PHYSICAL REVIEW B, VOLUME 64, 134415

Zone-boundary excitations in coupled Haldane spin chain compounds PbNi2 V2 O8
and SrNi2 V2 O8
A. Zheludev*
Physics Department, Brookhaven National Laboratory, Upton, New York 11973

T. Masuda and K. Uchinokura
Department of Advanced Materials Science and Department of Applied Physics, The University of Tokyo, Tokyo 113-8656, JapanS. E. Nagler
Oak Ridge National Laboratory, Bld. 7692, MS 6393, P.O. Box 2008, Oak Ridge, Tennessee 37831
共Received 26 March 2001; published 12 September 2001兲
Magnetic excitations in the quasi-one-dimensional quantum antiferromagnets PbNi2 V2 O8 and SrNi2 V2 O8
are measured all the way up to the zone-boundary energy using inelastic neutron scattering from powder
samples. An estimate for next-nearest-neighbor in-chain interactions is obtained. The role played by theseinteractions in spin-vacancy-induced magnetic ordering in PbNi2 V2 O8 is discussed.
DOI: 10.1103/PhysRevB.64.134415

PACS number共s兲: 75.30.Ds, 75.50.Ee, 75.40.Gb

I. INTRODUCTION

The quasi-one-dimensional 共quasi-1D兲 S⫽1 antiferromagnet 共AF兲 PbNi2 V2 O8 was recently shown to be the firstexample of a Haldane-gap system1 to undergo long-range
magnetic ordering upon doping with nonmagnetic
impurities.2,3 This unique behavior was understood throughinelastic neutron scattering measurements, which suggested
that three-dimensional 共3D兲 interchain interactions in
PbNi2 V2 O8 are almost 共but not quite兲 strong enough to destroy the Haldane singlet ground state even in the undoped
material.2,4 In fact, in the very similar isostructural compound SrNi2 V2 O8 , interchain coupling exceeds the criticalvalue, and, unlike the Pb-based material, the undoped Sr-Ni
vanadate orders magnetically at T⫽7 K.2,4
The introduction of nonmagnetic vacancies into gapped
quantum spin chains produces free S⫽1/2 degrees of freedom on either side of each impurity site.5– 8 In the presenceof arbitrarily weak interchain interactions, these free spins
will order in three dimensions at sufficiently low temperature. However, next-nearest-neighbor 共NNN兲 interactions
within the chains can prevent this from happening.9 Naively,in the presence of AF NNN coupling, one can expect theliberated spins around each impurity to bind into nonmagnetic singlets 共dimers兲, preserving the disordered groundstate of the system.
There are at least two indications that NNN interactions
may be active in PbNi2 V2 O8 and SrNi2 V2 O8 . First, the spin
chains in both materials are not linear, but are formed byNi2⫹ ions arranged in four-step spirals propagating along thec axis of the orthorhombic lattice 共for a detailed discussionof chain geometry and interchain interactions see Ref. 4兲. In
this ‘‘twisted’’ structure NNN superexchange pathways involving equatorial oxygens of the NiO6 tetrahedra can bequite effective. Second, there remains some discrepancy between the in-chain exchange constant J deduced from the
Haldane gaps measured with neutron scattering, assuming
only nearest-neighbor 共NN兲 interactions (J⫽9.0 meV兲, and0163-1829/2001/64共13兲/134415共7兲/$20.00

