Modified Spin Wave Analysis of Low Temperature Properties of Spin-1/2 Frustrated Ferromagnetic Ladder

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by  Kazuo Hida

JournalofthePhysicalSocietyofJapan FULL PAPERS
Modified Spin Wave Analysis of Low Temperature Properties of
Spin-1/2 Frustrated Ferromagnetic Ladder
Kazuo Hida∗ and Takashi Iino†
Division of Material Science, Graduate School of Science and Engineering,
Saitama University, Saitama, Saitama, 338-8570
2 (Received November28,2011)
1
Low temperature properties of the spin-1/2 frustrated ladder with ferromagnetic rungs and legs, and
0
two different antiferromagnetic next nearest neighbor interaction are investigated using the modified spin
2
wave approximation in the region with ferromagnetic ground state. The temperature dependence of the
n
magnetic susceptibility and magnetic structure factors is calculated. The results are consistent with the
a
numerical exact diagonalization results in the intermediate temperature range. Below this temperature
J
range, the finite size effect is significant in the numerical diagonalization results, while the modified spin
0
waveapproximationgivesmorereliableresults.Thelowtemperaturepropertiesnearthelimitofthestability
2
of the ferromagnetic ground state are also discussed.

] KEYWORDS: frustration,ferromagneticladder,modifiedspinwave,magneticsusceptibility,magneticstructure
l
e factor
-
r
t
s j=2
.
t
a 1. Introduction
m
Among a variety of models and materials with strong
- quantum fluctuation, quantum spin ladders have been j=1
d i−1 i i+1 i+2
attracting the interest of many condensed matter physi-
n
o cists. Recently, a spin-1/2 ferromagnetic ladder com- Jl Jd1=Jd(1+δ)
c pound CuIICl(O-mi)2(µ-Cl)2 (mi = 2-methylisothiazol- J J =J (1−δ)
[ 3(2H)-one)is synthesized with possible frustrated inter- r d2 d

1 rung interaction.1 Fig. 1. Structure of frustrated ladder studied in the present
v Although frustrated and unfrustrated spin ladders work.
2 with nonmagnetic ground states have been extensively
4 studied,2–5 those with ferromagnetic ground states have
2
been less studied. This wouldbe due to the simplicity of
4
. the ferromagneticgroundstate.Evenifthe groundstate value of the total magnetization Sz vanishes as
1 tot
is ferromagnetic, however, the excited states and finite
0 Sz =0 (1)
2 temperature properties should be influenced by frustra- h toti
1 tion. Especially, near the stability limit of the ferromag- taking account of the Mermin-Wagner theorem.12
: netic groundstate,weexpectcharacteristictemperature This procedure is quite successful for one and two-
v
i dependence of physical quantities. In the present work, dimensional ferromagnets6–8 and two-dimensional anti-
X
we investigate the finite temperature properties of frus- ferromangets.9–11 Recently, this method has been suc-
r tratedspin ladderswith ferromagneticgroundstates us- cessfully applied to dimerizedferromagnetic chains.13 In
a
ing the modified spin wave (MSW) approximation6–11 thepresentwork,weemploythismethodtocalculatethe
and the numerical exact diagonalization (ED) calcula- low temperature magnetic susceptibility of the present
tion for short ladders. model with ferromagnetic ground states. We also calcu-
The MSW approximationwas firstproposedby Taka- late the magnetic structure factor to confirm that the
hashi6 to investigate the low temperature properties of short range correlation is properly described within the
ferromagnetic chains. If the conventional spin wave ap- MSW approximation.
proximationis applied to one-dimensionalferromagnets, Thispaperisorganizedasfollows.Inthenextsection,
the thermal fluctuation of magnetization diverges at fi- themodelHamiltonianisintroduced.TheMSWanalysis
nite temperatures. This is natural, considering the ab- is explainedin 3.In 4,the MSW equationsarenumer-
§ §
sence of finite temperature long range order in one di- ically solved and the results are compared with the ED
mension.12 This divergence, however, prevents the cal- calculation for short chains. The last section is devoted
culation of physical properties such as magnetic suscep- tosummaryanddiscussion.Thelowtemperatureexpan-
tibility and magnetic structure factor at finite tempera- sion of the MSW equations is explained in Appendix.
tures. To circumvent this difficulty, Takahashi proposed
to introduce the chemical potential µ for magnons and
to fix µ by imposing the constraint that the expectation
∗E-mailaddress:[email protected]
J.Phys. Soc. Jpn. FULL PAPERS
2. Hamiltonian (ak,2 eikak,1)(ak′,2 eik′ak′,1)+h.c (7)
× − −
We consider the frustrated ferromagnetic ladder with o
up to the quartic orderin a anda† .Here, E andE
spin-1/2 described by the Hamiltonian k,i k,i 1 2
are defined by
L
= [J(S S +S S ) E1 2Jl(1 cosk) Jr Jd1 Jd2 S,
H l i,1 i+1,1 i,2 i,2 ≡{− − − − − } (8)
Xi=1 E2 (Jr+eikJd1+e−ikJd2)S.
