Crossover behavior of the J1-J2 model in a staggered magnetic field

4 Pages·0.17 MB·English

by  H. Otsuka

Crossover behavior of the J -J model in a staggered magnetic field
1 2
∗
Hiromi Otsuka
Department of Physics, Tokyo Metropolitan University, Tokyo 192-0397, Japan
(Received February 2, 2008)
3 The ground states of the S = 1 Heisenberg chain with the nearest-neighbor and the next-nearest-
2
0
neighbor antiferromagnetic couplings are numerically investigated in a staggered magnetic field.
0
Whilethestaggered magneticfieldmayinducetheN´eel-typeexcitationgap,anditischaracterized
2
bytheGaussian fixedpointin thespin-fluidregion, thecrossover tothebehaviorcontrolled bythe

n Ising fixed point is expected to be observed for the spontaneously dimerized state at finite field.
a Treating a low-lying excitation gap by the phenomenological renormalization-group method, we
J
numerically determine the massless flow connecting the Gaussian and Ising fixed points. Further,
7 to check thecriticalities, we perform the finite-size-scaling analysis of theexcitation gap.
1
PACS number(s): 75.10.Jm, 64.60.Fr
]
h
c
e Theoretically and experimentally, there have been in- system is in the spin-fluid phase described by the Gaus-
m tensive investigations on critical phenomena observed in sianfixedpoint(c=1CFT);atJ =0.5(theMajumdar-
2
low-dimensionalquantumspinsystems. Inparticular,re- Ghosh point) the direct products of spin singlet pairs
-
t cent investigations on this subject have focused not only formed either on bonds 2k,2k+1 or 2k+1,2k+2
a h i h i
t oncriticalfixedpointsandtheirneighboringregions,but become twofold degenerated ground states (i.e., sponta-
s alsoontheglobalbehaviorsoftherenormalization-group neouslydimerizedstates),3,4,5 andafiniteexcitationgap
.
at (RG) flows connecting them. For one-dimensional (1D) exists there.6 According to Haldane7, and Kuboki and
m quantumsystems (also for 2D classicalsystems),besides Fukuyama,8 the gapformationcausedby the frustration
the exact solutions available for some cases, the confor- canbewelldescribedbythequantumsine-Gordonmodel
-
d malfieldtheory(CFT)providesthemostefficientwayto obtained via the bosonization procedure;9 the effective
n characterizethefixedpoints,wherethevaluesofthecen- Hamiltonian of Eq. (1) is given as
o
tral charge c specify their universality classes. Further,
c v 1
[ with respect to the RG flows observed in the continuum 1= dx K(∂xθ)2+ (∂xφ)2
and unitary models, Zamolodchikov’s c-theorem serves H Z 2π (cid:20) K (cid:21)

1 fortheexplanationsoftheirgeneralproperties.1 Sincein 2g
v + dx φ cos√8φ, (3)
investigations of the quantum spin chains and interact- Z (2πα)2
6
ing electron systems, the c = 1 CFT has close relevance
1
to their criticality,2 it is very important to understand wherethebosonicoperatorθisthedualfieldofφsatisfy-
3
1 its instability and a crossover to behaviors controlled by ingthecommutationrelation∂y[φ(x),θ(y)]=iπδ(x−y).
0 other critical fixed points. K andvaretheGaussiancouplingandthespin-waveve-
3 In this paper, we shall study the ground states of locity,andgφstandsforthespinUmklappscatteringbare
0 the S = 1 Heisenberg chain with the nearest-neighbor amplitude.10 Althoughthe propertyofthiseffective con-
t/ and the 2next-nearest-neighbor antiferromagnetic cou- tinuummodelhasbeenwellunderstoodandactuallythe
a g termmaybecome relevant,11 forthe determinationof
m plings (the so-called J1-J2 model) in a staggered mag- φ ∗
thefluid-dimertransitionpointJ ,numericaltreatments
netic field; the Hamiltonian to be considered is given by 2
- H =H +H with of Eq. (1) were required. By carefully investigating the
d 1 2
effect of the marginal g term on the critical fixed point
n φ
L throughlower-energyexcitationlevels observedin finite-
co H1 = (2J1Sj ·Sj+1+2J2Sj ·Sj+2), (1) sizesystems,OkamotoandNomurapreciselydetermined
: Xj=1 J∗ 0.2411.12 While it is knownthatthe excitationgap
iv L al2so≃exists in the case of J2 >0.5,13 we, in the following
X H2 = −hs(−1)jSjz, (2) discussion, restrict ourselves to the region 0 ≤ J2 ≤ 0.5
r Xj=1 for simplicity.