from high-temperature (T⬇J) susceptibility data (J⫽8.2
meV兲. The former technique probes low-energy excitations
around q 储 ⫽ ␲ 共AF zone center兲, while the latter is sensitive
to excitations in the entire Brillouin zone. The mismatch may
be interpreted in terms of an additional coupling with a characteristic interaction length of twice the spin-spin separationin the chains, which is known to have different effects onzone-center and zone-boundary excitation energies.
The most direct way to resolve this issue would be tomeasure the dispersion of magnetic excitations in the entire
Brillouin zone in a single-crystal sample. Unfortunately, nosingle-crystal or aligned powder samples of either
PbNi2 V2 O8 or SrNi2 V2 O8 , suitable for inelastic neutron
scattering studies, are available to date. As will be discussed
below, some information on NNN coupling can be obtainedusing powder samples if the measurements are extended all
the way to the zone-boundary energy. Previous neutron scattering studies employed cold neutrons to investigate lowenergy excitations in the two compounds.2,4 In the present
paper we report thermal-neutron studies specifically aimed atmeasuring high-energy excitations. We then engage in a
semiquantitative discussion of the role NNN interactions
play in vacancy-induced ordering of weakly coupled
Haldane spin chains and apply the results to the particularcase of PbNi2 V2 O8 .
II. EXPERIMENTAL PROCEDURES AND RESULTS

Both vanadates crystallize in tetragonal structure, spacegroup I41cd, with lattice constants a⫽12.249(3) Å and c⫽8.354(2) Å for PbNi2 V2 O8 共Ref. 10兲 and a⫽12.1617 Åand c⫽8.1617 Å for SrNi2 V2 O8 共Ref. 11兲, respectively. All
measurements were performed on large (⬇5 g兲 powder
samples of PbNi2 V2 O8 . and SrNi2 V2 O8 , similar to thoseused in previous cold-neutron studies.4 The experiments
were carried out at the HB3 three-axis spectrometer installedat the High Flux Isotope Reactor at Oak Ridge National
Laboratory. Neutrons with a fixed final energy of 13.5 meVwere used with a 5-cm-thick pyrolitic graphite 共PG兲 filter

64 134415-1©2001 The American Physical Society

ZHELUDEV, MASUDA, UCHINOKURA, AND NAGLERPHYSICAL REVIEW B 64 134415

FIG. 1. Typical background-corrected constant-Q scans measured in PbNi2 V2 O8 at T⫽1.5 K. Arrows roughly indicate the gap
and zone-boundary energies. Solid lines are simulations based on
the result of a semiglobal fit to the data, which allows for nextnearest-neighbor in-chain interactions, as described in the text.
Dashed lines are a best fit assuming that only nearest-neighbor interactions are present. Statistical error bars are smaller than thesymbol size.

positioned after the sample to eliminate higher-order beam
contamination. PG共002兲 reflections were used for monochromator and analyzer, in combination with 48⬘ -40⬘ -40⬘ -120⬘
collimators. Sample environment was in all cases a standard
4
He-flow cryostat. The data were taken at T⫽1.5 K.
Typical data sets collected in constant-Q scans for
PbNi2 V2 O8 and SrNi2 V2 O8 are shown in Fig. 1 and Fig. 2,
respectively. Figure 3 shows the bulk of the data collected
for the Sr system 共only five scans, at 兩 Q 兩 /c * ⫽2.5, 3, 3.5, 4,
and 5, were collected for the Pb-based material兲. The measured intensities were scaled to correct for the previously
determined energy-dependent ␭/2 contribution to the incident beam, which affects the monitor rate. Within the incident neutron energy range used, the monitor efficiencychanges by roughly 30%. In addition, a flat background 共17
counts/min兲, measured at 30 meV energy transfer 共above themagnetic excitation band兲, was subtracted from the data.
For both materials all scans have the characteristic

FIG. 2. Typical background-corrected constant-Q scans measured in SrNi2 V2 O8 at T⫽1.5 K. Arrows and solid lines are as inFig. 1. Statistical error bars are smaller than the symbol size.

M shape. In a powder sample one measures the spherical
average of the dynamic structure factor S(Q, ␻ ). For large
momentum transfers, a large portion of Q space is sampled
simultaneously, and the powder cross section is in many
ways similar to the density-of-states function n( ␻ ). In particular, one expects to see a peak in the powder cross section
wherever the density of states is large, i.e., at the gap energy
and at the zone boundary. As discussed previously,4 the peakaround 5 meV energy transfer corresponds to excitations
close to the Haldane gap at a momentum transfer q 储 ⫽ ␲FIG. 3. The bulk of constant-Q scans measured in SrNi2 V2 O8 at
T⫽1.5 K. Solid lines are as in Fig. 1. Statistical error bars are
smaller than the symbol size.