+J S S +J S S +J S S ], (2) ≡
r i,1 i,2 d1 i,1 i+1,2 d2 i+1,1 i,2 We introduce the unitary transformation
where S (i = 1,...,L,j = 1,2) are the spin opera-
tors withi,jmagnitude S. For the numerical calculation, α†k ≡ √12 a†k1eiφ(2k) +a†k2e−iφ(2k) ,
we only consider the case of S = 1/2. The number of (cid:16) (cid:17) (9)
the unit cells is denoted by L.In this paper,we focus on β† 1 a† eiφ(2k) a† e−iφ(2k) ,
the case J,J <0 (ferromagnetic) and J ,J >0 (an- k ≡ √2 k1 − k2
l r d1 d2
(cid:16) (cid:17)
tiferromagnetic). The lattice structure is schematically and the corresponding magnon number operators
depicted in Fig. 1. We also use the parameterization
nˆ =α†α , nˆ =β†β . (10)
αk k k βk k k
J =J (1+δ), J =J (1 δ) (3)
d1 d d2 d
− The phase φ(k) is determined afterwards to minimize
if convenient. the free energy. We assume the density matrix for the
magnonsin the formofa product ofsingle-magnonden-
3. Modified Spin Wave Analysis
sity matrices as
3.1 Formulation
We employ the standard Holstein-Primakoff transfor- ρ= Pαk(nαk)Pβk(nβk) nαk,nβk nαk,nβk ,
{ } { }
mation, {nαXk,nβk}Yk (cid:12) (cid:11)(cid:10) (cid:12)
(cid:12) (11) (cid:12)
S+ =Sx +iSy =√2Sf (S)a ,
i,j i,j i,j i,j i,j
where
Si−,j =Six,j −iSiy,j =√2Sa†i,jfi,j(S), |{nαk,nβk}i= (nαk!nβk!)−12(α†k)nαk(βk†)nβk|0i (12)
Siz,j =S−a†i,jai,j, Yk
is the eigenstate of nˆ and nˆ with eigenvalues n
αk βk αk
f (S)= 1 (2S)−1a† a (i=1,...,L;j =1,2), and n . In the following, the expectation value with
i,j − i,j i,j βk
respect to (11) is denoted by ... . The probability that
q (4)
h i
the state withwavenumber k is occupiedby n bosonsis
wherea†i,j andai,j aremagnoncreationandannihilation denotedbyPνk(n)(ν =αorβ).Therefore,thefollowing
operators on the i-th rung and the j-th leg. After the normalization conditions are required for all k.
Fourier transformation with respect to i, the Hamilto-
∞ ∞
nian is rewritten as P (n)=1 , P (n)=1. (13)
αk βk
= 0+ 1, (5) nX=0 nX=0
H H H
Accordingly, the free energy is given by
E E a
H0 =Xk (cid:16)a†k,1a†k,2(cid:17)(cid:18) E21∗ E21 (cid:19)(cid:18) akk,,12 (cid:19), (6) F =hH0i+hH1i+T ∞ Pνk(n)lnPνk(n)
= Jl 2 e−i(k′−q)(1 eik′)(1 eik) νX=α,βXk nX=0 (14)
1
H −4L − −
kXk′qXj=1n where the expectation values 0 and 1 are ex-
hH i hH i
+eik(1 e−i(k+q))(1 e−i(k′−q)) pressed as follows
− −
1
×a†k+q,ja†k′−q,jak′,jak,j o hH0i= k (cid:26)E1n˜k+ 2(E2e−iφ(k)+E2∗eiφ(k))δn˜k(cid:27),
X
J (15)
r a† a†
− 4L k+q,1 k′−q,2 2
kXk′qn Jl
= (1 cosk)n˜
1 k
×(ak,1−ak,2)(ak′,1−ak′,2)+h.c} hH i 2L Xk − !
