Critical exponent in the magnetization curve of quantum spin chains

4 Pages·0.12 MB·English

by  Toru Sakai

Critical exponent in the magnetization curve of quantum spin chains
Tˆoru Sakai1 and Minoru Takahashi2
1Faculty of Science, Himeji Institute of Technology, Kamigori, Ako-gun, Hyogo 678-12, Japan
2Institute for Solid State Physics, University of Tokyo, Roppongi, Minato-ku, Tokyo 106, Japan
(January 98)
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ThegroundstatemagnetizationcurvearoundthecriticalmagneticfieldH ofquantumspinchains
9 c
withthespingapisinvestigated. Weproposeasizescalingmethodtoestimatethecriticalexponent
1
δ definedasm∼|H−H |1/δ from finiteclustercalculation. Theapplications ofthemethodtothe
c
n S = 1 antiferromagnetic chain and S = 1/2 bond alternating chain lead to a common conclusion
a δ = 2. The same result is derived for both edges of the magnetization plateau of the S = 3/2
J
antiferromagnetic chain with thesingle ion anisotropy.
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2 PACS Numbers: 75.10.Jm, 75.40.Cx, 75.45.+j

]
h
c
e Themagnetizationcurveofquantumspinchainsshows Forthe S =1/2bondalternatingchainthe bosonization
m various nontrivial behaviors due to quantum effects. In method gave the same result. [15] δ = 2 was also de-
- the spin-gap systems, where a finite energy gap exists rived from the fermionic excitation with the dispersion
t in the spin excitation spectrum, the gap is controlled k2 which was numerically verified to be a good picture
a
t by applied magnetic field H through the Zeeman term for both systems. [3,4] In addition the argument of the
s
. in the Hamiltonian. The typical examples are the an- equivalence between the magnetization process of anti-
t
a tiferromagnetic chain with integer spin called Haldane ferromagnetic chains and some integrable models of the
m magnets [1], spin Peierls systems and spin ladders etc. crystal-shape profile lead to the same conclusion. [16] In
- In these systems a phase transition occurs at the criti- any theories giving δ = 2, however, the original spin
d cal field H corresponding to the amplitude of the gap Hamiltonians were mapped into other solvable models
c
n
[2–4]; the system has the nonmagnetic ground state and with some crucial approximations. Thus it would be im-
o
a finite gap for H < H , while the magnetic ground portant to estimate δ for the original Hamiltonian di-
c c
[ state and no gap for H > Hc. The transition was rectlyinsomenumericalways,totesttheseeffectivethe-
observed in the magnetization measurements on some ories and to investigate unknown systems.
1
quasi-one-dimensional materials; for example, an S = 1 In this paper we propose a size scaling method to es-
v
8 antiferromagnet Ni(C2H8N2)2NO2(ClO4) [5,6], abbrevi- timate the critical exponent δ of quantum spin chains
8 ated NENP,and a spin-PeierlscompoundCuGeO3. [7,8] using the result of the finite cluster calculation. In or-
2 In our previous work [2] we presented a method to der to examine the validity of the method, we apply it
1 derive the ground-state magnetizationcurve in the ther- to the S = 1 antiferromagnetic chain and the S = 1/2
0 modynamic limit from the finite-cluster calculation by bondalternatingchains. Inadditionarecenttopiconthe
8 the size scaling based on the conformal invariance. [9] magnetization plateau of the S = 3/2 antiferromagnetic
9
The obtainedcurveofthe S =1antiferromagneticchain chain with anisotropy is investigated by the method.