a
In the case of nonzero field [this is approximately re-
where Sν is the νth component of the spin operator on
j alized in the quasi-1D antiferromagnets with the alter-
the jth site S and couplings J , J > 0 (we take J as
j 1 2 1 nating gyromagnetic tensors, e.g., Cu-benzoate14 and
theenergyunitinthefollowing). Weassumetheperiodic
Yb As 15 and the field may be also generated as the in-
boundary condition S = S and an even number of 4 3
L+1 1 trinsiconeoriginatedfromtheN´eelorderedsublattice16]
L.
since the bosonized form of Eq. (2) is given as
In the zero-field case (h = 0), there are two special
s
points where the exact ground states have been known: h
= dx s cos√2φ, (4)
atJ2 =0wheretheBetheansatzsolutionisavailable,the H2 Z − πα
1
and the scaling dimension of this perturbation term is
J=0.32 J=0.40 J=0.46 J=0.50
x = 1 (1/ν = 2 x = 3/2) on the fixed point,17 the 2 2 2 2
2 2 − 2 5.2
second-order phase transition occurs for systems in the 2.8 1.6
stipoinn-lfleunigdthreξgionh−J22/≤3 (Ja2∗sidweitfhrotmhethdeivleorggaernitchemoifcccoorrrreelac-- L∆E
s
∝ ∗
tionduetotheg term). Ontheotherhand,forJ >J ,
φ 2 2 1
due to the relevant g term, the crossover of the transi-
φ
tiondrivenbythe staggeredmagneticfieldoccurs. Since 5 2.4 1.2
the dimer gap may survive in a weak field, the critical
∗
point h (J ) takes nonzero values depending on J , and
s 2 2
further the universality class is changed. Recently, Fab-
rizio et al.,18 on the basis of the double-frequency sine-
0.8
Gordon (DSG) theory given by Delfino and Mussardo,19 2
∗ 4.8 0.5
aIsrignugedfixtehdatptohinetsywsittehmco=n h1si(nJ2a)cicsorrdenworimthaltihzeed“dtoowthne- 0.01 hs 0.02 0.02 0.04 0.06 0.08 0.12 0.14
2 FIG. 1. The L and hs dependences of L∆E(J2,hs,L).
hill” condition of the c-theorem, so the criticality in the
From left to right, the next-nearest-neighbor coupling
vicinity ofthis line is relatedtothe divergentcorrelation
length of the form ξ [h h∗(J )]−1 (explainedbelow) J2=0.32, 0.40, 0.46, and 0.5. Data for systems of L = 18
described by the ϕ4 ∝theorsy−. s 2 (crosses),20(triangles),22(squares),24(diamonds),26(cir-
cles), and 28 (double circles) are plotted with the fitting
From the viewpoint that the Gaussian fixed point is
curves.
perturbed by two relevant operators of g and h , we
φ s
∗
can qualitatively estimate hs(J2) around the point ac- 1 µ, 2 I+ǫ, (8)
cording to the crossover argument.20 However, to evalu- O → O →
ate its precise value especially near the Ising fixed point, where µ is the disorder field (Z2 odd) and ǫ is the en-
a numerical treatment of H should be required. Here, ergy density operator (Z2 even) with scaling dimensions
it should be noted that the criticality on the N´eel-phase x = 1 andx =1,respectively. Accordingtothiscorre-
µ 8 ǫ
boundariesintheS =1bond-alternatingXXZchainalso spondence,wecanobtainarelevantexcitationinnonzero
belongs to the Ising universality and they were precisely field by taking the limit hs 0 and assigning it to one
ց
determined by the numerical method.21 Thus, we shall of well-characterized states. From Eqs. (4) and (8), the
perform our calculations by analyzing the lower-energy staggered magnetic field plays a role of the “thermal”
∗
excitations observed in the finite-size systems. Let us scalingvariableonh (J ),andthusthecriticalexponent
s 2
first focus our attention to the excitations in the spin- 1/ν =2 x =1 asalreadymentioned(see Ref. 18). On
ǫ
−
fluid region described by the Gaussian fixed point,22,23 one hand, since the “magnetic” excitation µ stemming
fromthedimerexcitationprovideslowerenergy(i.e.,the
=√2sin√2φ, (5) most divergent fluctuation), we will thus focus attention
1
O ∗
=√2cos√2φ, (6) on it for the determination of hs(J2).
