Spin excitations under fields in an anisotropic bond-alternating quantum S=1 chain: contrast with Haldane spin chains

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by  M. Hagiwara

Spin excitations under fields in an anisotropic bond-alternating quantum S=1 chain:
contrast with Haldane spin chains
M. Hagiwara1,∗, L. P. Regnault2, A. Zheludev3, A. Stunault4,
N. Metoki5, T. Suzuki6, S. Suga6, K. Kakurai5, Y. Koike5, P. Vorderwisch7, and J. H. Chung8∗
1RIKEN(The Institute of Physical and Chemical Research), Wako, Saitama 351-0198, Japan
2CEA-Grenoble, DRFMC-SPSMS-MDN, 17 rue des Martyrs, 38054 Grenoble Cedex 9, France
5 3Condensed Matter Sciences Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831-6393, USA
0 4Institut Laue Langevin, 6 rue J. Horowitz, 38042 Grenoble Cedex 9, France
0 5JAERI, Advanced Science Research Center, Tokai, Ibaraki 319-1195, Japan
2 6Department of Applied Physics, Osaka University, Suita, Osaka 565-9871, Japan
7BENSC, Hahn-Meitner Institut, D-14109 Berlin, Germany
n
a 8NIST Center for Neutron Research, National Institute of Standards and Technology, Gaithersburg, Maryland 20899, USA
J (Dated: February 2, 2008)

0 Inelastic neutron scattering experiments on theS=1 quasi-one-dimensional bond-alternating an-
1
tiferromagnetNi(C9D24N4)(NO2)ClO4 havebeenperformedundermagneticfieldsbelowandabove

a critical field Hc at which the energy gap closes. Normal field dependece of Zeeman splitting of
]
l the excited triplet modes below Hc has been observed, but the highest mode is unusually small
e and smears out with increasing field. This can be explained byan interaction with a low-lying two
-
r magnoncontinuumatqk =π thatispresentindimerizedchainsbutabsentinuniformones. Above
st Hc, we find only one excited mode, in stark contrast with three massive excitations previously ob-
. servedin thestructurally similar Haldane-gap material NDMAP[A.Zheludevet al., Phys. Rev. B
t
a 68, 134438 (2003)].
m
PACSnumbers: 75.40.Gb,75.10.Jm,75.50.Ee
-
d
n
o Recentexperimentaladvancesallowedstudiesofanew as shown by magnetization and ESR measuren-
c exciting phenomenon, namely field-induced Bose con- ments [8], and neutron scattering experiments [13,
[
densation of magnons in gapped quantum magnets [1]. 14]. It is structurally similar to the Haldane ma-