/
t successfullyrealizedtheexperimentalresultsofthemag- At first we consider the S = 1 antiferromagnetic
a
netization measurements on NENP qualitatively. The Heisenbergchainfortheexplanationofthemethod. The
m
method was also applied to get theoretical magnetiza- followingargumentiseasilytobe appliedto moregener-
-
d tion curves of some other one-dimensional spin systems. alized models. To investigate the magnetization process
n [10,11]However,the critical behavior near Hc cannot be we consider the Hamiltonian
o investigatedbythismethod, becauseitcanyieldtoofew
c points near Hc to determine the critical exponent of the H =H0+HZ,
:
v magnetization curve by the standard curve fitting. L
Xi In general, except for the Kosterlitz-Thouless transi- H0=XSj ·Sj+1, (2)
tion [12], the magnetization m near the critical field be- j
r
a haves like L
m∼(H −H )1/δ, (1) HZ=−HXSjz,
c
j
for the second-order phase transition. The critical ex-
undertheperiodicboundarycondition. Werestrictuson
ponent δ is an important quantity to determine the uni-
even-site systems to avoid the frustration. Throughout
versalityclass of the phase transitionwhich does not de-
we use the unit such that gµ = 1. For L-site systems,
pend on any detailed properties of each system. For the B
S =1antiferromagneticchaintheexponentwasdeduced the lowest energy of H0 in the subspace where PjSjz =
asδ =2fromsomeeffectiveHamiltoniantheories.[13,14] M (the macroscopic magnetization is m = M/L) is de-
1
noted as E(L,M). We assume the asymptotic form of The convergence of the size correction is guaranteed by
the size dependence of the energy as the condition (7). Thus the extrapolation of the L-
dependent exponent δ(L,L+2) defined by the left hand
1 1
E(L,M)∼ǫ(m)+C(m) (L→∞), (3) side of (8) gives an estimation of δ.
L Lθ
Notethatthe methodcanbeeasilygeneralizedforthe
where ǫ(m) is the bulk energy and the second term de- behavior around a finite magnetization m0, which is de-
scribes the leading size correction. We also assume that scribed as m−m0 ∼ (H −Hc)1/δ. In this case we have
C(m) is an analytic function of m. For gapless cases only to change the form (5) into
the conformal field theory predicted θ = 2. [9] Since the
method works better for faster convergence of the size f(L)≡E(L,M0+2)+E(L,M0)−2E(L,M0+1), (9)
correction as shown later, we can also accept the expo-
nentialdecaylikee−L/ξ whichisreasonablyexpectedfor whereM0 =Lm0. Inadditionthemethodcanbeapplied
even to gapless cases where H might be zero. In the
thegroundstateofthespingapsystemsinsteadof1/Lθ. c
following argument we don’t mention the value of H
We neglect the m-dependence of θ because it gives only c
but we concentrate on the estimation of δ.
higher ordercorrectionswhichdoes notchangethe main
For the behavior of the magnetization curve around
result. If the bulk system has the critical behavior de-
m = 1/2 of the S = 1 antiferromagnetic chain, the L-
scribedbytheform(1),m-dependenceoftheenergyǫ(m)
dependent exponent δ(L,L+2) derived from the form
near m=0 should have the form
(9) using the finite cluster results of E(L,M) up to
ǫ(m)∼ǫ(0)+H m+Amδ+1, (4) L = 18 calculated by Lanczos algorithm is plotted ver-
c
sus 1/(L+1) in Fig. 1. Fitting the quadratic function
whereAisapositiveconstantandweassumeδ >0. Now δ(L,L+2)∼δ+a/(L+1)+b/(L+1)2tothedata,theex-
weputM =0,1and2intotheform(3)anduse(4). IfL trapolated value is determined as δ =0.99±0.01, based
is sufficiently large, C(m) can be expanded with respect on the standard least-square method. The result leads
to m as C(1/L) ∼ C(0)+C′(0)1/L+1/2C′′(0)1/L2··· totheconclusionδ =1,whichisreasonablyexpectedfor
for M =1. Thus we get the forms the gapless point m = 1/2. To check the condition (7)
E(L,M)−ǫ(m) for m=1/2 of the S =1 antiferromag-
1
E(L,0)∼Lǫ(0)+C(0) , netic chain is plotted versus 1/L2 in Fig. 2, where the
Lθ−1
value of ǫ(m) was estimated by fitting of the quadratic
1
E(L,1)∼Lǫ(0)+H +A function of 1/L. The plot suggests θ =2 in the form (3)
c Lδ
which is consistent with the prediction of the conformal
1 1 1 1
+C(0) +C′(0) + C′′(0) ···, field theory. [9] Then the condition (7) is satisfied.
