PHYSICAL REVIEW B

VOLUME 62, NUMBER 13

1 OCTOBER 2000-I

Magnetic excitations in coupled Haldane spin chains near the quantum critical point
A. Zheludev
Department of Physics, Brookhaven National Laboratory, Upton, New York 11973-5000

T. Masuda,* I. Tsukada,† Y. Uchiyama,‡ and K. Uchinokura*
Department of Applied Physics, The University of Tokyo, 6th Engineering Building, 7-3-1 Bunkyo-ku, Tokyo 113-8656, Japan

P. Böni
Paul Scherrer Institut, 5232 Villigen PSI, Switzerland

S.-H. LeeNIST Center for Neutron Research, National Institute of Standards and Technology, Gaithersburg, Maryland 20899
共Received 26 January 2000; revised manuscript received 2 May 2000兲
Two quasi-one-dimensional S⫽1 quantum antiferromagnetic materials, PbNi2 V2 O8 and SrNi2 V2 O8 , arestudied by inelastic neutron scattering on powder samples. A simple model Hamiltonian is proposed for thetwo systems and used to analyze the data. Quantitative information on in-chain and interchain interactions
obtained from this analysis is discussed in the context of qualitatively different ground-state properties: magnetic long-range order in SrNi2 V2 O8 and disordered ‘‘spin liquid’’ Haldane-gap state in PbNi2 V2 O8 .

I. INTRODUCTION

After two decades of intensive theoretical and experimental studies, one-dimensional 共1D兲 Heisenberg quantum antiferromagnets 共AF兲 are now rather well understood. A great
deal of work has been done on integer-spin systems that havea spin-liquid ground state and a famous Haldane gap in themagnetic excitation spectrum.1,2 The focus in quantum magnetism has now shifted towards studies of more complex
phenomena, that include interchain interactions, spin-latticecoupling, and/or spin-vacancies and substitutions. Of particular current interest is the quantum phase transition between spin-liquid 共non-magnetic兲 and ordered states. This
type of transition in gapped 1D systems occurs as 3D magnetic interactions and/or magnetic anisotropy are increased
beyond certain threshold values. Their effect is to lower theenergy of excitations at certain points in reciprocal space,and ultimately induce a soft-mode transition to a Néel-likeordered structure. An example of such behavior is found in
the extensively studied CsNiCl3 compound.3–6 The corresponding phase diagram has been worked out by several authors, including Sakai and Takahashi7 共Fig. 1兲.
The most direct way to observe such a transition experimentally is by looking at a series of isostructural compoundswith slightly different interchain coupling constants or anisotropy terms. The problem is that most known quasi-1DS⫽1 AF materials are otherwise deep inside the spin-liquidarea of the phase diagram 共good 1D systems兲, or obviously
in the 3D Néel-like or XY -like ordered phases 共Fig. 1兲.
CsNiCl3 and related compounds3,5,8 are perhaps the only systems close to the phase boundary that have been extensively
studied to date. Unfortunately, these compounds order inthree dimensions at low temperature, and have no isostructural counterpart with a spin-liquid ground state. Only abouta year ago the first quasi-1-D integer-spin AF that is still inthe spin-liquid state, but is on the verge of a 3D ordering
0163-1829/2000/62共13兲/8921共10兲/$15.00

PRB 62

instability, was characterized.9 This Haldane-gap material,
PbNi2 V2 O8 is so close to the phase boundary that LRO,
absent in the pure compound, can be induced by spinvacancy substitution.9 Moreover, an isostructural undoped
system, namely SrNi2 V2 O8 does order in three dimensions at
low temperatures. This opens a unique opportunity for comparative studies of two very similar quasi-1D materials withvastly different ground-state properties. Regrettably, sufficiently large single crystals of either material are not available to date, and for now one has to content with measurements on powder samples.

FIG. 1. D⫺J 1 Phase diagram for weakly coupled Haldane spin
chains 共Ref. 7兲, showing the location of some well-characterized
quasi-1D S⫽1 systems. For each material J 1 is scaled by the coordination number. CsNiCl3 data from Ref. 4; Y2 BaNiO5 data from
Ref. 27; NENP data from Ref. 28; AgVP2 S6 data from Ref. 29.
PbNi2 V2 O8 and SrNi2 V2 O8 are placed based on the results of thiswork. The vertical bar indicates that the SrNi2 V2 O8 compound
must be actually deeper in the ordered phase than suggested by our
analysis.
8921©2000 The American Physical Society

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A. ZHELUDEV et al.

PRB 62magnet distortion of this Néel 共collinear兲 spin arrangement is
attributed to the presence of weak Dzyaloshinskii-Moriya
off-diagonal exchange interactions in the noncentric crystal.
Preliminary inelastic neutron-scattering studies9 provided
an estimate for the interchain interaction strength. Only alimited amount of neutron data are available for PbNi2 V2 O8 ,and, to date, none for SrNi2 V2 O8 . The present paper deals
with more extensive comparative inelastic neutron-scatteringstudies of both materials.
II. A MODEL FOR MAGNETIC INTERACTIONS

The main problem with interpreting inelastic neutronscattering data from powder samples is an effective loss of
information upon spherical averaging. Indeed, the quantity ofinterest is the dynamic structure factor S(Q, ␻ ) that for eachchannel of spin polarization is a scalar function in fourdimensional E⫺Q space. In a powder sample one measures
the spherical average of this function, a scalar function defined in two-dimensional space:
S powder共 Q, ␻ 兲 ⫽14␲

冕 ␾ ␪␪dsin d S 共关 Q sin ␪ cos ␾ ,Q sin ␪ sin ␾ ,Q cos ␪ 兴 , ␻ 兲 .
FIG. 2. 共a兲 Crystal structure on PbNi2 V2 O8 and SrNi2 V2 O8 . The
magnetic chains are formed by edge-sharing NiO6 octahedra 共darkgray兲 and are bridged by VO4 tetrahedra 共light gray兲. 共b兲 A perspective view of the spiral-shaped Ni2⫹ -chains in PbNi2 V2 O8 andSrNi2 V2 O8 .