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ZONE-BOUNDARY EXCITATIONS IN COUPLED . . .

PHYSICAL REVIEW B 64 134415⫹2␲n (n integer兲 along the chain axis. The weaker higherenergy peaks represent excitation near the top of the band,i.e., in the vicinity of q 储 ⫽ ␲ /2⫹ ␲ n (n integer兲.
III. DATA ANALYSIS
A. Model cross section

The convoluted nature of inelastic powder data makes it
notoriously difficult to interpret 共for an example see Ref. 12,
particularly Sec. IV兲. The only way to extract information on
the dynamic structure factor from such measurements is to
assume a ‘‘reasonable’’ model and refine the parameter values to best fit the data. The main assumption of the model
used for PbNi2 V2 O8 and SrNi2 V2 O8 to date is that thesecompounds are described as weakly interacting S⫽1 AF
chains with magnetic anisotropy. This picture is based on
susceptibility, high-field, and specific heat measurements andis supported by the observed unique shape of constant-E
neutron scans.2 In the present work we shall adopt this model
as well.
The form of the neutron scattering cross section for thismodel was derived in Ref. 4 and is written in the singlemode approximation 共SMA兲. Interchain interactions were accounted for within the random phase approximation 共RPA兲.
This cross section function performed rather well in describing low-energy neutron data. Previous studies of other
Haldane-gap compounds, such as NENP 共Ref. 13兲 and
CsNiCl3 共Ref. 14兲, have demonstrated that the SMA is, infact, applicable in most of the Brillouin zone. The previously
derived SMA cross section 关Eqs. 共5兲–共13兲 of Ref. 4兴 could
thus be safely used in interpreting our high-energy results.
However, the expression for the dispersion relation 关Eqs.
共14兲 and 共15兲 of Ref. 4兴 had to be slightly modified. These
modifications are discussed in the following paragraphs.
1. Isolated chains with nearest-neighbor interactions

In Ref. 4 the dispersion of excitations in noninteracting
Haldane spin chains was assumed to be sinusoidal and symmetric around the classical AF zone boundary q 储 ⫽ ␲ /2. In
fact, since translational symmetry is not broken in the
Haldane ground state, the excitation energy extrapolates to alarger value at q 储 →0 than at q 储 → ␲ .15 This effect can betaken into account empirically by including an extra term in
the dispersion relation.13 The dynamic susceptibility of asingle Haldane spin chain is then written as␹␤␤

共 q 储 , ␻ 兲 ⫽ ␹ ␲␤␤2⌬ ␤␤1⫺ cos q 储
, 共1兲
2
2
2
共 ប ␻ ␤␤q 兲 ⫺ 共 ប ␻ ⫹i ⑀ 兲
储

q
2
2
2
2
2
2 储
.
共 ប ␻ ␤␤q 储 兲 ⫽⌬ ␤␤ ⫹ v sin q 储 ⫹ ␣ cos
2

共2兲

In these formulas, which replace Eq. 共22兲 in Ref. 4, ⌬ ␤␤ isthe gap energy for the Haldane-gap mode of a particularpolarization ␤␤ , ␹ ␲␤␤ is the static staggered susceptibility,and v is the spin-wave velocity. The spin chains PbNi2 V2 O8
and SrNi2 V2 O8 contain four equivalent spins per crystallographic period along the c axis. Throughout this work theactual wave vector transfer will be denoted as Q⫽ha*⫹kb* ⫹lc* . The symbol q 储 ⬅Qc/4 in Eq. 共2兲 will stand forthe reduced wave vector transfer along the chains.
The energetics of a Haldane-gap AF is by now very wellestablished through numerical work by several authors.16 –18
For an S⫽1 Heisenberg chain with single-ion anisotropy oftype D(S z ) 2 the approximate values of parameters are as
follows:2⌬ xx ⫽⌬ y y ⬇ 具 ⌬ 典 ⫺ D,
3