2
O Now,weshallexplainthenumericalcalculationproce-
3 =exp( i√2θ). (7) dure. Since the system is massive unless it is located on
O ±
the critical line, the phenomenological renormalization-
and denote dimer and N´eel excitations, respec-
1 2 group (PRG) method is expected to work efficiently for
O O
tively, while 3 is the doublet excitation changing an our aim:26 We numerically solve the PRG equation for
O
amount of the total spin. According to the finite-size-
the systems of L and L+2,
scaling (FSS) argument based on CFT, corresponding
energylevels∆Ei fortheseoperators(takingtheground- (L+2)∆E(J2,hs,L+2)=L∆E(J2,hs,L) (9)
state energy as zero) are expressed by the use of their
scaling dimensions xi as ∆Ei ≃ 2πvxi/L.24 For the with respect to hs for a given value of J2. Since the
case of hs = 0, we can calculate ∆Ei according to the equation can be satisfied by the gap having the size de-
level-spectroscopy method,23 where discrete symmetries pendence of ∆E(J ,h ,L) 1/L, the obtained value
2 s
∝
ofthelatticeHamiltonian(seebelow)areutilizedtospec- canberegardedastheL-dependenttransitionpoint,say
ify excitation levels. On the other hand, for the case h∗(J ,L+1). Then, extrapolating them to the thermo-
s 2
of hs 6= 0, the usable symmetry becomes lower, and dynamic limit, we estimate h∗s(J2). Alternatively, as ex-
more importantly, the universality class is changed to plainedinRefs.22and23—towhichwereferthe inter-
∗
the Ising one on the line hs(J2) so that we should em- estedreadersforadetailedexplanation—aths =0,the
ploy other criterion to characterize the levels. Here, we dimer excitation is in the subspace of total spin Sz =0,
T
will use the so-called UV-IR (ultraviolet-infrared) oper- wave number k = π, space inversion P = +1, and spin
ator correspondence,18,25 i.e., along the critical RG flow reversal T = +1 for L 0 (mod 4) [Sz = 0, k = 0,
≡ T
theoperatorsontheGaussianfixedpoint(UV)aretrans- P = 1, and T = 1 for L 2 (mod 4)]. Thus, we can
− − ≡
muted to those on the Ising fixed point (IR) as calculate ∆E(J ,h ,L) = E (J ,h ,L) E (J ,h ,L)
2 s µ 2 s g 2 s
−
2
fromE (J ,h ,L)connectingtothedimerexcitationand L=28 sites are treated, where the Lanczos algorithm is
µ 2 s
E (J ,h ,L), i.e., the ground-state energy. usedtoobtaineigenvaluesoftheHamiltonianinspecified
g 2 s
subspaces. We can see that the L dependence of the
crossingpointisalmostabsentforlargevalueofJ (near
2
0.15
J =0.5), while it is visible for the small value case.
2
∗
hs After evaluating hs(J2,L+1),we extrapolatethemto
0.1 L ; here we assume the following formula:
→∞
0.1 h*s(J2) h∗s(J2,L)=h∗s(J2)+aL−2(1+bL−ω) (ω >0), (10)
0.05 where the b term stands for a correction to the leading
∗
one, and four parameters h (J ), a, b, and ω are deter-
s 2
minedaccordingtotheleast-square-fittingcondition. We
0.05 0
N’eel used the data of L=18 28, and extrapolated them to
0 0.002 L−20.004 h∗(J ) as shown in the i−nset of Fig. 2, where from bot-
s 2
SF tom to top a series of data with fitting curves are given
"dimer"
in the increasing order of J . Consequently, we obtain
0 2
0.2 0.3 0.4 J 0.5 the critical line h∗s(J2) as shown in the figure. The RG
2
eigenvalueofthescalingvariableg is“almostzero”(i.e.,
φ
FIG.2. TheboundarylineoftheN´eeland“dimer”phases marginal), while that of the staggered magnetic field is
(for an explanation of “dimer” phase, see Ref. 18). In- ∗
3/2 on the point (J ,h )=(J ,0). The phase boundary
2 s 2
cseatl fisheoldwsstsoomtheeotfheerxmtroadpyonlaatmioincsliomfitt,hewhLe-rdeepfietntidnegntcucrrviteis- is thus expected to behave as h∗s(J2) ∝ (J2−J2∗)1.5/0+,
whichagreeswith a weak J dependence of the line near
2
are given. The double circle shows the fluid-dimer transi-
the point.