− J4dL1 e−i(k′−q)a†k+q,1a†k′−q,2 + Jr (n˜ δn˜ cosφ(k)) 2
kXk′qn 4L" k− αk #
k
(ak,1 eikak,2)(ak′,1 eik′ak′,2)+h.c X 2
× − − J
d1
− J4dL2 e−i(k′−q)a†k+q,2a†k′−q,1 o + 4L"Xk (n˜k−δn˜kcos(φ(k)−k))#
kXk′qn
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2
J
d2
+ (n˜ δn˜ cos(φ(k)+k)) , (16) Substituting (21) and (22) into (24) and (25), we find
k k
4L" − #
Xk 1
S (q)=S+ n˜ n˜ , (26)
where intra 4L k+q/2 k−q/2
k
X
n˜ = n = nP (n) (ν =α,β). (17)
νk h νki n νk Sinter(q)=Sicnter(q)+iSisnter(q), (27)
X
1
We also define Sc (q)= cos(φ(k+q/2) φ(k q/2))
inter 4L − −
n˜ =n˜ +n˜ , δn˜ =n˜ n˜ (18) k
k αk βk k αk βk X
−
δn˜ δn˜ , (28)
for convenience. The condition (1) reduces to × k+q/2 k−q/2
1 Ss (q)= 1 sin(φ(k+q/2) φ(k q/2))
S = 2L n˜k. (19) inter 4L − −
k Xk
X
Introducing the Lagrangianmultipliers µνk and µ which ×δn˜k+q/2δn˜k−q/2. (29)
accountfortheconstraint(13)and(19),weminimizethe
It should be noted that S (q) has an imaginary part
inter
following quantity W with respect to P and φ(k).
νk because ofthe absenceofspaceinversionsymmetry x
↔
∞ ∞ x.
W =F µ P (n) µ nP (n)−.
νk νk νk
− −
ν=α,β k n=0 ν=α,β k n=0 3.4 Lowest order approximation
X X X X XX
(20)
To the lowest order approximation, we only consider
the Hamiltonian and impose the constraint (1). In
0
3.2 Spin-Spin Correlation Function and Magnetic Sus- H
the following, we call this approximationthe MSW0 ap-
ceptibility
proximation. Minimizing
The spin-spin correlation function is expressed in
∞
terms of the magnon occupation numbers as
W = T µ P (n)
0 0 νk νk
2 hH i− S−
hSi1Si′1i=hSi2Si′2i= 21L cosk(xi′ −xi)n˜k! , ∞νX=α,βXk nX=0
Xk µ nPνk(n) (30)
(21) −
ν=α,β k n=0
X XX
2
1 with respect to φ, we find
Si1Si′2 = cos(φ(k) k(xi′ xi))δn˜k ,
h i 2L k − − ! ∂W0 = ∂hH0i = i(E e−iφ(k) E∗eiφ(k))δn˜ =0.
X (22) ∂φ(k) ∂φ(k) −2 2 − 2 k
(31)
where x is the position of the i-th site. Using these ex-
i
pressions, the magnetic susceptibility per spin χ is ex- This leads to
pressed as
J sinφ(k)+J sin(φ(k) k)+J sin(φ(k)+k)=0,
r d1 d2
−
(gµ )2 2 2 (32)
χ= 2LBT hSizjSiz′j′i which determines the phase φ(k) as
ii′ j=1j′=1
XXX
(J J )sink
(gµ )2 2 2 tanφ(k)= d1− d2 ( π/2<φ<π/2).
= B Sij Si′j′ Jr+(Jd1+Jd2)cosk −
6LT h · i
ii′ j=1j′=1 (33)
XXX
= (gµB)2 (n˜2 +n˜2 +n˜ +n˜ ), (23) Minimizing W0 with respect to Pνk, we find
6LT αk βk αk βk ∂W
Xk 0 =nεν(k)+T(lnPνk+1) nµ µνk =0 (ν =α,β),
∂P − −
where µ is the Bohr magneton and g is the gyromag- νk
B (34)
netic ratio.
where
3.3 Magnetic Structure Factor 1∂
0
ε (k)= hH i (35)
We define the intralegandinterlegmagnetic structure ν
n ∂P
νk
factors as
are two branches of the magnon excitation energy given
1
S (q)= S S exp(ix q) by
intra 0,1 i,1 i
L h i
i 1
= 1 X S S exp(ix q), (24) εαβ(k)=E1± 2(E2e−iφ(k)+E2∗eiφ(k))
0,2 i,2 i
L h i =S 2J(1 cosk) J (1 cosφ(k))
Xi {− l − − r ∓
Sinter(q)= 1 S0,1Si,2 exp(ixiq). (25) −Jd1(1∓cos(k−φ(k))−Jd2(1∓cos(k+φ(k))}.
L h i (36)
X
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The Hamiltonian is rewritten as is the dressed single magnon excitation energy given by
0
H
H0 = α†k βk† εα0(k) εβ0(k) αk βk . (37) ε˜αβ(k)=−2Jl(1−cosk)S1′ −Jr(1∓cos(φ(k))S2′
Xk (cid:0) (cid:1)(cid:18) (cid:19)(cid:0) (cid:1) −Jd1(1∓cos(φ(k)−k))S3′ −Jd2(1∓cos(φ(k)+k))S4′.