The magnetic properties of both PbNi2 V2 O8 andSrNi2 V2 O8 are due to spin S⫽1 octahedrally coordinatedNi2⫹ ions, while the V5⫹ sites are presumed to be nonmagnetic. The crystal structure, visualized in Fig. 2共a兲, istetragonal, space group I41cd, with lattice constantsa⫽12.249(3) Å, c⫽8.354(2) Å for PbNi2 V2 O8 ,10 anda⫽12.1617 Å, c⫽8.1617 Å for SrNi2 V2 O8 ,11 respectively.
The magnetic Ni sites are arranged in peculiar spiral-shaped
chains that run along the unique crystal axis, as shown in
Fig. 2共b兲. Even though all nearest-neighbor Ni-Ni bonds are
crystallographically equivalent, the spin spirals have a step-4
periodicity. The dominant magnetic interaction is antiferromagnetic, between nearest-neighbor Ni2⫹ spins within eachchain. The corresponding exchange constant, J⬇8.2 meV inboth systems, was deduced from the high-temperature part ofthe experimental ␹ (T) curves.9 The ground state of
PbNi2 V2 O8 is a Haldane singlet, and the excitation spectrum
has an energy gap, as unambiguously shown in lowtemperature ␹ (T) and C(T) measurements. The energy
gaps, 1.2 and 2.2 meV, for excitations polarized along, andperpendicular to the chain axis, respectively, were accurately
determined in high-field magnetization studies. The observed
anisotropy of the spin gap is attributed to single-ion easyaxis magnetic anisotropy on the Ni sites D⬇⫺0.23 meV.
Unlike PbNi2 V2 O8 , SrNi2 V2 O8 orders magnetically inthree dimensions at T N⫽7 K. The magnetic structure has notbeen determined to date, but, according to bulk measurements, is of a weak-ferromagnetic type, with a dominant antiferromagnetic component. The ordered staggered moment
is along the unique c crystallographic axis. The weak-ferro-

共1兲

The transformation S(Q, ␻ )→S powder(Q, ␻ ) is not reversible.
The only way the full correlation function can be extracted
from the experiment is by assuming some parametrized
model and fitting it to the measured S powder(Q, ␻ ). Prior toreporting our experimental findings, we shall briefly discuss
the magnetic interactions that may play an important role in
the physics of PbNi2 V2 O8 and SrNi2 V2 O8 , and construct amodel spin Hamiltonian for these systems.
As mentioned in the introduction, all previous studies
point to that the dominant magnetic interaction is the antiferromagnetic coupling J between nearest-neighbor spins(2.78 Å) within each spiral-shaped chain. It is also clear thattaking the expectedly weaker interchain interactions into account is crucial to understanding the static and dynamic
properties. In our previous work 共Ref. 9兲 we assumed that thedominant interchain coupling is between pairs of Ni2⫹ ionsfrom adjacent chains, such that both interacting spins havethe same c-axis fractional cell coordinate, and that each Ni
site is coupled to four other sites. The resulting coupling
topology is schematized in Fig. 3共a兲. Below we make use ofa different model, visualized in Fig. 3共b兲. Here one assumes
interactions along the shortest interchain Ni-Ni bond
(5.0 Å). Coupled Ni2⫹ ions are offset relative to each otherby c/4 along the chain axis. We shall denote the corresponding exchange constant as J 1 . It is clear that in a powder
experiment on a quasi-1-D system, one cannot hope to unambiguously determine the actual interchain coupling routes,
as for low-energy excitations only the effective mean-fieldrandom-phase approximation 共RPA兲 chain-coupling constantwill be relevant. The two models will result in virtually identical dispersion relations, and the only difference will be in
how one assigns microscopic exchange constants based on
the measured transverse bandwidth of magnetic excitations.
While it makes no real difference which model is actuallyused, we chose to switch to the second model, as it appears
to be somewhat better justified from the structural point of

MAGNETIC EXCITATIONS IN COUPLED HALDANE . . .

PRB 628923

unlike that of J, is a relevant parameter and should be considered. The interchain coupling term in the Hamiltonian will
have the form:Ĥ inter–chain⫽zz
S i ⬘ ,k ⬘
关 J 1,储 S i,k兺
i,i ,k,k

⬘

⬘

x

y

x
y
⫹J 1,⬜ 兵 S i,k
S i ⬘ ,k ⬘ ⫹S i,k
S i ⬘ ,k ⬘ 其 兴 .

共4兲Here the sum is taken over pairs of nearest-neighbor spinsfrom adjacent chains.
Once again, we stress that one cannot hope to determine
all the details of magnetic interactions in a system from apowder experiment. Any data analysis will not be a uniqueinterpretation, and will depend on the model used. As will be
discussed below, however, the simple Hamiltonian proposed
above can reproduce the previously published bulk data, as
well as the new inelastic neutron-scattering results with aminimal number of adjustable parameters.
III. EXPERIMENTAL
FIG. 3. Topology of magnetic interactions PbNi2 V2 O8 andSrNi2 V2 O8 . Interchain bonds are shown as dashed lines. Solid lines
are in-chain Ni-Ni bonds. 共a兲: Model used in Ref. 9. 共b兲: Model
used in the present paper.

view. Indeed, it assumes coupling between nearest-neighbor
共as opposed to next-nearest neighbor兲 Ni sites from adjacent
chains. In addition, unlike in the previous model, where theactual exchange route is not clear, the coupled spins in the
new model are bridged by VO4 tetrahedra, suggesting a possible superexchange pathway.
Magnetic anisotropy is clearly manifest in bulk susceptibility and magnetization measurements, and must be explicitly included in the spin Hamiltonian. Because of the strong
dispersion along the chain axis, all the low-energy responseof each chain is concentrated at wave vectors close to the
1-D AF zone center. In this long-wavelength limit, two-ion
anisotropy of in-chain interactions and single-ion anisotropy
associated with individual spins cannot be distinguished. In
our model we shall therefore combine the two effects into aneffective single-ion term:Ĥ single–ion⫽Dz 2
兲 .
共 S i,k兺
i,k共2兲Here D is the anisotropy constant (D⬍0⫽easy-axis) andSi,k is the spin operator for site i in chain k. The choice of theanisotropy axis along the chain direction is based on previous bulk magnetic studies. Two-ion anisotropy being effectively included in the single-ion term, in-chain exchange interactions will be assumed to be isotropic:Ĥ in–chain⫽J