tion point (J2,hs) = (J2∗,0) at which the criticality of the Atthisstage,weshallcheckthecriticalitiesbytheuse
boundary changes from the Gaussian to the Ising type. The
spin-fluid(SF) state exists on the x axis of J2 ≤J2∗. oftheFSSanalysisinthevicinityoftheboundarylines,27
i.e., we assume the following one-parameterscaling form
of the excitation gaps in the finite system of L:
3 ∆E(J ,h ,L)=L−1Ψ(L(h h∗(J ))ν), (11)
2 s s− s 2
]
E (a) Gaussian (ν=2/3) where ν = 2/3 for J J∗ and ν = 1 for J > J∗. Fur-
∆ 2 ≤ 2 2 2
L (b), (c) Ising (ν=1) ther,theasymptoticbehaviorsofthescalingfunctionare
[ expected as Ψ(x) x for large x and Ψ(x) const for
n ∝ ≃
l (a) x 0. Usingthe obtainedtransitionpoints,weplotEq.
2 (1→1)forthe Gaussian(J =J∗)andtheIsingtransitions
2 2
(J =0.46 and 0.5) in Fig. 3. The results show that the
2
data of different system sizes are collapsed on the sin-
gle curve in both cases, and the asymptotic behaviors of
Ψ agree with the expected ones (dotted lines) despite of
thesmallL. Fromtheseplots,wecancheckthecrossover
1
behavior of the transitions driven by the staggeredmag-
(b) L=20 netic field in the present frustrated quantum spin chain
22 systemandalso the accuracyof the phase boundary line
24 in Fig. 2. This goodFSS behaviorsmay rely on the con-
26 ditions that the marginaloperatorthat bringsabout the
(c)
0 28 multiplicative logarithmic correction to the pure power-
lawsingularityofξisabsentonthefluid-dimertransition
−2 0 ν 2 point,28 and that the crossover region for J2 > 0.4 may
ln[L(h −h*) ] be large enough to be detected. On the other hand, the
s s
FSS nature becomes obscure in the weak and intermedi-
FIG. 3. The FSS plots of the excitation gap for systems
of L = 20−28 in log-log scale. We have used ν = 2/3 for ate values of J2 since our system sizes are too small to
∗ reach the region.
the Gaussian transition [(a) J2 = J2], and ν = 1 for the
Ising transition [(b) J2 =0.46, (c) J2 =0.5]. The slope of the Finally, we evaluate the values of the central charge
dotted line is 1showing theexpectedasymptotic behaviorof through the L dependence of the ground-state energy:29
thescaling functions. πvc
∗
E (J ,h (J ),L) e L (12)
g 2 s 2 ≃ 0 − 6L
In Fig. 1, we demonstrate L and h dependences of
s
(e is the energy density in the thermodynamic limit).
L∆E(J ,h ,L) at several values of J . Systems up to 0
2 s 2
For this calculation, we should estimate v in advance;
3
thheerelewaestu-ssqeutahree-FfiSttSinrgelaptrioocned∆uEreµf≃or2tπhvexdµa(t=a.18)T/hLeaonbd- 1A(1.9B8.6Z)a[JmEoTloPdcLheitkto.v4,3P,is7’3m0a(1Z9h8.6E)]k.sp. Teor. Fiz. 43, 565
tained results agree well with the Ising one for the con- 2F.D.M. Haldane, J. Phys. C 14, 2585 (1981).
siderably large values of J , i.e., (v,c) (1.439,0.494)at 3C.K.Majumdar,J.Phys.C3,911(1970);C.K.Majumdar
2
J =0.46 and (1.248,0.491) at J =0.50≃. For comparison, and D.K. Ghosh, J. Math. Phys. 10, 1399 (1969).
2 2
we also estimate them at J = J∗ and obtain (v,c) 4P.M. van den Broek, Phys.Lett. 77A, 261 (1980)
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2 8K.KubokiandH.Fukuyama,J.Phys.Soc.Jpn. 56,3126
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and A.M. Tsvelik, Bosonization and Strongly Correlated
ground-state phase diagram of the one-dimensional J -
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s 2
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∗
Email address: [email protected]
4