Eliminating µ from (34) using the condition (13), we (47)
νk
find
ExpressionsforP (n)andn˜ areobtainedbyreplacing
νk νk
P (n)= 1 e−(εν(k)−µ)/T e−n(εν(k)−µ)/T. (38) ε byε˜ in(38)and(39),respectively.Thephaseφ(k)
νk νk νk
−
and chemical potential µ are determined by solving (42)
n o
The expectation value of n is given by
νk under the condition (19) numerically.
∞
1
n˜νk = nPνk(n)= e(εν(k)−µ)/T 1. (39) 3.6 Low Temperature Behavior
nX=0 − We employ the MSW0 approximation to investigate
The chemical potential µ is determined using the condi- the low temperature behavior. Among two branches of
tion (19). themagnonexcitation,ε (k)hasanenergygapatk =0.
β
Therefore, we only consider ε (k) at low temperatures.
α
3.5 O(S0) approximation Near k =0, the dispersion relation is given by
In this approximation, we include the terms of O(S0)
ε (k) JSk2, (48)
in the Hamiltonian i.e. 1. We call this approximation α ≃
H
the MSW1 approximation.Minimizing where
∞ Jr(Jd1+Jd2)+4Jd1Jd2
J J + (49)
W1 =hH0i+hH1i−TS− µνk Pνk(n) ≡−(cid:20) l 2(Jr+Jd1+Jd2) (cid:21)
ν=α,β k n=0
X X X is the effective ferromagnetic exchange interaction be-
∞ tween the ferromagnetic rung dimers consisting of S
µ nP (n) (40) i,1
− νk and Si,2. For negative (ferromagnetic) Jl and Jr, J is
ν=α,β k n=0
X XX positive as long as the frustrating antiferromagnetic in-
with respect to φ(k), we find teraction J and J are small. However, with the in-
d1 d2
crease of J and J , J decreases and vanishes at
∂W ∂ ∂ d1 d2
1 0 1
= hH i + hH i =0 (41)
∂φ(k) ∂φ(k) ∂φ(k) J (J +J )+4J J J +2J (1 δ2)
r d1 d2 d1 d2 r d
J = = − J .
which yields ls − 2(Jr+Jd1+Jd2) − Jr+2Jd d
(50)
J sinφ(k)S′ +J sin(φ(k) k)S′
r 2 d1 − 3 Thispointisthelimitofthestabilityoftheferromagnetic
+Jd2sin(φ(k)+k)S4′ =0, (42) ground state.
ForJ >0,thelowtemperatureexpansioncanbecar-
where
ried out in the same manner as the ferromagnetic chain.
1
S′ = coskn˜ , The details are explained in Appendix. As a result, we
1 2L k
k obtain
X
S2′ = 21L cos(φ(k))δn˜k, χ 8(gµB)2S4J 1 3 ζ 12 T + 3 ζ2 21 T .
Xk ≃ 3T2 − 4S √(cid:0)π(cid:1)sJS 16S2 π(cid:0) (cid:1)JS!
1 (51)
S′ = cos(φ(k) k)δn˜ ,
3 2L − k
k Namely, the susceptibility is proportional to T−2 at low
X
1 temperatures. Defining J˜ J/2,Seff 2S, the sus-
S4′ = 2L cos(φ(k)+k)δn˜k. (43) ceptibility per rung is given≡by ≡
k
X χ =2χ
rung
From eq. (42), φ(k) is given as
tanφ(k)= (JrS2′(J+d1(SJ3d′1−S3′J+d2SJ4d′)2Ssi4′n)kcosk). (44) ≃ 2(gµB3)T2S2e4ffJ˜ 1− 3√ζ(cid:0)2π12(cid:1)s2JT˜Se3ff + 3ζ22π(cid:0)21(cid:1)2JT˜Se3ff!.