Si,k Si⫹1,k .
兺
i,k

共3兲

The magnitude of dispersion perpendicular to the chain axis
is expected to be rather small, so the entire range of wavevector transfers in the (a,b) plane will influence the lowenergy properties. For this reason, two-ion anisotropy of J 1 ,Inelastic neutron-scattering studies of PbNi2 V2 O8 andSrNi2 V2 O8 powder samples 共about 10 g each兲 were carriedout in two series of experiments. The conventional three-axis
technique was used to measure certain characteristic constant-E and constant-Q scans. These measurements were
performed at the TASP and Druchal spectrometers at thecontinuous spallation source SINQ at Paul Scherrer Institute.
Neutrons of a fixed final energy of 5 meV or 8 meV wereused with pyrolytic graphite monochromator and analyzer,
and a typical (open)⫺80⬘ ⫺80⬘ ⫺(open) ⬘ collimation setup.
For some scans a horizontally focusing analyzer was used toincrease the useful scattered intensity, in which case no collimators were used after the sample. In most of the measurements a flat analyzer was employed instead. In the 5 meVfinal configuration a Be filter was inserted after the sample tosuppress higher-order beam contamination. The spectrum ofneutrons emerging from the guide is such that no higherorder filter was required for the 8 meV-final measurements.
In the second series of experiments we took advantage ofthe area-sensitive detector 共ASD兲 three-axis setup available
for the NG-5 ‘‘SPINS’’ spectrometer installed at the National Institute of Standards and Technology. In special cases
when a large domain of E⫺Q space needs to be surveyed ina spherically symmetric sample 共such as powder兲 this technique can provide an almost tenfold increase of data collection rate, as compared to the standard three-axis geometry,without any penalty in resolution. In our experiments on
PbNi2 V2 O8 and SrNi2 V2 O8 we utilized a PG monochromator and a (open)⫺80⬘ ⫺80⬘ ⫺80⬘ (radial) array of collimators. The measurements were done in the fixed-finalenergy mode, with the central blade of the composite
analyzer tuned to E f⫽3.125 meV. In the experiment scattering events with final neutron energies in the range E f⫾0.4 meV are registered simultaneously. To suppress
higher-order beam contamination we used a Be-O filter after
the sample. The data were taken for momentum and energy
transfers of up to 2.5 Å ⫺1 and 7 meV, respectively.
In both series of experiments the sample environment wasa standard ‘‘ILL-orange’’ cryostat, and the temperature range
1.5⫺30 K was covered. The background was measured by

A. ZHELUDEV et al.

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FIG. 4. False-color plot of inelastic neutron-scattering intensities measured in PbNi2 V2 O8 共a兲 and SrNi2 V2 O8 共b兲 at T⫽1.5 K
using an area-sensitive detector setup. The background has been
subtracted, as described in the text. Shaded areas indicate the location of possible spurions.

repeating some scans with the analyzer moved away from its
elastic position by 10°. In the SPINS experiment the background due to air scattering was separately measured withthe sample removed from the spectrometer.
IV. EXPERIMENTAL RESULTS
A. Results obtained at low temperatures

The inelastic neutron-scattering intensities measured in
PbNi2 V2 O8 and SrNi2 V2 O8 at T⫽1.2 K with the ASD setup

FIG. 5. Constant-energy scans measured in PbNi2 V2 O8 共a兲 and
SrNi2 V2 O8 共b兲 powder samples at T⫽2 K in the standard three-axis
mode. The solid lines are simulations based on parameters obtainedin a global fit to the ASD data, as described in the text. Dashed lines
are similar simulations for noninteracting spiral-shaped spin chains.

FIG. 6. Typical constant-Q scans measured in PbNi2 V2 O8 powder sample at T⫽2 K in the standard three-axis mode. The solidlines are as in Fig. 5.

are shown in the false-color plot in Fig. 4 共background subtracted兲. The resolution of the area-sensitive detector isgreater than the actual energy and wave vector resolution of
the spectrometer 共0.11 meV and 0.023 Å ⫺1 at ប ␻ ⫽0). The
customary procedure is to resample the mesh data to acoarser grid. Instead, for visualization purposes, the data in
Fig. 4 were smeared using a fixed-resolution 0.5 meV⫻0.06 Å ⫺1 full width at half maximum Gaussian filter. As
reported previously,9 constant-energy scans measured in
PbNi2 V2 O8 at 4 meV energy transfer have a peculiar andvery characteristic shape. Such scans were collected at T
⫽2 K for both materials, using the 8 meV-final/flat analyzer
setup SINQ. These scans are shown in Fig. 5. In addition,
representative constant-Q scans were also measured in thisconfiguration, and are shown in Figs. 6 and 7.
Despite the different ground states, the main features ofthe powder-averaged dynamic cross section for the two materials are quite similar. This, however, is not particularly
surprising. The dynamic structure factor S(Q, ␻ ) is severely
smeared out by the spherical averaging that occurs in a powder sample. The main features are defined by the form factorof each spiral-shaped chain, and by the steep dispersion
along the chain axis, expected to be almost identical in thetwo systems. The significant difference between the two materials is expected to be in the magnitude of the interchaininteractions, and possibly single-ion anisotropy. These effects are much more difficult to observe, as they only influence the weak spin-wave dispersion in the (a,b) crystallographic plane. Our data do in fact contain relevant
information on the transverse dispersion of spin excitations,

MAGNETIC EXCITATIONS IN COUPLED HALDANE . . .