共3兲

4
⌬ zz ⬇ 具 ⌬ 典 ⫹ D,3

共4兲具 ⌬ 典 ⬇0.41J,

共5兲v ⬇2.49J,

共6兲v
␹ ␲␤␤ ⬇1.26 2 .
⌬ ␤␤

共7兲

The parameter ␣ in Eq. 共2兲 characterizes the asymmetryof the dispersion relation. Existing experimental data for
NENP 共Ref. 13兲 and CsNiCl3 共Ref. 14兲 give ␣ ⫽1.45J⬇0.58v and ␣ ⫽1.1J⬇0.44v , respectively. The latter result
inspires more confidence, as it was obtained with a muchhigher experimental resolution and gives a dispersion relation very similar to that obtained by direct diagonalization in
finite-length chains.15 Throughout the rest of the paper weshall assume ␣ fixed at ␣ ⬅0.44v .
2. Next-nearest-neighbor interactions

Next-nearest-neighbor in-chain interactions, which we believe can be relevant for PbNi2 V2 O8 and SrNi2 V2 O8 , can be
accounted for within the RPA. It is straightforward to verify
that isotropic NNN coupling of magnitude J ⬘ gives an additional term on the right-hand side 共RHS兲 of Eq. 共2兲 that canbe written as2
␹ ␲␤␤ ⌬ ␤␤J⬘1⫺ cos q 储
cos 2q 储 .
2

共8兲The primary effect of NNN interactions is to modify the gap
energies:
2
2
␤␤
⫽⌬ ␤␤
⌬ ␤␤
,0共 1⫹ ␹ ␲ J ⬘ 兲 ,共9兲

where ⌬ ␤␤ ,0 are the gaps in the absence of NNN coupling.
The relative effect of NN interactions on the zone-boundary
energy and spin-wave velocity is considerably smaller, andcan be ignored. To a good approximation, in analyzing theexperimental data one can use the dispersion relation 共2兲, but
treat the gap energies and v as independent parameters,
rather than assuming that the relation 共6兲 is valid only in thecase J ⬘ ⫽0. The ratio ␰ ⫽ v / 具 ⌬ 典 can then be determined andused to obtain an estimate for J ⬘ . From Eqs. 共3兲–共7兲 and 共9兲
for J ⬘ ⰆJ one gets

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ZHELUDEV, MASUDA, UCHINOKURA, AND NAGLER

冉

冊

J⬘
␰ ⬇ ␰ 0 1⫺9.33 .
J

PHYSICAL REVIEW B 64 134415

共10兲

Here ␰ 0 ⫽ v / 具 ⌬ 0 典 ⬇6.07 is equal to the equal-time correlation
length in the absence of NNN coupling.
3. Interchain interactions

For the coupling geometry found in PbNi2 V2 O8 andSrNi2 V2 O8 , switching on interchain interactions within the
RPA gives an additional term2
J 1 cos共 ␲ l/2兲关 cos共 ␲ h 兲 ⫹ cos共 ␲ k 兲兴
⫺ ␹ ␲␤␤ ⌬ ␤␤

共 1⫺ cos ␲ l/2兲2
共11兲

on the RHS of Eq. 共2兲. In this formula, which corresponds tothe last terms in Eqs. 共14兲 and 共15兲 of Ref. 4, J 1 is themagnitude of nearest-neighbor interchain interactions. For
our present purposes, the possible small anisotropy of J 1 willbe ignored.
B. Parameters for PbNi2 V2 O8 and SrNi2 V2 O8

