is recently been synthesized [5] and their low-temperature
because the quantity S s fills the role of suppressing thermal properties were well elucidated by the modified
−
the divergence in Eq. (2.38). On the other hand, the spin-wave theory [34]. However, it was unfortunate that
Schwinger-boson approach constantly gives highly pre- the additional constraint was imposed on the uniform
cise estimatesofthe low-energyproperties. Figure1 fur- magnetizationandthereforethehiger-temperatureprop-
ther demonstrates that the Schwinger-boson mean-field erties were much less illuminated. Controlling the stag-
theoryis highlysuccessfulindescribingthelow-lyingex- geredmagnetizationinsteadbasedonthenewscheme,we
citations. Both the bosonic languages well interpret the canfully investigatesuch polymeric chaincompounds as
ferromagnetic excitations, whereas the linear spin waves well.
4.0
QMC
SB
3.0 LMSW
PIMSW
k2.0
w
1.0
(a) (1,1/2) (b) (3/2,1/2) (c) (3/2,1)
0.0
0.0 0.2 0.4 0.6 0.8 1.00.0 0.2 0.4 0.6 0.8 1.00.0 0.2 0.4 0.6 0.8 1.0
2ak/p

FIG. 1. The Schwinger-boson (SB), linear-modified-spin-wave (LMSW), perturbational interacting-modified-spin-wave
−
(PIMSW), and quantum Monte Carlo (QMC) calculations of the dispersion relations of the ferromagnetic (ω ) and anti-
k
ferromagnetic (ω+) elementary excitations for thespin-(S,s) ferrimagnetic Heisenberg chains at zero temperature.
k
5
2.0
(a) (1,1/2) (b) (3/2,1/2) (c) (3/2,1)
QMC
B SB
Nk1.0 LMSW(Takahashi)
C/ PLIMMSSWW((YTaamkaahmaoshtoi))
PIMSW(Yamamoto)
0.0
0.0 1.0 2.0 3.00.0 1.0 2.0 3.00.0 1.0 2.0 3.0
k T/J
B
FIG. 2. The Schwinger-boson (SB), linear-modified-spin-wave (LMSW), perturbational interacting-modified-spin-wave
(PIMSW),andquantumMonteCarlo(QMC)calculationsofthespecificheatC asafunctionoftemperatureforthespin-(S,s)
ferrimagnetic Heisenberg chains. The modified spin waves are constructed in two different ways, theTakahashi scheme (Taka-
hashi) and thenew scheme (Yamamoto).
4.0
(a) (1,1/2) (b) (3/2,1/2) (c) (3/2,1)
SB
PIMSW(Takahashi)
3.0
0
k=2.0
w
1.0
0.0
0.0 1.0 2.0 3.0 0.0 1.0 2.0 3.00.0 1.0 2.0 3.0
k T/J
B
FIG.3. Theferromagnetic(ω− )andantiferromagnetic(ω+ )excitationgapsasfunctionsoftemperatureforthespin-(S,s)
k=0 k=0
ferrimagnetic Heisenberg chains calculated by the Schwinger bosons (SB) and the perturbationally interacting modified spin
waves (PIMSW) based on theTakahashi scheme.
IntheSchwingerrepresentationandthemodifiedspin- function of temperature.
wave treatment based on the Takahashi scheme, the en-
Figure 4 shows the temperature dependence of the
ergy spectrum depends on temperature. Since the low-
magnetic susceptibility-temperature product, which elu-
energy band structure is well reflected in the thermal
cidates ferromagnetic and antiferromagnetic features co-
behavior and can directly be observedthrough inelastic-
existinginferrimagnets[25]. χT divergesatlowtempera-
neutron-scatteringmeasurements,we investigatethe fer-
turesinaferromagneticfashionbutapproachesthehigh-
romagnetic (ω− ) and antiferromagnetic (ω+ ) excita-
k=0 k=0 temperature paramagnetic behavior showing an antifer-
tion gaps as functions of temperature in Fig. 3. The
romagneticincrease. Themodifiedspinwavesmuchbet-
Schwinger-boson mean-field theory claims that the an-
ter describe the magnetic behavior than the Schwinger
tiferromagnetic gap should first decrease and then in-
bosons. The spin waves modified along with the Taka-
crease with increasing temperature, while the modified
hashi scheme better work at high temperatures, while
spin-wave theory predicts that the excitation energies
those along with the new scheme precisely reproduce
of both modes should be monotonically increasing func-
the low-temperature behavior. Both calculations con-
tionsoftemperature. We findasimilarcontrastbetween
verge into the paramagnetic behavior χk T/N(gµ )2 =
the twolanguagesappliedto ladderferrimagnets[35,58]. B B
[S(S+1)+s(s+1)]/3athightemperatures,whereasthe
In the case of Haldane-gap antiferromagnets, both the
Schwinger-bosonmean-fieldtheoryagainbreaksdownat
Schwinger-boson and modified-spin-wave [41] findings,
the particular temperature (3.1). Considering that nu-
togetherwiththenonlinear-σ-modelcalculations[59,60],
merical tools less work at low temperatures, we realize
commonlysuggestthatthe Haldanegapisasimplyacti-
the superiority of the new-scheme-based modified spin-
vated function of temperature. Extensive measurements
wave theory all the more.