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8925

␺ 1⫽

␲d
h,
a

共8兲

␺ 2⫽

␲d
k.
a

共9兲

In these formulas the argument ␻ has been dropped. The
phases ␺ 1 and ␺ 2 represent the 3D structure factor of the
spiral-shaped spin chains, and d⫽0.08a is the offset of eachNi2⫹ -ion relative to the central axis of the corresponding
spiral chain, along the a or b axis. The polarization factors
for longitudinal 共polarization along the c axis兲 and transverse
关polarization in the (a,b) plane兴 spin excitations are definedasˆ兲 ,P 储 共 Q兲 ⫽sin2 共 Q,z

共10兲

ˆ兲 .
P⬜ 共 Q兲 ⫽1⫹cos2 共 Q,z

共11兲

The dynamic structure factors S ⬘储 and S⬜⬘ for straight 共as opposed to spiral-shaped兲 Haldane chains are written in the
single-mode approximation共SMA兲:12–14
S ⬘储 共 Q, ␻ 兲 ⫽FIG. 7. Typical constant-Q scans measured in SrNi2 V2 O8 powder sample at T⫽2 K in the standard three-axis mode. The solidlines are as in Fig. 5.

which can be extracted from a quantitative analysis, as described in the following section.
B. Data analysis

The experimental data for PbNi2 V2 O8 and SrNi2 V2 O8
were analyzed using a parametrized model magnetic dynamic structure factor S(q, ␻ ) derived within the randomphase approximation for the spin Hamiltonian 2-4, as discussed in the theory section below:
共5兲

2S 储 共 Q兲 ⫽2 cos2 ␺ 1 cos2 ␺ 2 S ⬘储 共 h,k,l 兲 ⫹cos2 ␺ 1 sin2 ␺ 2⫻ 关 S ⬘储 共 h⫹1,k,l⫹1 兲 ⫹S ⬘储 共 h⫹1,k,l⫹3 兲兴⫹sin2 ␺ 1 cos2 ␺ 2 关 S ⬘储 共 h,k⫹1,l⫹1 兲 ⫹S ⬘储 共 h,k⫹1,l⫹3 兲兴 ⫹2 sin2 ␺ 1 sin2 ␺ 2 S ⬘ 共 h⫹1,k⫹1,l⫹2 兲 , 共6兲
2S⬜ 共 Q兲 ⫽2 cos2 ␺ 1 cos2 ␺ 2 S⬜⬘ 共 h,k,l 兲 ⫹cos2 ␺ 1 sin2 ␺ 2⫻ 关 S⬜⬘ 共 h⫹1,k,l⫹1 兲 ⫹S⬜⬘ 共 h⫹1,k,l⫹3 兲兴⫹sin ␺ 1 cos ␺ 2 关 S⬜⬘ 共 h,k⫹1,l⫹1 兲 ⫹S⬜⬘ 共 h,k⫹1,l2

共12兲

1⫺cos共 Qc/4兲
Zv
␦ „ប ␻ ⫺ប ␻⬜ 共 Q兲 ….
2ប ␻⬜ 共 Q兲
共13兲

Here v ⫽2.49J is the spin-wave velocity, and Z⫽1.26 关Ref.
15, Eq. 共8.2兲. Note that in this notation Z is denoted as g兴.
Finally, the dispersion relation for weakly coupled chains are
given by1关 ប ␻ 储 共 Q兲兴 2 ⫽⌬ 2储 ⫹ v 2 sin2 共 ␲ l/2兲 ⫺ Z v J 1,储 cos共 ␲ l/2兲2

1. Model cross section

S 共 Q兲 ⫽ P 储 共 Q兲 S 储 共 Q兲 ⫹ P⬜ 共 Q兲 S⬜ 共 Q兲 ,S⬜⬘ 共 Q, ␻ 兲 ⫽1⫺cos共 Qc/4兲 Z v
␦ „ប ␻ ⫺ប ␻ 储 共 Q兲 …,
2ប ␻ 储 共 Q兲2⫹3 兲兴 ⫹2 sin2 ␺ 1 sin2 ␺ 2 S ⬘ 共 h⫹1,k⫹1,l⫹2 兲 , 共7兲⫻关 cos共 ␲ h 兲 ⫹cos共 ␲ k 兲兴共 1⫺cos ␲ l/2兲 , 共14兲1
关 ប ␻⬜ 共 Q兲兴 2 ⫽⌬⬜2 ⫹ v 2 sin2 共 ␲ l/2兲 ⫺ Z v J 1,⬜ cos共 ␲ l/2兲2
⫻关 cos共 ␲ h 兲 ⫹cos共 ␲ k 兲兴共 1⫺cos ␲ l/2兲 , 共15兲
where ⌬ 储 and ⌬⬜ are the longitudinal and transverse intrinsic
Haldane gaps for noninteracting chains, respectively. The
splitting of the triplet in our model is caused by single-ion
anisotropy:16⌬⬜ ⫽ 具 ⌬ 典 ⫺0.57D,

共16兲⌬ 储 ⫽ 具 ⌬ 典 ⫹1.41D.

共17兲

An important parameter is the mean intrinsic Haldane gap

具 ⌬ 典 ⬇(⌬ 储 ⫹2⌬⬜ )/3. This parameter is, to a good approximation, defined by J alone: 具 ⌬ 典 ⬇0.41J. It is also useful todefine the actual gaps 共excitation energies at the 3D AF zone
center兲:2⫽⌬⬜2 ⫺2Z v 兩 J 1,⬜ 兩 ,E min,⬜

共18兲

A. ZHELUDEV et al.

8926

FIG. 8. Simulated inelastic scattering cross section based onparameters obtained in a global fit to the ASD data for PbNi2 V2 O8
共a兲 and SrNi2 V2 O8 共b兲 at T⫽1.5 K.

E 2min, 储 ⫽⌬ 2储 ⫺2Z v 兩 J 1,储 兩 .