Our understanding of the physics of PbNi2 V2 O8 andSrNi2 V2 O8 is based on knowledge of the following characteristics: the intrinsic gap energies ⌬ zz and ⌬ xx ⫽⌬ y y , themagnitude of interchain interactions J 1 , and the spin-wave
velocity v . Can these four parameters be unambiguously determined from the available experimental data?
First, consider those quantities that characterize lowenergy excitations near q 储 ⫽ ␲ . For PbNi2 V2 O8 the actual
minima of the 3D dispersion, E min, ␤␤ , were independently
measured in high-field magnetization studies of aligned powder samples.2,4 These energies are directly related to ⌬ ␤␤ andJ 1 . The third experimental value needed to unambiguously‘‘untangle’’ these three quantities was probed in previous
high-resolution cold-neutron measurements.4 Indeed, thewell-defined lower-energy 5 meV peak in constant-q powder
scans is located approximately at the top of the dispersion
band perpendicular to the chains at q 储 ⫽ ␲ , and its position is
directly related to ⌬ ␤␤ and J 1 . From the combined high-fieldand neutron measurements ⌬ ␤␤ and J 1 can thus be reliably
determined within the established model. For PbNi2 V2 O8
these parameters are ⌬ zz ⫽3.1(3) meV, ⌬ xx ⫽⌬ y y ⫽4.0(3)meV, and J 1 ⫽⫺0.17(2) meV.4
For SrNi2 V2 O8 , which is ordered at low temperature, themodel of weakly coupled Haldane spin chains is, strictly
speaking, not applicable. Nevertheless, as the ordered moment is expected to be very small, the model is still a good
approximation, except in the direct proximity of the magnetic Bragg peaks. From the analysis of the low-energy powder spectrum for SrNi2 V2 O8 the following parameters werepreviously determined: ⌬ zz ⫽2.8(4) meV, ⌬ xx ⫽⌬ y y⫽3.9(3) meV, and J 1 ⫽⫺0.17(3) meV.4
The exact value of the remaining ‘‘high-energy’’ parameter v has little impact on the low-energy powder spectrum
and was not treated as an independent variable in Ref. 4. In
contrast, v determines the high-temperature magnetic susceptibility of the system, which, provided k BTⰇ⌬, is insen-

sitive to the details of the low-energy spectrum. This type ofexperiments yields v ⬇20.4 meV for PbNi2 V2 O8 . The
present study provides the most direct measure of the zoneboundary energy, and hence the spin-wave velocity v , defined by the location of the higher-energy peak and cutoff inthe constant-q powder scans. In analyzing these data weshall use the cross section defined in Eqs. 共5兲–共13兲 of Ref. 4
in combination with the dispersion relation given by Eqs. 共2兲
and 共11兲 above, using only v as an adjustable parameter ofthe model.
C. Fits to experimental data

In the data analysis, the powder average of the parametrized cross section was calculated numerically using aMonte Carlo routine. The result was further numerically convoluted with the Q-E resolution function of the three-axis
spectrometer,
calculatedinthe
Cooper-Nathansapproximation.19 Performing a true global fit to the data, using a single set of parameters to describe all measured scans,
was not possible because of the scattering-angle dependenceof neutron transmission in the sample. In the experiments thecylindrical samples were mounted with the symmetry axis
horizontal. This admittedly less-than-ideal geometry wasused for purely technical reasons. In this low-symmetry configuration neutron absorbtion effects, due to appreciable incoherent cross sections of Ni and V nuclei, are difficult totake into account accurately. Transmission in the sample wasstudied by measuring the intensity of incoherent elastic scattering and found to change by ⬇30%, depending on scattering geometry. Within each measured constant-Q scan
共scattering-angle range of about 30°) this correction is small
and could be ignored. However, in analyzing the entire data
sets, collected in a broad range of momentum and energy
transfers 共scattering-angle range of over 60°), a separate intensity prefactor had to be used for each constant-Q scan.
Thus, for each compound, the data were analyzed in a semiglobal fit: a single adjustable parameter v was used to simultaneously describe all measured scans, each with its own
adjustable prefactor. The parameters were refined to best fit
the data using a least-squares algorithm.
Excellent fits were obtained with v ⫽20.6(7) meV for
PbNi2 V2 O8 and v ⫽21.2(4) meV for SrNi2 V2 O8 . Simulations based on the refined parameter values are shown as
solid lines in Figs. 1–3. The fits correspond to ␹ 2 ⫽4.7 for
PbNi2 V2 O8 , and ␹ 2 ⫽3.8 for SrNi2 V2 O8 . Given that theoverall shape of the measured scans is reproduced by ourmodel quite well, the large values of ␹ 2 are most likely due
to the fact that experimental statistical error bars are lesssignificant than unavoidable systematic errors, such as multiphonon scattering in the sample. While such effects canlead to an inaccurate background subtraction, they are not
likely to influence the measured zone-boundary energy, as
the latter is accurately pinpointed by the well-defined higherenergy cutoff in the energy scans. The relevant parameters
used to fit the data, both fixed and adjustable, are summarized in Table I for both systems. The refined values for v are
to be compared to previous estimates based on the measured
gap energies and Eq. 共5兲, assuming only nearest-neighbor