onspin-1antiferromagneticHeisenbergchaincompounds
[61–63] also report that the Haldane massive mode is Finally we calculate another type of ferrimagnet in
shifted upward with increasing temperature. We en- order to demonstrate the constant applicability of the
courageneutron-scatteringexperimentsonferrimagnetic present new scheme. Figure 5 shows the thermody-
chain compounds to solve the present disagreement be- namic properties of the ferromagnetic-ferromagnetic-
tween the Schwinger-boson and modified-spin-wave cal- antiferromagnetic-antiferromagnetic bond-tetrameric
culations of the antiferromagnetic excitation gap as a spin-1 Heisenberg chain,
2
6
2.0 6.0 3.0
QMC
SB
LMSW(Takahashi)
2)B 4.0 PLIMMSSWW((YTaamkaahmaoshtoi)) 2.0
mg PIMSW(Yamamoto)
N(1.0
T/
kB 2.0 1.0
c
(a) (1,1/2) (b) (3/2,1/2) (c) (3/2,1)
0.0 0.0 0.0
0.0 1.0 2.0 3.0 0.0 1.0 2.0 3.0 0.0 1.0 2.0 3.0
k T/J
B
FIG. 4. The Schwinger-boson (SB), linear-modified-spin-wave (LMSW), perturbational interacting-modified-spin-wave
(PIMSW), and quantum Monte Carlo (QMC) calculations of the susceptibility-temperature product χT as a function of
temperatureforthespin-(S,s)ferrimagneticHeisenbergchains. Themodifiedspinwavesareconstructedintwodifferentways,
theTakahashi scheme (Takahashi) and the new scheme(Yamamoto).
1.2 2.0 1.0 0.4
(a) QMC (b) (a) QMC (b)
0/CNkB .9 m2T/N(g)BB LPIMMSSWW((YYaammaammoototo)) 0/LkCB .8 m2J/L(g)B SLPFBIDMMISMSWWS(W(TTa(aTkkaaahkhaaashshahis)ih)i)
0.3 k c
c 0.2
0.0 0.0 0.0 0.0
0.0 1.0 kBT / J 2.0 3.0 0.0 1.0 kBT / J 2.0 3.0 0.0 1.0 kBT / J 2.0 3.0 0.0 1.0 kBT / J 2.0 3.0
FIG. 5. The linear-modified-spin-wave (LMSW), pertur- FIG. 6. The
bationalinteracting-modified-spin-wave(PIMSW),andquan- linear-modified-spin-wave (LMSW), perturbational interact-
tum Monte Carlo (QMC) calculations of the specific heat C ing-modified-spin-wave (PIMSW), full-diagonalization inter-
and the susceptibility-temperature product χT as functions acting-modified-spin-wave (FDIMSW), and quantum Monte
of temperature for the spin-1 bond-tetrameric ferrimagnetic Carlo (QMC) calculations of thespecificheat C andthesus-
2
Heisenberg chain of JF = JAF. The modified spin waves are ceptibility χ as functions of temperature for the spin-1 anti-
constructed on thenew scheme. ferromagneticHeisenbergchain. Themodifiedspinwavesare
constructed on theTakahashi scheme.
N
= J (S S +S S )
AF 4n−3 4n−2 4n−2 4n−1
H · ·
n=1
X(cid:2) IV. SUMMARY AND DISCUSSION
J (S S +S S ) , (3.2)
F 4n−1 4n 4n 4n+1
− · ·
where we have set all the g-factors equal fo(cid:3)r simplicity. We have demonstrated the Schwinger-boson mean-
The new modified spin-wave scheme again successfully field representation and the modified spin-wave treat-
reproduces the Schottky peak of the Specific heat. The ment of one-dimensional Heisenberg ferrimagnets. The
interactingmodifiedspinwavesfurtherinterpretthelow- Schwingerbosonsformanexcellentlanguageatlowtem-
temperature Shoulder-like structure. The characteris- peratures but rapidly lose their validity with increasing
ticminimumofthesusceptibility-temperatureproductis temperature. The modified spin-wave theory is more re-
unfortunately less reproduced but the calculation again liable in totality provided the number of bosons is con-
correctly gives the paramagnetic susceptibility at suffi- trolled without modifying the native energy structure.