共19兲

The spherical average of Eq. 共5兲 was calculated numerically using a Monte Carlo algorithm that also eliminates the␦ functions in Eqs. 共12兲 and 共13兲. The parameters were thenrefined by a standard least-squares routine to best-fit the data.
To accelerate the fitting process the ASD data were resampled to a rectangular 0.046 Å ⫺1 ⫻0.1 meV resolution. An
additional benefit of this resampling is that it allowed us not
to worry about resolution effects. The lower 0.5 meV energy
transfer range, that contains the elastic-incoherent and possibly phonon scattering, was excluded from the fits.
2. Analysis of PbNi2 V2 O8 data

For the Pb compound the adjustable parameters were thetwo intrinsic gap energies ⌬ 储 and ⌬⬜ , and the doublet 3D
gap E min,⬜ . The singlet 3D gap was fixed to E min,储
⫽1.2 meV, as determined in high-field bulk measurements.9
The only additional parameter was an overall scale factor.
The least-squares refinement yields ⌬⬜ ⫽4.0⫾0.25 meV,⌬ 储 ⫽3.1⫾0.3 meV, and E⬜(min)⫽2.4⫾0.2 meV (J 1 ⬍0), with
␹ 2 ⫽2.3. The resulting fit is shown in Fig. 8. Substituting theobtained values into the expression for S(Q, ␻ ), performing
a powder average and convoluting the result with the spectrometer resolution function, reproduces the measured threeaxis const-E and const-Q scans rather well, as shown in solidlines in Figs. 5共a兲 and 6. The relatively large ␹ 2 of the global
fit to the ASD data is to be attributed to systematic error,
primarily due to ‘‘spurious’’ scattering. In particular, thedata are partially contaminated by spurions of type ‘‘k i →k i ’’and ‘‘k f →k f ’’ originating from the stronger Bragg powderlines from the sample. Areas that are affected by these spurious processes are shown as shaded ‘‘streaks’’ in Fig. 4.
Another prominent streak in the data is to the left from, andparallel to, the two shaded areas. The origin of this feature is
not known. It is however resolution-limited and temperature
independent, and is thus almost certainly spurious and ofnonmagnetic origin. Considering that this type of systematic

PRB 62error is unavoidable in powder experiments, the obtainedmodel fit to the data is quite acceptable.
To illustrate that the J 1 is in fact a relevant parameter, wehave performed simulations of the dynamic structure factors
using the same values ⌬⬜ ⫽4.0 meV, ⌬ 储 ⫽3.1 meV, as determined in the analysis of the ASD data, but assuming J 1,⬜
⫽J 1,储 ⫽0. This results is represented in Fig. 5 by a dashedline. The well-defined intensity maximum seen in the data at
兩 Q 兩 ⬇1.2 Å ⫺1 , is replaced with a broad monotonous featurefor uncoupled chains. In contrast, the interacting chain model
reproduces the peak rather well.
The parameters E min,⬜ and E min,储 are dependent only on
the absolute values of J 1,⬜ and J 1,储 . However, the sign ofthese coupling constants explicitly enters Eqs. 共14兲 and 共15兲,and thus directly affects S(q, ␻ ). Separate fits to the data
were performed assuming positive or negative signs . The
values quoted in the previous paragraph correspond to J 1,⬜
⬍0 and J 1,储 ⬍0. Assuming positive signs leads to a total
incompatibility between the experimental data and modelcross section. A negative sign of J 1 , that links pairs of spins
offset relative to each other along the chain axis, means thatat the mean-field level the effective inter-chain coupling is
antiferromagnetic, and suggest that the 3D magnetic zone
center 共the global minimum of the 3D dispersion relation兲 islocated at the (1,1,2) reciprocal-space point. Using the fit
results described above, from Eqs. 共18兲, 共19兲 we get J 1,⬜ ⫽⫺0.18 meV and J 1,储 ⫽⫺0.14 meV. Note that 兩 J 1 兩 is larger bya factor of 1.5–2, compared to our previous estimate( 兩 J⬜ 兩 ⫽0.096⫾0.003 meV) in Ref. 9. This discrepancy
should be partly attributed to a difference in the definition of
J 1 . In our previous model each spin was coupled to four
spins in adjacent chains 共coordination number 4兲. In thepresent model the interchain coordination number is 2. This
automatically translates into a factor of 2 for 兩 J 1 兩 within the
random phase approximation.
From the refined ⌬⬜ and ⌬ 储 we can also get the in-chain
coupling constant: J⬇9.0 meV. This value is in a better
agreement with the high-temperature susceptibility estimate
J⬇8.2 meV in Ref. 9, than our preliminary neutron result J⫽9.5 meV, from the same reference. The anisotropy constantD can be estimated from ⌬⬜ and ⌬ 储 using Eqs. 共16兲 and 共17兲.
For PbNi2 V2 O8 we get D⫽⫺0.45 meV.
To determine the relevance of the two-ion anisotropy ofJ 1 , we performed additional fits to the data, assuming J 1,⬜
⬅J 1,储 , and varying this parameter together with the intrinsic
gap energies. The best fit with this model corresponds to␹ 2 ⫽2.7 and is thus in considerably worse agreement with
our measurements than the anisotropic-J 1 model. At aglance, it may appear surprising that our refinement gives
兩 J 1,⬜ 兩 ⬎ 兩 J 1,储 兩 共easy-plane type兲, while D⬍0 共easy axis兲. This,
however, is not necessarily a contradiction. Indeed, D, whichis expected to have a large contribution from single-ion anisotropy, is primarily determined by spin-orbit interactions
on the magnetic sites themselves. In contract, the anisotropyof J 1 is influenced by spin-orbit coupling through the entire
exchange route, that in our model involves the VO4 tetrahedra. Single-ion and two-ion anisotropies can thus be considered as independent parameters.
3. Analysis of SrNi2 V2 O8 , data

Similar data analysis was performed for SrNi2 V2 O8 .
E min,储 in this case was fixed at zero value 共see discussion inMAGNETIC EXCITATIONS IN COUPLED HALDANE . . .