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PHYSICAL REVIEW B 64 134415

TABLE I. Parameters of the model Hamiltonian used to analyze
the experimental data.
Parameter

PbNi2 V2 O8

SrNi2 V2 O8⌬ zz⌬ xx ⫽⌬ y yJ1
␣v

3.1 共fixed兲
4.0 共fixed兲
⫺0.17 共fixed兲
1.1 共fixed兲
20.6共7兲

2.8 共fixed兲
3.9 共fixed兲
⫺0.17 共fixed兲
1.1 共fixed兲
21.2 共4兲

interactions, v ⫽22.5(1.0) meV and v ⫽21.5(0.9) meV, for
PbNi2 V2 O8 and SrNi2 V2 O8 , respectively.4 For comparison,
the dashed lines in Figs. 1 are simulations based on theselatter values for PbNi2 V2 O8 .
It was important to verify that any uncertainty that may
exist for the dispersion asymmetry parameter ␣ does nothave a significant impact on our determination of v . This wasdone by analyzing the PbNi2 V2 O8 data using a larger value␣ ⫽0.58v , as previously experimentally determined for
NENP,13 rather than ␣ ⫽0.44v . Essentially the same qualityof fits was obtained, and the refined spin-wave velocity v⫽19.9(8) meV is within the error bar of that obtained with␣ ⫽0.44v , as described above. This demonstrates that thedata analysis procedure employed is fairly insensitive to dispersion asymmetry.
We see that any deviations from the model involving only
nearest-neighbor interactions are small in either material.
While for SrNi2 V2 O8 , within the error bars, the discrepancy
is undetectable, for PbNi2 V2 O8 it appears to be more substantial. It is gratifying to see that for the latter system, thebandwidth measured in this work is in excellent agreementwith estimates based on the high-temperature susceptibility
共see above兲. As explained in the previous section, the reduction of zone-boundary energy is a sign of antiferromagnetic
NNN in-chain interactions. Using Eq. 共10兲 for PbNi2 V2 O8
we obtain J ⬘ /J⬇0.05(2) and J ⬘ ⬇0.7(3) meV. Detecting
such small values of J ⬘ is only possible due to the large
staggered susceptibility of a Haldane spin chain 共the gap isrelatively small compared to v ) and, hence, the large prefactor in Eq. 共10兲.
IV. DISCUSSION

To summarize, the previously proposed nearest-neighbormodel for PbNi2 V2 O8 and SrNi2 V2 O8 should be an adequate
description of these compounds in most cases, in the entire
Brillouin zone. For PbNi2 V2 O8 the data suggest, barely outside the error bars, a small antiferromagnetic NNN in-chain
coupling of a few tenths of a meV. But how strong should
this coupling be to have an impact on long-range ordering ina doped material? We shall now discuss the general significance of NNN interactions to spin-vacancy-induced order inweakly coupled integer-spin quantum chains and then consider the particular case of PbNi2 V2 O8 .
A. Interactions between end-chain spins

The problem of spin-vacancy-induced ordering in coupledinteger-spin quantum chains with only nearest-neighbor in-