ciently high temperatures. A recent experiment [64] on On the other hand, the Schwinger-boson representation
asingle-crystalsampleofCu(C H NCl) (N ) [7],which can be extended to anisotropic systems [67] more rea-
5 4 2 3 2
maybedescribedbytheHamiltonian(3.2),hasreported sonably because it is rotationally invariant in contrast
that the specific heat exhibits a double-peakedstructure to the modified spin-wave theory. While the tempera-
as a function of temperature. There is indeed a possi- turedependence ofthe anitiferromagneticexcitationgap
bility of an additional peak appearing at low tempera- ω+ islefttosolveexperimentally,wearenowconvinced
k=0
tures as the ratio J /J moves away from unity [65]. that the bosonic languages remain effective in low di-
F AF
However, no parameter assignment has yet succeeded in mensions and may be applied to extensive ferrimagnets
interpreting all the observations consistently. There are [68]. Besidesground-statepropertiesandthermodynam-
further chemical attempts to synthesize novel ferrimag- ics, quantum spin dynamics [69,70] can be investigated
nets. Organic ferrimagnets [6,66] are free from magnetic through the modified spin-wave scheme.
anisotropyandthussuitableforanalyzingintermsofthe We further mention our findings in the antiferromag-
modified spin waves. netic limit with the view of realizing the close relation
7
between the two bosonic languages. We equalize s with fermioniclanguage[74,75],whichisinprinciplecompact,
S and set 2N, the number of spins, equal to L for the being superior to any bosonic representation.
Hamiltonian (2.1). At S = s, the ground-state sublat- In the case of ferromagnets, the Holstein-Primakoff
tice magnetization(2.38)diverges and therefore the new bosons are already diagonal in the momentum space
modified spin-wave scheme is no more applicable. We [37,56],suggestingnoquantumfluctuationintheground
havetosettlethequantum,aswellasthermal,divergence state, and therefore the present new scheme turns out
inevitably employing the Takahashi scheme. Besides the equivalent to the Takahashi scheme. The new-scheme-
perturbationaltreatmentof ,wemayconsiderthefull based modified spin-wave theory is the very method for
0
H
diagonalization of + , where the ground-state en- low-dimensional ferrimagnets and is ready for extensive
1 0
H H
ergy and the dispersion relations are still given by Eqs. explorations.
(2.32) and (2.33), respectively, but with θ satisfying
k
γ (k)+γ (k)=0. (4.1)
1 0 ACKNOWLEDGMENTS
Such an idea applied to ferrimagnets ends in gapped
e
ferromagnetic excitations and misreads the low-energy The authors are gratefulto M. Takahashiand H. Hori
physics. The perturbational series-expansion approach for useful discussions. This work was supported by the
is highly successful in the case of ferrimagnets [32,33]. Ministry of Education, Culture, Sports, Science, and
Focussing ourinterestonHaldane-gapantiferromagnets, Technology of Japan, and the Nissan Science Founda-
welistthebosoniccalculationsoftheground-stateprop- tion.
erties in Table II. The bosonic languages interpret
the ground-state correlation very well but underesti-
mate the Haldane gap considerably. Indeed they can-
not detect the topological terms responsible for vanish-
ing gap [72], but they are still qualitatively consistent
with the nonlinear-σ-model quantum field theory, yield-
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9
TABLE I. The Schwinger-boson (SB), linear-modified-spin-wave (LMSW), perturbational interacting-modified-spin-wave
(PIMSW),andnumericaldiagonalization (Exact)calculations oftheground-stateenergyEg andthezero-temperatureantifer-
romagnetic excitation gap ∆0 for thespin-(S,s) ferrimagnetic Heisenberg chains.
(S,s)=(1,1) (S,s)=(3,1) (S,s)=(3,1)
Approach 2 2 2 2
Eg/NJ ∆0/J Eg/NJ ∆0/J Eg/NJ ∆0/J
SB −1.45525 1.77804 −1.96755 2.84973 −3.86270 1.62152
LMSW −1.43646 1 −1.95804 2 −3.82807 1
PIMSW −1.46084 1.67556 −1.96983 2.80253 −3.86758 1.52139
Exact −1.4541(1) 1.759(1) −1.9672(1) 2.842(1) −3.861(1) 1.615(5)
TABLE II. The Schwinger-boson (SB), linear-modified-spin-wave (LMSW), perturbational interacting-modified-spin-wave
(PIMSW),full-diagonalization interacting-modified-spin-wave(FDIMSW),andquantumMonteCarlo(QMC)[71]calculations
of the ground-stateenergy Eg and thelowest excitation gap ∆0 for thespin-S antiferromagnetic Heisenberg chains.
S =1 S =2 S =3
Approach
Eg/LJ ∆0/J Eg/LJ ∆0/J Eg/LJ ∆0/J
SB −1.396148 0.08507 −4.759769 0.00684 −10.1231 0.00295
LMSW −1.361879 0.07200 −4.726749 0.00626 −10.0901 0.00279
PIMSW −1.394853 0.07853 −4.759760 0.00655 −10.1231 0.00287
FDIMSW −1.394617 0.08507 −4.759759 0.00684 −10.1231 0.00295
QMC −1.401481(4) 0.41048(6) −4.761249(6) 0.08917(4) −10.1239(1) 0.01002(3)
10