PRB 628927

the theory section below兲. The best fit to the ASD data is
obtained with ⌬⬜ ⫽3.9⫾0.3 meV, ⌬ 储 ⫽2.8⫾0.4 meV andE min,⬜⫽2.35⫾0.3 meV with ␹ 2 ⫽2.0, and is shown in Fig.
8共b兲. These values correspond to interchain coupling constants J 1,⬜ ⫽⫺0.18 meV and J 1,储 ⫽⫺0.15 meV, respectively,
almost identical to the corresponding PbNi2 V2 O8 values. An
important consistency check is that using the gap values toestimate J gives almost the same value as for PbNi2 V2 O8 , as
expected: J⫽8.6 meV. The easy-axis anisotropy constant in
SrNi2 V2 O8 is larger than in PbNi2 V2 O8 : D⫽⫺0.56 meV.
Simulations for constant-E and constant-Q scans measured
for SrNi2 V2 O8 in the standard three-axis mode are shown insolid lines in Figs. 5共b兲 and 7.
4. Scattering at the 3D magnetic zone center

From the point of view of spin dynamics, the main distinction between the singlet-ground-state PbNi2 V2 O8 and themagnetically ordered SrNi2 V2 O8 is expected to be the presence of a spin gap in the former system, and a gapless excitation spectrum in the latter. In principle, the gap is directly
accessible experimentally at the 3D AF zone center, where
the dispersion of magnetic excitations is a global minimum.
As mentioned above, for both vanadates the 3D zone centeris at Q(0) ⫽(1,1,2), which corresponds to a momentum transfer 兩 Q(0) 兩 ⬇1.67 Å ⫺1 . Constant-Q scans extracted from ourASD data for this momentum transfer, as well as a standard
E f ⫽5 meV constant-Q scan at this wave vector, are shownin Fig. 9. The solid line is a simulation based on the parameter values cited above. To better understand these data, wenote that near the 3D zone center the single-mode contribution to the dynamic structure factor, independently of thedetails of the spin Hamiltonian, should be of the followingform:
S 共 Q, ␻ 兲 ⬀1
␦ 共 ␻ ⫺ ␻ Q兲 ,
␻Q

2
(0) 2
2
共 ប ␻ Q兲 ⫽⌬ ⫹ v 2储 共 Q 储 ⫺Q (0)储 兲 ⫹ v⬜ 共 Q⬜ ⫺Q⬜ 兲 .
2

2

共20兲

In this formula ⌬ is the gap energy, and v 储 and v⬜ are spinwave velocities along and perpendicular to the chain axis c,respectively. The data shown in Fig. 9共c兲 were collected witha horizontally focusing analyzer. These intensities were normalized by incoherent scattering from the two systems, andthus can directly compared. At energy transfers below 1 meVin this mode we are picking up a great deal of diffuse andphonon scattering 关 (1,1,2) is an allowed nuclear Bragg
peak兴. Above this contaminated region though, in a powder
sample we effectively observe a Q integral of the cross section around the 3D zone center. The same applies to thescans in Figs. 9共a兲 and 9共b兲, where integration was performed in the range 1.6–1.7 Å ⫺1 . For the Q-integrated intensity Eq. 共20兲 gives
S 共 ␻ 兲 ⬀ 冑␻ 2 ⫺⌬ 2 .

共21兲

The linear increase of intensity seen for SrNi2 V2 O8 in Fig. 9
is thus consistent with having a gapless spin wave, but ofcourse the presence of a gap of less than 1 meV cannot be
excluded. In contrast, for PbNi2 V2 O8 , the threshold behavior, and a smaller absolute intensity between 1 and 2 meV

FIG. 9. Constant-Q scans extracted from the ASD data sets for
PbNi2 V2 O8 共a兲 and SrNi2 V2 O8 共b兲 at a momentum transfer that
corresponds to the 3D antiferromagnetic zone-center at T⫽1.5 K
共symbols兲. Unlike in SrNi2 V2 O8 , the measured inelastic intensity in
PbNi2 V2 O8 extrapolates to zero at a nonzero energy transfer 共arrows兲. This is even better seen in constant-Q scans measured in thestandard three-axis mode at T⫽2 K 共c兲. Open and solid symbols
correspond to SrNi2 V2 O8 and PbNi2 V2 O8 , respectively. In the latter plot the data for the two compounds were brought to a single
scale using the elastic incoherent signal. In all cases the solid lines
are as in Fig. 5.

energy transfer, are in agreement with our expectations for a
gap spectrum. These effects are also seen in constant-Q
scans extracted from the ASD data by resampling the pixels
in a 0.1 Å ⫺1 Q-range. Clearly, further measurements on
single crystal or, at least, aligned-powder samples, are required to unambiguously resolve the gap/no gap issue for
PbNi2 V2 O8 and SrNi2 V2 O8 .
C. SrNi2 V2 O8 : Temperature dependence

To better understand the mechanism of long-range ordering of SrNi2 V2 O8 we studied the temperature dependence of
inelastic scattering at the momentum transfer 兩 Q(0) 兩⫽1.67 Å ⫺1 in this compound. Typical data are shown in Fig.
10. The solid lines were obtained by fitting our model crosssection to the data at each temperature. The lower 0.75 meVenergy transfer range was excluded from this analysis, as thedifficult-to-estimate phonon contribution in this range is expected to increase dramatically with increasing T. The set ofindependent parameters was slightly modified. The anisotropy splitting of the triplet D, as well as both interchain
coupling constants J 1,⬜ and J 1,储 were fixed at the values determined at T⫽1.5 K. The mean gap 具 ⌬ 典 and an intensity
prefactor were refined to best-fit the scans at each temperature. The obtained temperature dependences are shown in

A. ZHELUDEV et al.

8928

PRB 62

FIG. 11. Measured temperature dependence of the mean intrinsic Haldane gap 具 ⌬ 典 共a兲 and intensity prefactor in the model crosssection described in the text 共b兲 for SrNi2 V2 O8 .
FIG. 10. Typical constant-Q scans measured in SrNi2 V2 O8
powder at different temperatures, at a momentum transfer that corresponds to the 3D antiferromagnetic zone-center. The solid lines
are fits to the model cross section, as described in the text.