FIG. 4. A schematic representation of magnetic interactions
around a nonmagnetic impurity site. End-chain S⫽1/2 degrees of
freedom 关共a兲 and 共d兲兴 appear to either side of the impurity site 共b兲.
For simplicity, the delocalized nature of these end-chain spins is notshown. The interaction J 1 of a liberated spin 共a兲 with a neighboring
chain locally polarizes the latter, inducing a ‘‘pocket’’ of staggeredmagnetization 共c兲. The induced magnetization, in turn, couples tothe other free spin 共d兲. The resulting effective coupling competes
with direct next-nearest-neighbor in-chain interactions J ⬘ .

chain interactions has been studied theoretically by Shenderand Kivelson.9 The introduction of a nonmagnetic impurity
in place of an S⫽1 ion produces two free S⫽1/2 spins located on the two ends of the severed chain.5– 8 In each of thetwo fragments the end-chain spin is delocalized on a lengthscale defined by the intrinsic equal-time correlation length␰ ⬇6 of a Haldane spin chain. In principle, existing 3D interactions in the system should lead to long-range magnetic
ordering of these liberated end-chain spins at a nonzero temperature.
The most interesting result of Ref. 9 is that if interchaininteractions are below the threshold value at which even the
undoped system orders in three dimensions, there must be a
critical spin-vacancy concentration x c below which the system remains disordered at T⫽0. The mechanism of this behavior is illustrated in Fig. 4, which show spins in the vicinity of one nonmagnetic impurity. For simplicity, thedelocalized nature of end-chain spins is not shown. By virtue
of interchain coupling J 1 the end-chain spin on one of thefragments of the severed chain induces a ‘‘pocket’’ of staggered magnetization on a neighboring chain. The size of thepocket is determined by the correlation length ␰ . The induced staggered moment interacts 共again via J 1 ) with theend-chain spin on the other fragment. This adds up to aneffective second-order coupling between end-chain spins of⬘ 兩 ⬇z ␹ ␲ J 21 , where z is the interchain coordithe order of 兩 J eff
nation number. In Ref. 9 it is argued that this coupling is
antiferromagnetic and will bind the two end-chain spins into
a nonmagnetic singlet. As a result, the system will remain ina singlet ground state despite the introduction of a vacancy.
Another way of looking at this phenomenon is that the ef⬘ ‘‘patches’’ the broken Haldane
fective NNN interaction J eff
chain at the impurity site with an antiferromagnetic bond,albeit a weak one.
We believe that this argument is, in general, incorrect.
Indeed, from Fig. 4 it is clear that, no matter what the sign of⬘ is actually ferromagnetic (J eff⬘ ⬍0) and, if anything,J 1 is, J eff
binds the two end-chain spins into a magnetic triplet, delo-

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ZHELUDEV, MASUDA, UCHINOKURA, AND NAGLERPHYSICAL REVIEW B 64 134415

calized over the length scale ␰ . Thus, in the absence of NNN
interactions, long-range ordering should occur for arbitrarysmall impurity concentration x. The ordering temperature
will, of course, be exponentially small with x.9
This conclusion has to be modified in the presence ofactual in-chain NNN AF interactions. For J ⬘ ⲏ 兩 J ⬘e f f 兩 the arguments of Ref. 9 can be repeated verbatim to show that thesystem will remain disordered at T⫽0 for sufficiently small
x. On the other hand, the doped system will order at a nonzero temperature for J ⬘ ⱗ 兩 J ⬘e f f 兩 . At any given small concentration there will be a quantum phase transition from a disordered to ordered state at a critical strength of J ⬘ given byexperimental error bars and the semiquantitative nature of
Eq. 共12兲, we can only conclude that, in PbNi2 V2 O8 , J ⬘ 关experimental value 0.4共2兲 meV兴 is of the same order of magnitude as the critical value J c⬘ . The question of whether there
is a critical doping level below which PbNi2 V2 O8 remains
disordered stays open. Hopefully, extending the T N(x) measurements to lower concentrations, more accurate singlecrystal neutron scattering measurements of the gap energies,
and the development of a quantitative theoretical model will
resolve this interesting issue in the future.

J c⬘ ⬇z ␹ ␲ J 21 .