(52)
Minimizing W with respect to P , we find
1 νk
∂W This coincides with the low temperature susceptibility
1
∂P =nε˜νk+T(lnPνk+1)−nµ−µνk =0, (45) of the ferromagnetic chain with spin Seff and effective
νk exchange interaction J˜.6
where
1 ∂
ε˜ = ( + ) (46)
νk 0 1
n∂P (n) hH i hH i
νk
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χχTT22[[ccmm−−33mmooll KK22]] 2
(a) ED L=6 MSW0
J=−14.47K J=−5.64K ED L=8 MSW1
r l
ED L=10
J =3.91K J =0
d1 d2 J=−14.47K J=−5.64K
4 r l
g=2.13
J =3.91K J =0
d1 d2
T=2K
S (q)
intra
1
2
MSW0
MSW1 Ss (q) Sicnter(q)
Extrapolation from L=6,8,10 inter
Experiment 0
0 1 2 3
0 q
0 1 2 T1/2[K1/2] 3
2
(b) ED L=6 MSW0
Fig. 2. Magnetic susceptibility for exchange parameters corre- ED L=8 MSW1
sponding to [CuIICl(O-mi)2(µ-Cl)2] calculated by MSW0 (open ED L=10
J=−14.47K J=−5.64K
squares) andMSW1 (filledsquares) approximations. Opencircles r l
are Shanks extrapolation from the ED data for L = 6,8 and 10. J =3.91K J =0
d1 d2
Thefilledcirclesareexperimental datatakenfromref.1.
T=3K
1 Sintra(q)
4. Numerical Results Sc (q)
inter
4.1 Results for the parameter set corresponding to
CuIICl(O-mi)2(µ-Cl)2 Ss (q)
inter
The magnetic susceptibility is calculated for the pa-
rameter set determined in ref. 1 for CuIICl(O-mi)2(µ- 00 1 2 3
q
Cl) as shown in Fig. 2. The susceptibility calculated
2
2
by two MSW approximations and the Shanks extrap-
(c) ED L=6 MSW0
olation14 from the ED data for L = 6,8 and 10 are ED L=8
shown.TheexperimentaldataforCuIICl(O-mi)2(µ-Cl)2 ED L=10
J=−14.47K J=−5.64K
are also shown. Although Fig. 2 contains only a limited r l
numberofexperimentaldata,therearemanydatapoints Jd1=3.91K Jd2=0
at higher temperatures and the exchange constants are T=4K
determined by fitting them with the ED calculation as 1 S (q)
intra
explained in ref. 1.
TheMSW0approximationreproducestheoveralltem-
perature dependence including relatively higher temper- Sc (q)
inter
atureregime.Atlowtemperatures,wheretheEDresults
are strongly size dependent, both MSW approximations
Ss (q)
give the consistent results. The result of the MSW1 ap- inter
proximationismoresmoothlyconnectedtotheEDdata. 00 1 2 3
q
However, it strongly deviates from the ED data as the
temperature is raised. Actually, the MSW1 equations
have no solution other than φ=0,S′ =S′ =S′ =0 for Fig. 3. Magnetic structure factors with exchange parameters
1 2 3 corresponding to CuIICl(O-mi)2(µ-Cl)2 calculated by MSW0
T T∗ 4K.ThisimpliesaphasetransitionatT =T∗.
≥ ≃ (small open circles) and MSW1 (small filled circles) approxima-
However, this transition is spurious because the present tions. The ED data for L = 6 (big filled squares), 8 (big open
systemisone-dimensionalandnophasetransitionshould squares), and 10 (big open circles)are also shown. Temperatures
take place at finite temperatures. Therefore, the MSW1 are(a)2K,(b)3K,and(c)4K.
approximation is unreliable at T>O(T∗). This spurious
transition takes place even in the∼absence of frustration
withintheMSW1approximation.Asimilarkindofspuri-
oustransitiontakesplaceinthesquarelatticeHeisenberg isappropriatelytakenintoaccountbytheMSWapproxi-
antiferromagnet.9Hence,thisisanartifactoftheMSW1 mations,themagneticstructurefactorsareshowninFig.
approximation rather than the effect of frustration. On 3. With the decrease of temperature, the short range
thecontrary,theMSW0approximationpredictsnophase ferromagnetic order develops as expected. The results
transition and gives the results qualitatively consistent of ED calculation are also presented in Fig. 3 for the
with the ED results up to relatively high temperatures. samechoiceofexchangeconstants.Althoughthe MSW1
To confirm that the magnetic short range correlation approximation reproduces the ED results better than
J.Phys. Soc. Jpn. FULL PAPERS
−1
Jl (a) δ=0.8 Jd=1 χ102 JJr==−18 δJ=l=0J.l8c(Jr,δ)≅−1.213333
nonmagnetic ____ d
(gµ )2 MSW0
B
MSW1
101 ED
100
Ferro
10−1
−2 10−2 10−1 T 100
−8 −6 −4 J −2
r
−1 Fig. 5. Magnetic susceptibility at the ferromagnetic-
Jl (b) δ=0.5 Jd=1 nonmagnetic transition point (Jr = −8,δ = 0.8,Jl =Jlc(Jr,δ) ≃
−1.213333) which coincides with the limit of the stability of the
nonmagnetic
ferromagneticgroundstateJls.TheMSW0andMSW1resultsare
shownbyopencirclesanddoublecircles,respectively.Theextrap-
olatedvalues fromtheEDresultswithL=6,8and10areshown
by open squares. The curves fitted by χ ≃ (A+BT1/4)T−4/3
−1.5 are also shown. For the fitting of the MSW0 data, A is fixed to
0.1795937 asobtainedinAppendix.