Fig. 11. As previously observed in other Haldane-gap systems, 具 ⌬ 典 increases with increasing T. Even though thischange is rather small, according to Eqs. 共18兲 and 共19兲, itcorresponds to an appreciable variation of the gap in thelongitudinal mode 共Fig. 12兲.
V. THEORY AND DISCUSSION
A. Derivation of the model cross section

To calculate the effect of weak interchain interactions on
dynamic spin correlations in PbNi2 V2 O8 and SrNi2 V2 O8 weshall use the random phase approximation 共RPA兲.17 This approach for directly coupled Haldane spin chains has beensuccessfully applied, for example, to CsNiCl3 共Ref. 4兲. Morerecently it was shown to also work well for Haldane chainscoupled via classical spins.18 The only additional difficulties
in the present case arise from the complicated 3D arrangement of magnetic ions in the PbNi2 V2 O8 structure, and fromthe rather nontrivial geometry of interchain coupling. To
somewhat simplify the task we chose to break it up into two
distinct problems. First, we concerned ourselves only withthe topology of magnetic interactions and consider anequivalent Bravais lattice of spins, assuming straight spin
chains and a system of inter-chain bonds shown in Fig. 3共b兲.
In a separate step we adapted the result to the more complex
structure of PbNi2 V2 O8 and SrNi2 V2 O8 .
The bare 共noninteracting兲 susceptibility for an isolated
Haldane spin chain is known to be rather well described by
the single-mode approximation 共SMA兲:12–14

␹ 0 共 Q, ␻ 兲 ⫽1⫺cos共 Q z c/4兲2⫻Zv
⌬ ⫹ v sin 共 Q z c/4兲 ⫺ 共 ប ␻ ⫹i ⑀ 兲 2222. 共22兲

This expression should be used separately for each channelof spin polarization, with appropriate values of gap energy ⌬for each particular mode. Writing down the exact RPA equations for the model shown in Fig. 3共b兲 is straightforward, butrather tedious, due to the fact that interchain bonds connect
sites with different chain-axis cell coordinates. For the particular crystal structure, for each momentum transfer q, thisleads to RPA coupling between ␹ (q) and ␹ (q⫹nc* ) (n⫽1,2,3). The resulting RPA susceptibility contains a multitude of spin-wave branches. The calculation can be mademuch simpler for low-energy excitations, that involve only

FIG. 12. Measured temperature dependence of the 3D gap energies in for SrNi2 V2 O8 . The solid lines are as in Fig. 11.

PRB 62

MAGNETIC EXCITATIONS IN COUPLED HALDANE . . .

momentum transfers close to the 1D AF zone center. At
these wave vectors the spectrum is dominated by a single
branch for each channel of spin polarization, and the RPA
dynamic sysceptibility given by⫺1
␹ RPA
共 Q兲 ⫽ ␹ ⫺1
0 共 Q 兲关 ␹ 0 共 Q 兲 J共 Q 兲 ⫹1 兴 ,

共23兲

where
J共 Q兲 ⫽J 1 cos共 Q z c/4兲关 cos共 Q x a/2兲 ⫹cos共 Q y a/2兲兴 共24兲is the combined Fourier transform of all interchain interactions involving magnetic sites in a crystallographic unit cell.
The above RPA equation is easily solved analytically. Bytaking the imaginary part of the thus obtained ␹ RPA(Q) we
arrive at Eqs. 共12兲–共15兲.
Note that at this level approximation 关Eq. 共23兲兴 the actual
geometry of interchain interactions becomes unimportant.
For example, for the geometry of Fig. 3共a兲, the result would
be the same except that the prefactor cos(Qzc/4) in Eq. 共24兲is replaced by a constant factor of 2 共all interchain bonds areperpendicular to the chains and the interchain coordination
number is twice as large兲. As low-energy excitations are all
located around Q z c/4⬇ ␲ , the two models are indistinguishable in our experiments.
The transformation from the Bravais spin lattice 共straightchains兲 to the spiral-chain structure of PbNi2 V2 O8 andSrNi2 V2 O8 , while tedious, is quite straightforward as well.
The final results is given by Eqs. 共6兲 and 共9兲. In each spiralshaped chain the transverse displacement of magnetic sites
from the chain-axis has a step-4 periodicity. As a result, at
any wave vector (h,k,l) one simultaneously observes
straight-chain scattering calculated at wave vectors (h,k,l,),(h⫹1,k,l⫾1), (h,k⫹1,l⫾1) and (h⫹1,k⫹1,l⫹2), eachwith a particular structure factor, depending on the momentum transfer perpendicular to the chain axis. In addition to(1,1,2), wave vectors (2,2,0), (1,0,1) and (0,1,1) will also
correspond to global minima of 3D spin-wave dispersion,and magnetic Bragg reflections can be expected to occur atthese reciprocal-space points in the ordered phase of
SrNi2 V2 O8 .
B. Temperature effects
1. Disordered phase

The RPA calculation above can be repeated almost verbatim for the case T⬎0. Temperature enters the RPA equations indirectly, through an intrinsic temperature dependenceof bare susceptibilities of individual chains 关Eq. 共22兲兴. Twoeffects at T⬎0 need be considered: 共i兲 the increase of the
Haldane gap, compared to its T⫽0 value, and 共ii兲 dampingof Haldane excitations. Both phenomena have been observedin a number of model quasi-1D systems 共see, for example,
Refs. 19–21兲. Unfortunately, it is almost impossible to extract meaningful information regarding excitation lifetimes
from powder inelastic data. As most of the observed inelastic
signal originates from excitations with energies greater than
2 meV, the temperature dependence of the intrinsic Haldane
gap is much easier to observe. For this reason we analyzed
the temperature dependence of the inelastic signal measured
in SrNi2 V2 O8 using the same single-mode approximation
关Eq. 共22兲兴, as at base temperature 共see fits in Fig. 10兲, to8929

obtain the T dependences of intrinsic Haldane gap energies
共Fig. 11兲. A self-consistency check is that the refined intensity prefactor is almost T independent, as shown in Fig.
11共b兲. In other words, the decrease of actual excitation intensity is entirely due to the increase of gap energy, and theintensity scales as 1/␻ .
From the mapping of the Heisenberg Hamiltonian on thequantum nonlinear ␴ model 共NLSM兲 one expects a rather
steep increase of the gap energy with increasing temperature:22