Lθ−1 Lθ 2 Lθ+1
1
E(L,2)∼Lǫ(0)+2H +2δ+1A
c Lδ 2.5
1 1 1 S=1 m=1/2
′ ′′
+C(0)Lθ−1 +2C (0)Lθ +2C (0)Lθ+1 ···. 2.0 S=1 m=0
If we define the quantity
2)1.5
+
L
f(L)≡E(L,2)+E(L,0)−2E(L,1), (5) L,
δ(1.0
the asymptotic behavior of f(L) becomes
0.5
1 1
f(L)∼A(2δ+1−2) +C′′(0) (L→∞). (6)
Lδ Lθ+1 0.00.00 0.02 0.04 0.06 0.08 0.10 0.12
1/(L+1)
When the second term of (6) converges faster than the
first one, the exponent δ can be estimated from the size FIG.1. L-dependent exponent δ(L,L+2) plotted versus
dependence off(L). Thusthe necessaryconditionunder 1/(L+1)form=1/2(solidcircle)andm=0(soliddiamond)
which the method gives the correct value of δ is of the S = 1 antiferromagnetic chain. The extrapolated re-
sults are δ=0.99±0.01 and δ=1.9±0.1, respectively.
θ >δ−1. (7)
Thereforewe haveto checkthe conditionthatE(L,0)/L
convergesfaster than 1/Lδ−1 after determining δ. Using
the calculatedvaluesoff(L)andf(L+2),the exponent
δ can be estimated by the form
f(L) L+2 1
ln /ln )∼δ+O . (8)
(cid:0)f(L+2)(cid:1) (cid:0) L (cid:0)Lθ−δ+1(cid:1)
2
forβ=2.0and−0.2inFig. 3. Thesameextrapolationas
0.00 theS =1chainresultsinδ =2.03±0.03and1.9±0.1for
β =2.0 and −0.2, respectively. The results are also con-
sistent with δ = 2 predicted by some theories discussed
m) above. We also have to check the condition (7) which is
εM)-( -0.01 θ > 1 because of δ = 2. In Fig. 4 E(L,M)−ǫ(m) for
m=M =0 of the system (10) with β =2.0 and −0.2 is
L,
E( plotted versus 1/L. It obviously shows a faster conver-

S=1 m=1/2 gence of the size correction for the ground state energy
S=1 m=0 per site than 1/L, which implies that the condition is
-0.02 satisfied.
0.00 0.01 0.02
1/L2
FIG. 2. Size correction of the ground state energy per 2.5
site E(L,M)−ǫ(m) of the S = 1 antiferromagnetic chain
plotted versus 1/L2 for m = 1/2 (solid circle) and m = 0 2.0
(solid diamond). It converges just like 1/L2 for m = 1/2,
while obviously faster than 1/L2. +2)1.5
L
L,
InFig. 1we alsoshowthe plotofδ(L,L+2)basedon δ(1.0 Bond Alternation m=0 β=2.0
Bond Alternation m=0 β=-0.2
the form (5) versus1/(L+1) for m=0 of the S =1 an- 0.5 S=3/2 m=1/2 D=8.0 δ+
tiferromagnetic chain which is more interesting because S=3/2 m=1/2 D=8.0 δ−
the system has the Haldane gap which vanishes at H .
c 0.0
The extrapolated result is δ = 1.9±0.1 which suggests 0.00 0.05 0.10
1/(L+1)
δ = 2, as predicted by the above effective Hamiltonian
theories. The plot of E(L,M)−ǫ(m) versus 1/L2 for FIG. 3. L-dependent exponent δ(L,L+2) plotted ver-
m=0 in Fig. 2 obviously shows that the size correction sus 1/(L+1) for the S = 1/2 bond alternating chain with
intheform(3)decaysfasterthan1/L2 incontrasttothe β = 2.0 (solid circle) and β = −0.2 (solid diamond). The
plot for m=1/2. It implies θ >2 and the condition (7) extrapolated results are δ = 2.03±0.03 and δ = 1.9±0.1,
which is θ >1 in this case is also satisfied for m=0. respectively. L-dependent exponents δ±(L,L+2) associated
Next we investigate the S = 1/2 bond alternating with the magnetization plateau at m = 1/2 are also plotted
versus1/(L+1)fortheS =3/2antiferromagneticchainwith
chain as another example with the spin gapbetween the
singlet ground state and the triplet first excited state. the single ion anisotropy D = 8.0; δ+: open circle and δ−:
The Hamiltonian is defined as opendiamond. Theextrapolatedresultsareδ+ =1.98±0.04
and δ− =1.99±0.04, respectively.