Ferro
MSW0approximationbehavesasT−4/3 asshowninAp-
pendix.AsanexampleofthecaseJ =J ,thetempera-
−2 lc ls
−6 −4 −2 turedependenceofthesusceptibilityforJ = 8,J =1
Jr r − d
andδ =0.8andJ =J 1.213333isshowninFig.5.
l ls ≃−
TheresultscalculatedbytheMSW0andMSW1approx-
Fig. 4. GroundstatephasediagramforJd=1with(a)δ=0.8
and (b) δ =0.5. The thick line is the ferromagnetic-nonmagnetic imations,andthoseextrapolatedfromtheEDresultsfor
transition line and the dotted line is the limit of the stability of L=6,8and10areshown.Allresultsshowthe behavior
theferromagneticstategivenby(50). χ T−4/3. However, the ampltude of the susceptibility
∼
is overestimated by the MSW approximations. We may
understand this discrepancy in the following way:
Even at J = J , the ferromagnetic state remains
l ls
MSW0atT =2K,itbecomespooratT =3Kwherethe one of the ground states. In addition, the nonmagnetic
susceptibility also deviates from the ED result. On the state which replace the ferromagnetic states for J >J
l ls
other hand, the MSW0 approximation gives reasonable comes into play. However, this state has no magnetic
agreement with the ED results even at T = 4K where moment and does not contribute to the susceptibility.
the MSW1 equations only have an unphysical solution. Also, the low-lying excited states around the nonmag-
netic states have small magnetic moments and do not
4.2 Results at the ground state phase transition point havesignificantcontributiontothesusceptibility.Never-
Withtheincreaseofthefrustration,theferromagnetic theless, these states havefinite statisticalweight.There-
ground state becomes unstable againstmagnon creation fore, the contribution from the ferromagnetic state and
atJ =J andthetransitiontothenonmagneticground the excitations around it, which is correctly described
l ls
state takesplace. However,the groundstate phase tran- by the MSW approximations, can reproduce the lead-
sitiondoesnotalwaystakeplaceatJ =J .Theground ing temperature dependence of magnetic susceptibility,
l ls
state can change from ferromagnetic to nonmagnetic at while its actual amplitude is reduced from the results of
J = J ( J ) where the ground state energy of the the MSW approximations.
l lc ≤ ls
nonmagnetic state becomes equal to that of the ferro- These nonmagnetic states have antiferromagnetic or
magnetic one. Examples of ground state phase bound- incommensurate short range correlations induced by
aries and stability limits of the ferromagnetic state are frustration.Togetmoreinsightintotheireffects,wecal-
presented in Fig. 4(a) for δ = 0.8 and in Fig. 4(b) for culate the magnetic structure factor for this parameter
δ = 0.5. The ferromagnetic-nonmagnetic phase bound- set as shown in Fig. 6. It is clearly observed that the
ary is determined by the ED method for L = 10. It is q = 0 component, which is responsible for the suscep-
checkedthatthesizedependenceisnegligibleinthescale tibility, is overestimated in the MSW0 approximation.
of this figure. It should be noted that the size dependence of the ED
First,letusexaminethebehaviorofthemagneticsus- resultsforS (q =0)andS (q =0)is almostnegli-
intra inter
ceptibility at J = J in the case J = J . On the gible.Therefore,thisdiscrepancyisnotattributedtothe
l lc lc ls
line J =J , the excitation energy ε (k) is proportional finite size effect. On the other hand, the large q compo-
l ls α
to k4. In this case, the susceptibility calculated by the
J.Phys. Soc. Jpn. FULL PAPERS
1
J=−8J=1 δ=0.8 MSW0 ED
Jr=J (δ,dJ) L=10 8 6 MSW0
≅l−1.lc2133r33 T=1 L=10
T=0.2 L=8
S (q) T=0.1
1 intra L=6
T1/3SMSW0(q=0)
intra
0.5
T1/3SED(q=0)
0.5 intra
Sc (q)
inter
Ss (q) T1/3∆Sintra(q=0)
inter
0
0 1 2 3 0
qqq 0 0.2 0.4
T
Fig. 6. Magnetic structure factors on the ground state phase
boundarycalculatedbyMSW0approximation(lines)andED(open Fig. 7. TemperaturedependenceofSintra(q=0)ontheground
statephaseboundarycalculatedbytheMSW0approximation(+)
symbols)methods.TemperaturesareT =1(dottedlines,circles),
0.2(brokenlines,squares)and0.1(solidlines,triangles).Thebig, andEDmethods(opensymbols).Thedifference∆Sintra(q=0)(=
medium and small symbols represent the system sizes L = 10, 8 SiMntSrWa0(q=0)−SiEnDtra(q=0))betweentheMSW0andEDresults
isalsoshown(filledsymbols).AllquantitiesaremultipliedbyT1/3.