冉⌬ 共 T 兲 ⬇⌬ 共 0 兲 ⫹ 冑2 ␲ 冑T⌬ 共 0 兲 exp ⫺冊⌬共 0 兲
.
k BT

共25兲

The actual increase of the gap energy observed experimentally, NENP „Ni共C2H8N2兲2NO2ClO4…, 19 NINAZ„Ni共C3H10N2兲2N3共ClO4兲…, 20 and Y2 BaNiO5 ,21 was found to
be consistently smaller than this NLSM prediction. The suppressed T dependence of the gap energy is also observed in
SrNi2 V2 O8 . For comparison with our data, Eq. 共25兲 is plottedin a dashed line in Fig. 11共a兲.
The temperature dependence of ⌬⬜ and ⌬ 储 , deduced fromthe fits to the data at each temperature, and shown in Fig. 12,
suggests a softening of the c-axis polarized gap mode between 5 and 10 K, i.e., approximately at T N ⫽7 K. This result
is very reassuring and suggests we are indeed dealing withthe same type of soft-mode mechanism of magnetic orderingin coupled gapped spin chains, as seen in the well-studied
CsNiCl3 compound.3–5 The fact that it is the c-axis modethat softens is consistent with the dominance of easy-axis
anisotropy in the system.
2. Placement on the phase diagram and magnetically orderedstate in SrNi2 V2 O8

The line of quantum phase transition separating the ordered and spin-liquid states can be derived from Eqs. 共16兲–
共19兲. The critical value for J 1 corresponds to the lower gap
energy 共in our case E min,储) being equal to zero. The resulting
phase diagram is very similar to the direct numerical calculations by Sakai and Takashi,7 and is shown in Fig. 1. It isclear that in our case one cannot unambiguously extract any
detailed information on interchain interactions from powder
experiments. What is relevant for the phase behavior thoughis the effective MF interchain coupling constant, i.e., theproduct of J 1 and the interchain coordination number. The
corresponding values obtained in our analysis of the powder
data are in very good agreement with those obtained in previous bulk studies, and thus inspire enough confidence toattempt placing PbNi2 V2 O8 on the phase diagram. Using experimental values for J, D, and J 1 , we can tentatively position this compound on Fig. 1 共solid circle兲, just below thespin liquid- Ising transition line.
Placing the Sr compound on the phase diagram is moredifficult. Our model cross section was derived under the implicit assumption that the system is in a nonmagnetic state,
which is not applicable to SrNi2 V2 O8 at T⬍T N . In this regime one expects an increase of the gap energy for the longitudinal mode, due to the presence of a mean static staggered field generated by the ordered staggered moment in thesystem.23,24 To adapt the MF-RPA calculation to this temperature regime one has to know the ordered moment in the

A. ZHELUDEV et al.

8930

PRB 62system. To date, powder diffraction experiments on
SrNi2 V2 O8 failed to detect any magnetic Bragg reflections inthis compound at low temperatures,25 and the magnetic orderparameter is thus expected to be very small. Provided theordered moment is small, within the accuracy of the powder
experiment, our analysis based on the assumption E min,储⫽0 isjustified to get the overall picture of interchain interactions.
However, the extracted values for J 1 and the intrinsic gap
energies cannot be considered sufficiently accurate to place
the material on the phase diagram. Indeed, the assumption
E min,储⫽0 implies that the system is right on the phase transition line and T N⬅0. In fact, SrNi2 V2 O8 must be deeper inthe ordered region of the phase diagram, than suggested by
the values for J 1 and ⌬ deduced from fits to the T⫽1.5 K
neutron data. This fact is indicated by an arrow in Fig. 1.
Note that the analysis for T⬎T N , where the excitations are
expected to have a nonzero energy gap, is not affected by theabove-mentioned artifact.

pound on the Sakai-Takahashi phase diagram. For the magnetically ordered SrNi2 V2 O8 the observed T dependence of
inelastic scattering indicates that long-range order is a resultof a soft-mode transition, much like in CsNiCl3 . A totallynew piece of information obtained in the present study is thesign of transverse dispersion of magnetic excitations. The
effective MF interchain coupling is antiferromagnetic, andthe 3D magnetic zone center is located at the (1,1,2)
reciprocal-space point. It is at this position that magnetic
Bragg scattering should occur in the ordered SrNi2 V2 O8
compound.
For a complete characterization of these new and exciting
quasi-1D systems, future aligned-powder and/or single-crystal experiments will be required. High-pressure studies, as
previously attempted for NENP,26 may lead to an observation of pressure-induced long-range ordering in a quantumdisordered magnet.

VI. SUMMARYACKNOWLEDGMENTS

The main result of this work is that a simple model with aminimal number of adjustable parameters is consistent withboth the previously published bulk data and new inelastic
neutron-scattering results. This brings strong support to theidea that PbNi2 V2 O8 and SrNi2 V2 O8 should be considered
as Haldane-gap antiferromagnets in which interchain interactions and magnetic anisotropy are just strong enough to bringthem close to the line of quantum phase transition between
the spin-liquid and Ising-like states. For the Pb-based compound, the knowledge that the ground state is indeed disordered, makes the data analysis more straightforward, and thedetermined parameters can be used to position this com-

We would like to thank Y. Sasago, who had suggested at
the early stage of this study that SrNi2 V2 O8 may be a
Haldane-gap antiferromagnet. We also thank R. Wickmann
for sending a copy of Ref. 11. Work at the University ofTokyo was supported in part by Grant-in-Aid for COE Research ‘‘Phase Control of Spin-Charge-Photon Coupled Systems’’ from the Ministry of Education, Science, Sports andCulture of Japan. Work at Brookhaven National Laboratory
was carried out under Contract No. DE-AC02-76CH00016,
Division of Material Science, U.S. Department of Energy.
Studies at NIST were partially supported by the NSF underContract No. DMR-9413101.

*Also at The Department of Advanced Materials Science, The Uni-

12

versity of Tokyo, Tokyo, Japan.
†
Present address: Central Research Institute of Electric Power Industry, 2-11-1, Iwato kita, Komae-shi, Tokyo 201-8511, Japan.
‡
Present address: ULSI Device Development Laboratory, NEC
Corporation.
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