H =H0+HZ,
L L
H0=XS2j−1·S2j −βXS2j ·S2j+1, (10)
0.001
j j Bond Alternation m=0 β=2.0
2L Bond Alternation m=0 β=-0.2
S=3/2 m=1/2
HZ=−HXSjz, m) 0.000
j ε(
L-
where2LS =1/2spinsareincludedinthesystems. The M)/
system has the gap except for β = −1 where it is the E(L, -0.001
uniform S =1/2 antiferromagnetic chain. We chose two
typical values of β; (i)β = 2.0 and (ii)β = −0.2, which
correspond to the ferromagnetic-antiferromagnetic and -0.002
0.00 0.05 0.10 0.15
antiferromagnetic-antiferromagnetic alternating chains, 1/L
respectively. Inthe lattercaseparticularlythe finite size
FIG.4. Sizecorrectionofthegroundstateenergypersite
effectislargerinthevicinityofthegaplesspointβ =−1.
E(L,M)−ǫ(m)oftheS =1antiferromagnetic chainplotted
Thuswestudyonlyforasmallerβ (=−0.2)thanrealistic
versus 1/L for the S = 1/2 bond alternating chain (m = 0)
cases. The universalityargument, however,justifies that
with β = 2.0 (open circle) and β = −0.2 (open square), and
the critical exponents are independent of β except for
the S = 3/2 antiferromagnetic chain (m = 1/2) with the
β =−1, because the system with m=0 is in a common
single ion anisotropy D = 8.0 (open diamond). It converges
phase for β 6= −1. In order to estimate the exponent δ
faster than 1/L in all the cases. We plot the original values
aroundm=0ofthe system(10),theL-dependentexpo- times104 only forthebondalternatingchain withβ =−0.2.
nent δ(L,L+2) upto L=12is plotted versus1/(L+1)
3
Finally we apply the method to the S = 3/2 anti- In summary a finite size scaling method to estimate
ferromagnetic chain with the single-ion anisotropy. The the critical exponent δ associated with the magnetiza-
system is described by the Hamiltonian (2) with the tion curve around the critical magnetic field correspond-
anisotropy term DPLj(Sjz)2 added to H0. Recently an ing to the amplitude of the spin gap of quantum spin
argument based on the analogy to the quantum Hall ef- chains was proposed and applied to the S =1 antiferro-
fectsuggestedthatthegroundstatemagnetizationcurve magnetic chain and S = 1/2 bond alternating chain. In
possiblyhadaplateaujustatm=1/2whichcorresponds additionthe behaviorofthe magnetizationcurvearound
to1/3ofthe saturationmomentandthe singularpartof the edges of the plateau of the anisotropic S =3/2 anti-
the magnetization near the plateau was proportional to ferromagneticchainwasinvestigatedbythemethod. All
p|H −Hc| where Hc is the critical field at either edge the results indicated the same conclusion δ =2.
of the plateau. [17] The plateau was verified to exist for WewouldliketothankDr. K.Totsukaforsendinghis
D > D = 0.93 by finite cluster analyses and size scal- preprint and interesting discussions. We also thank the
c
ing techniques. [18] However,the form of the singularity Supercomputer Center,Institute for SolidState Physics,
near the edge of the plateau has not been derived by University of Tokyo for the facilities and the use of the
anynumericalstudiesontheoriginalHamiltonian. Thus Fujitsu VPP500. This research was supported in part
this problem is one of interesting examples to investi- by Grant-in-Aid for the Scientific Research Fund from
gatebythemethodpresentedinthispaper. Weconsider the Ministry of Education, Science, Sports and Culture
a sufficiently large D so that the magnetization curve (08640445).