and6,respectively
χ
____ J=−4 J=−1.41
nents are underestimated in the MSW0 approximation. (gµ1B0)26 Jrd=1 δ=l0.5
Considering the sum rule MSW0
ED
S (q)= S S =S(S+1), (53)
intra h 0,1 0,1i 104
q
X
the enhancement of q = 0 components suppresses the
q =0 component for S 6 (q). For S (q), we have 102
intra inter
S (q)= S S . (54)
inter 0,1 0,2
q h i 100
X
In this case, lhs is not a constant. However, this should
be close to 1/4 if J is strongly ferromagnetic as in 10−4 10−2 T 100
r
the present case. Hence, the similar explanation is nat-
urally valid for S (q). Thus, we may conclude that Fig. 8. Magnetic susceptibility at the ferromagnetic-
inter
this discrepancy comes from the frustration induced an- nonmagnetic transition point (Jr = −4,Jl = −1.41,δ = 0.5
)whichisawayfromthelimitofthestabilityoftheferromagnetic
tiferromagnetic or incommensurate short range order
ground state. The MSW0 results are shown by open circles and
which is not appropriately described by the MSW0 ap- extrapolated values from the ED results with L = 6,8 and 10
proximation. Fig. 7 shows the temperature dependence are shown by open squares. The solid line is the fitted line by
of S (q = 0) calculated by the MSW0 approxima- χ∝const.×T−2
intra
tion (SMSW0(q = 0)) and that calculated with the ED
intra
method (SED (q =0)). The difference ∆S (q =0)(=
intra intra
SMSW0(q = 0) SED (q = 0)) is also plotted. All
intra − intra
quantities are multiplied by T1/3. This plot shows that the MSW andED methods. The magnetic susceptibility
T1/3∆S (q = 0) does not diverge in the low temper- andstaticmagneticstructurefactorsarecalculated.Itis
intra
ature limit. Hence, the correctionto SMSW0(q =0) does foundthattheMSWmethodgivesreasonableagreement
intra
not diverge with the power stronger than T−1/3 in the withtheEDcalculationintheintermediatetemperature
lowtemperaturelimit.Thismeansthatthe powerofthe regimebelowwhichtheEDresultsshowconsiderablesize
leading term of χ T−4/3 is unaffected by this correc- dependence.Onthecontrary,theMSWmethodbecomes
∼
tion. more reliable in the low temperature regime. Therefore,
If the limit of stability of the ferromagnetic phase J we conclude that the MSW method is useful as a com-
ls
is away from the ferromagnetic-nonmagnetic transition plementary method to the ED analysis even in the pres-
pointJ ,thesusceptibilitybehavesasT−2asinthecase ence of moderate frustration as far as the ground state
lc
of ferromagnetic ground state. An example is shown in remains the ferromagnetic state.
Fig. 8 for J = 4,J = 1 and δ = 0.5 and J = J It is also predicted that the MSW approximation is
r − d l lc ≃
1.41. reliable even at the limit of the stability of the ferro-
−
magnetic ground state J = J insofar as the exponent
l ls
5. Summary and Discussion
of the leading temperature dependence is concerned.Al-
Low temperature properties of the frustrated ferro- though we have explicitly demonstrated χ T−4/3 on
∼
magnetic ladders with spin-1/2 are investigated using the ferromagnetic-nonmagneticphaseboundary,thisbe-
J.Phys. Soc. Jpn. FULL PAPERS
haviorshouldbeobservedevensomewhatawayfromthe noninteger α.
transitionpointsatfinitetemperatures.Asthetempera-
∞
tureislowered,thecrossovertothetruelowtemperature F(α,v)=Γ(1 α)vα−1+ (l!)−1( v)lζ(α l).