共i兲 This work provides a direct and accurate measurementof the zone-boundary energy in PbNi2 V2 O8 and SrNi2 V2 O8 .
The previously proposed model, based on weakly interactingS⫽1 quantum spin chains with nearest-neighbor interactions, is shown to be an adequate description of these compounds in the entire dynamic range of magnetic excitations.
共ii兲 Unlike what was thought previously, weakly coupled
Haldane spin chains should order in three dimensions at arbitrary small concentrations of nonmagnetic impurities. 共iii兲
Next-nearest-neighbor AF interactions can inhibit long-rangeordering at small impurity concentrations, but only if theyexceed a certain critical value J c⬘ . 共iv兲 In the particular caseof PbNi2 V2 O8 , NNN interactions, if at all present, are nolarger than several tenths of a meV and may be of the sameorder of magnitude as J ⬘c .

共12兲These estimates are semiquantitative at best, yet they provide
a simple physical picture and emphasize the relevant energy
scales.
B. Application to PbNi2 V2 O8

In PbNi2 V2 O8 , which remains in a gapped state at T→0,
long-range ordering can be induced by substituting some of
the S⫽1 Ni2⫹ ions with nonmagnetic Mg2⫹ ions.2 The transition was observed for all Mg concentrations x ranging from
0.1 down to 0.01.3 Though the Néel temperature T N was
found to decrease rapidly with decreasing x, there is, to date,no clear evidence of a critical concentration x c below which
the system remains disordered. The fact that in PbNi2 V2 O8
doping-induced ordering is related to end-chain spins has
been elegantly demonstrated by comparing the effects of
Mg2⫹ (S⫽0) and Cu2⫹ (S⫽1/2) substitution on the Nisites. The latter is exactly half as effective, in terms of theeffect on magnetic susceptibility, since the Cu spin and thetwo end-chain spins become bound in a doublet state. In
contrast with S⫽0 impurity substitution, the introduction ofa S⫽1/2 impurity produces only one, not two, free spins inthe system.2
To estimate the critical strength of NNN AF interactions
that would interfere with long-range ordering in weakly⬘ . Using the exdoped PbNi2 V2 O8 , we have to estimate J eff
perimental values of J and assuming z⫽2,4 from Eqs. 共12兲⬘ 兩 ⫽0.18(6) meV. Given the
and 共5兲–共7兲 we obtain J c ⫽ 兩 J eff*Present address: Oak Ridge National Laboratory, Bld. 7692, MS
6393, P.O. Box 2008, Oak Ridge, TN 37831.
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F. D. M. Haldane, Phys. Lett. 93A, 464 共1983兲; Phys. Rev. Lett.
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Y. Uchiyama, Y. Sasago, I. Tsukada, K. Uchinokura, and A. Zheludev, Phys. Rev. Lett. 83, 632 共1999兲.
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K. Uchinokura, Y. Uchiyama, T. Masuda, Y. Sasago, I. Tsukada,A. Zheludev, T. Hayashi, N. Miura, and P. Boni, Physica B 284–
288, 1641 共2000兲.
4A. Zheludev, T. Masuda, I. Tsukada, Y. Uchiyama, K. Uchinokura, P. Boni, and S.-H. Lee, Phys. Rev. B 62, 8921 共2000兲.
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M. Hagiwara, K. Katsumata, and I. Affleck, Phys. Rev. Lett. 65,
3181 共1990兲.

V. CONCLUSIONSACKNOWLEDGMENTS

We would like to thank Y. Uchiyama and I. Tsukada forthe collaboration in the early stage of this study, A. Tsvelik
for fruitful discussions, and B. Taylor, R. Roth, and K. Mohanty for technical support. The work at Brookhaven National Laboratory was carried out under Contract No. DEAC02-98CH10886, Division of Material Science, U.S.
Department of Energy. The work at the University of Tokyo
was partially supported by a Grant-in-Aid for COE Research‘‘SCP Project’’ from the Ministry of Education, Science,Sports and Culture. Oak Ridge National Laboratory is managed for the U.S. DOE by Lockheed Martin Energy Research
Corporation under Contract No. DE-AC05-96OR22464.

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