has a clear plateau at m = 1/2. The two critical fields
Hc± are denoted such that the curve has a plateau for
Hc− <H <Hc+. They can be given by
L L
E(L, ±1)−E(L, )→±Hc± (L→∞), (11)
2 2
[1] F. D. M. Haldane, Phys. Lett. 93A,464 (1993); Phys.
although we don’t consider the value of Hc± here. To Rev. Lett.50,1153 (1983).
investigate the singularity of the magnetization curve, [2] T. Sakai and M. Takahashi, Phys. Rev. B 43, 13383
the critical exponents δ± are defined as (1991).
[3] T. Sakai and M. Takahashi, J. Phys. Soc. Jpn. 60, 3615
1
m− 2 ∼(H −Hc+)1/δ+, (12) [4] T(1.9S91a)k.ai, J. Phys. Soc. Jpn. 64, 251 (1995).
12 −m∼(Hc−−H)1/δ−. (13) [[56]] KY..AKjaitrsouemtaatla. Pethyasl..PRheyvs..LRetetv..6L3e,tt1.46234,(81698(91)9.89).
[7] M. Hase et al. Phys. Rev. B48, 9616 (1993).
To estimate δ± we have only to change f(L) into f±(L) [8] H. Nojiri et al. Phys. Rev. B52, 12749 (1995).
definedasf±(L)≡±[E(L,L2 ±2)+E(L,L2)−2E(L,L2 ± [9] J. L. Cardy, J. Phys. A 17, L385 (1984); H. W. Bl¨ote,
1)]andextrapolatetheL-dependentexponentsδ±(L,L+ J. L.Cardy and M. P. Nightingale, Phys. Rev.Lett. 56,
2)defined by the left handside ofthe equation(8) using 742 (1986); I. Affleck,Phys. Rev.Lett. 56, 746 (1986).
f±(L) instead of f(L). In Fig. 3 we show the plot of [10] T. Tonegawa, T. Nakao and M. Kaburagi, J. Phys. Soc.
δ±(L,L+2) versus 1/(L+1) up to L=14 for D =8.0. Jpn. 65, 3317 (1996).
The extrapolated results are δ+ = 1.98±0.04 and δ− = [11] M. Hagiwara et al. Technical Report of ISSP, No. 3291
1.99±0.04, which imply δ+ = δ− = 2 as suggested by (1997).
the analogy to the quantum Hall effect. We also check [12] J. M. Kosterlitz and D. J. Thouless, J. Phys. C 6, 1181
the condition (7) by the plot of E(L,M)−ǫ(m) versus (1973).
1/L for M =L/2 and m=1/2 in Fig. 4 which suggests [13] M. Takahashi and T. Sakai, J. Phys. Soc. Jpn. 60, 760
thatthe sizecorrectiondecaysfasterthan1/L. Toavoid (1991).
large finite size effects we considered only a large value [14] I. Affleck,Phys. Rev. B43, 3215 (1991).
[15] R. Chitra and T. Giamarchi, Phys. Rev. B55, 5816
ofD (=8.0)whichisnotrealistic. Butitisexpectedthe
(1997).
result δ = 2 is always true for D > D (= 0.93) because
c
[16] Y.Akutsuetal.contributiontotheJPSmeetingautumn
the transition at the critical field belongs to a common
1997, at Kobe.
universality class.
[17] M. Oshikawa, M. Yamanaka and I. Affleck, Phys. Rev.
Recently the magnetization plateau was also investi-
Lett. 78, 1984 (1997).
gated on the S = 1/2 bond alternating chain with the
[18] T. Sakai and M. Takahashi, to appear in Phys. Rev. B
next-nearest neighbor interaction [19] by the bosoniza-
Rapid Comm. (SISSAcond-mat/9710327 preprint).
tion technique which lead to δ = 2 at the edge of the
[19] K. Totsuka, to appear in Phys. Rev.B.
plateau at m = 1/4. The result suggests the transi-
tion belongs to the same universalityclass as that of the
anisotropic S =3/2 chain.
4