− − −
behaviorin ferromagneticornonmagneticphasesshould
l=0
X
takeplace.Suchacrossoverbehaviorcanbe regardedas (A7)
·
aprecurserofthe groundstatephasetransitionifexper-
For the calculationof the susceptibility, the partially in-
imentally observed.
tegrated form
Inthepresentwork,wehaveconcentratedonthefinite
temperaturepropertiesintheparameterregimewithfer- ∞ uα−1eu+v
du=Γ(α)F(α 1,v) (A8)
romagnetic ground state. However, our preliminary cal- (eu+v 1)2 − ·
Z0 −
culationsuggeststhatthe nonmagneticphaseconsistsof
isuseful.Theequations(A3)and(A4)canbeexpressed
several different exotic phases with and without sponta- · ·
as
neous symmetry breakdown. The investigation of these
phases will be reported elsewhere. S Tn1 Γ 1 Γ 1 1 vn1−1+ζ 1 +O(v1) ,
The authorsthank M.KatoandA.Nagasawaforpro- ≃2πnAn1 (cid:18)n(cid:19)(cid:26) (cid:18) − n(cid:19) (cid:18)n(cid:19) (cid:27)
viding their experimental data and for discussion. The (A9)
·
numericaldiagonalizationprogramisbasedonthe pack-
ageTITPACKver.2codedbyH.Nishimori.The numer- χ (gµB)2 Tn1−1 Γ 1
icalcomputationin this workhasbeen carriedoutusing ≃ 3 2πnAn1 (cid:18)n(cid:19)
the facilities of the Supercomputer Center, Institute for 1 1
Solid State Physics, University of Tokyo and Supercom- Γ 2 vn1−2+ζ 1 +O(v1) .
× − n n −
putingDivision,InformationTechnologyCenter,Univer- (cid:26) (cid:18) (cid:19) (cid:18) (cid:19) (cid:27)
(A10)
sity of Tokyo, and Yukawa Institute Computer Facility ·
inKyotoUniversity.This workis supportedby a Grant- Solving (A9) with respect to v and substituting into
·
in-AidforScientificResearch(C)(21540379)fromJapan (A10), we find
·
Society for the Promotion of Science. χ (gµB)2S n−1 2nSAn1 sinπ nn−1 T−(n−n1)
Appendix ≃ 3 n n
(cid:16) (cid:17)
Let us define the density of states per site by 1 2n 1 1 Tn1
1+ − ζ Γ
w(x)= 1 δ(x ε (k))+δ(x ε (k)) . (A1) ×( n−1 (cid:18)n(cid:19) (cid:18)n(cid:19)2πnSAn1
α β
2L { − − } ·
2
k n(1 2n) 1 1 1
X − ζ2 Γ2 Tn2 .
At low temperatures, only the gapless magnon mode α −2(n−1)2 (cid:18)n(cid:19) (cid:18)n(cid:19)(cid:18)2πnSAn1 (cid:19) )
contributes. We consider the case where the dispersion
(A11)
relation of this gapless mode is given by ε (k) = Akn. ·
α
The density of states is then approximated as Within the ferromagnetic phase, n = 2. Setting A =
JS, we have
1 1
Rewwr(ixti)n≃g (21L9)Xakndδ((x2−3)εuαs(ikn)g)t≃he2πdnenAsin1tyxn1o−f1s.tate(sA, ·w2e) χ≃8(gµ3TB)22S4J (1− 43ST12ζπJ(cid:0)21(cid:1)S + 163πζS22(cid:0)J21(cid:1)ST).
(A12)
have p ·
∞ w(x)dx Tn1 1 1 At the stability limit of the ferromagnetic phase, n=4.
S = = F ,v Γ ,
Z0 exp(xT−1+v)−1 2πnAn1 (cid:18)n (cid:19) (cid:18)n(cid:19) Hence, we have
(A·3) χ (gµB)2S37A13234 1 7ζ 14 Γ 41 T14
(gµB)2 ∞ exp(xT−1+v)w(x)dx ≃ T43 ( − 3 (cid:0) 8(cid:1)πS(cid:0)A41(cid:1)
χ=
3T (exp(xT−1+v) 1)2
Z0 − 7 ζ 1 Γ 1 2
= (gµ3B)22Tπnn1A−1n1 Γ(cid:18)n1(cid:19)F (cid:18)n1 −1,v(cid:19), (A·4) +9 (cid:0)84π(cid:1)SA(cid:0)414(cid:1)! T21. (A·13)
where we define In the low temperature limit, we have
v µ/T (A5) χT4/3 A13
≡− · lim = (A14)
andF(α,v)istheBose-Einsteinintegralfunctiondefined T→0(gµB)2 2 ·
by for S = 1/2. Numerically A 0.04634074 for J =
r
≃
χT4/3
1 ∞ uα−1du 8,J = 1,δ = 0.8,J = J , 0.1795937. This
F(α,v)= . (A6) − d l lc (gµ )2 ≃
Γ(α) eu+v 1 · B
Z0 − value is used in the fitting of the MSW0 data in Fig. 5.
The behavior of this function for small v is known.15
For the present purpose, we only need the formula for
J.Phys. Soc. Jpn. FULL PAPERS
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