1 Particularly interesting is the case of antiferromagnetic terials NENP (Ni(C2H8N2)2(NO2)ClO4) and NDMAP
v S =1spinchains. Foruniformchainsthegroundstateis (Ni(C5H14N2)2N3(PF6)) whose fieldbehaviorwasprevi-
7 anexotic quantum spin liquid with only short-rangespin ously extensively investigated [15, 16, 17]. NTENP also
0
correlations and a gap in the excitation spectrum [2]. has comparable anisotropy and energy scales, which al-
2
Antiferromagnetic S = 1 chains that are not uniform, lows a direct comparison between these systems. In the
1
0 but instead feature alternating strong and weak bonds, present study we performed neutron scattering experi-
5 are also gapped spin liquids except at a quantum criti- mentson98%deuteratedNTENPinawiderangeofap-
0 cal point [3]. However, for sufficiently strong alternation plied fields. We clearly observedZeeman splitting of the
/
t theirso-calleddimerizedgroundstateisqualitatively dis- excited triplet states below Hc, a field-induced softening
a
tinctfromtheHaldanestate[4,5,6,7,8]. Thedifferences ofthespingap,andtheresultingemergenceofantiferro-
m
aresignificantyetsubtle,andinvolvethe breakingofthe magnetic long range order (LRO) above Hc [18], similar
-
d so-called “hidden” symmetry. This symmetry is related to what is seen in NDMAP. However, certain features
n toanon-localtopological“string”orderparameter[9,10] of the spin excitation spectrum in NTENP are substan-
o that is in principle not observable experimentally. For tiallydifferent. Notabledifferencesinthespincorrelation
c
example, thermodynamic properties are expected to be function are present already at H = 0, with even more
:
v almost identical for Haldane and dimerized cases. More- striking discrepancies becoming prominent at H >H .
c
i
X over,field-induced Bosecondensationofmagnonsoccurs First we summarize the crystal and magnetic prop-
inbothtypesofspinchainsandisexpectedtoleadthem
r erties of NTENP. This compound crystallizes in the tri-
a into the same (in terms of preserved symmetries of the clinicsystem(spacegroupP¯1)[12]withlatticeconstants
wavefunction) magnetized high-field state [11]. Thus, at a=10.747(1)˚A,b=9.413(2)˚A,c=8.789(2)˚A,α=95.52(2)◦,
a first glance, for the Haldane and the dimerized spin β=108.98(3)◦ and γ=106.83(3)◦ [12]. The Ni2+ ions are
chains one can expectthe spindynamics to be similar in
bridgedbynitritogroupsalongtheaaxishavingtwodif-
a wide range of external fields. But is it indeed true? ferent bond distances of 2.142(3) and 2.432(6) ˚A. These
The recently discovered [12] nickel chain material chains are weakly coupled via intervening perchlorate
Ni(C9H24N4)(NO2)ClO4, abbreviated to NTENP, of- counter anions. The inversion centers are situated not
fers an ideal experimental approach to the above prob- on the Ni2+ ions but on the nitrito groups, so that no
lem. NTENP features bond-alternating S=1 anti- staggered components of the magnetic moments are ex-
ferromagnetic chains in the gapped dimerized phase, pected to be retained, thus resulting in occurence of the
2
longrangeorderaboveHcatlowtemperatures[18]likein
NDMAP[20,21]. Themodelspinhamiltionianiswritten
as:
H = X[JS2i−1·S2i+δJS2i·S2i+1−µBSig˜H
i
+D(Sz)2], (1)
i
where J is the large exchange constant, S2i−1, S2i and
S2i+1 the S=1 spin operators, δ the bond-alternating
ratio, g˜ the g tensor of Ni2+, µB the Bohr magneton
and H an external magnetic field, and D is a single ion
anisotropy constant. From the analysis of the magnetic
susceptibility data, the following parameter values were
evaluated from a comparison with numerical calcula-
tions[8]: thelargeexchangeconstantJ/kB =54.2K,the
bond alternating ratio δ=0.45, the single ion anistropy
constant D/J = 0.25 (D/kB=13.6 K), and gk=2.14 for
thechaindirection. Asimilarbond-alternatingratiowas
alsoevaluatedinInelasticNeutronScattering(INS)mea-
surements [13, 14]. Such a bond alternating ratio corre-
spondstothegroundstateinthedimerphase. Thiscon-
clusionwasdirectlyconfirmedbymagnetization,ESR[8]
and INS [13] experiments on NTENP.
ThreelargesinglecrystalsamplesNTENPwithatyp-
ical size of 16×12×4mm3 were prepared for the current
study by the method described in Ref. 12. The mosaic FIG. 1: (a) Field dependence of the magnetic Bragg peak
spread of the sample was typically about 1.2◦. The INS intensity at q=(1,0,0). The solid line is a fit of the experi-
measurementswereperformedonseveralthree-axisspec- mental data to the expression a(H-Hc)2βc with Hc=11.35 T
andβc=0.316between11.3Tand12T.Theβc valueisclose
trometersinstalledatdifferentcoldandthermalneutron
to the expected 3D-XY value. The dashed line is a guide for
sources (IN8, IN12, IN14 and IN22 at ILL-Grenoble, V2
the eyes. (b) Field dependenceof the gap energies measured
at HMI-Berlin, SPINS at NIST center for neutron re- in NTENP. Solid lines are calculated triplet branches in a
search and LTAS at JAERI). Specific details will be re- magnetic field and a vertical thin broken line indicates the
ported elsewhere. The scattering plane was the (a∗,c∗) critical field Hc. The line above Hc is a guide for the eyes.
plane and the external magnetic field was applied to the
b axis. Hence, the field direction was not perpendicu-
lar to the chain direction (a axis) (γ = 106.83(3)◦). In as shown by a solid line in the main panel of Fig. 3. It
this configuration, the a∗ was tilted from the a axis by is well described by the form (~ω)2 =∆2+ωm2axsin2πh,
≈30◦ and the field dependence of the (1,0,0) magnetic wherethewavevectortransferq=(h,k,l),∆=1.1meV
Braggpeakmeasuredat100mKclearlyindicatedafield- andωmax =7.2meV.The higher-energymode (abroken
induced ordering transition at H ≈11.3 T (Fig.1(a)). line)followsasimilardispersioncurvewith∆=2.2meV
c
Asafirststepinourinvestigationwecharacterizedthe andωmax =7.2meV.Apartfroma somewhatlargergap
system at zero field. In agreement with previous studies energyandbandwidth,theobservedbehaviorappearsto
[13], we observed three distinct branches of gap excita- be very similar to that previously seen in NDMAP[22].
tionsthatappearasresolution-limitedpeaksinconstant- Thefirstkeynewfindingofthepresentstudyisthatat
q scansatthe 1DAF zone-center. These dataareshown H =0therelativeintensityofthehigher-energymember
in symbols in the upper panel of Fig. 2. Lines represent of the triplet is anomalously weak. This fact is consis-
contributions of each mode, as calculated in the Single tent with the data shown in Ref. 13, but was previously
Mode Approximation (SMA), assuming a parabolic dis- overlooked. Polarization effects aside, the intensity of
persionalongthechainaxisandfullytakingintoaccount all three components of the triplet in NDMAP scale as
resolution effects, as was previously done in Ref. [13]. 1/ω to a very good approximation. Throughapplication
Each of the three observed peaks corresponds to a par- of the 1-st moment sum rule for S(q,ω) [23], it can be
ticular member of the S =1 excitation triplet that even shownthatsuchbehaviordirectlyfollowsfromtheSMA,
at H =0 is split by single-ionanisotropy. The three gap provided that (D/J)2 ≪1. Experimentally, for NTENP
energies were estimated to be 1.06±0.01, 1.15±0.01 and (where D/J is actually smaller than in NDMAP) this
1.96±0.01meV,respectively. Thedispersionofthelower scalingisviolated,andtherelativeintensityoftheupper
modes was measured all the way to the zone-boundary, mode is lower than expected by about a factor of 3. We
3
came to this conclusion after performing measurements
in several Brellouin zones maintaining q = π, to elim-
k
inate the effect of polarization-dependent coefficients in
the unpolarized neutron scattering cross section.
The suppression of the upper mode in NTENP be-
comes even more apparent in applied magnetic fields.
The two lower modes in NTENP behave almost exactly
as in NDMAP for H applied perpendicular to the spin
chains. However, unlike in NDMAP, the upper mode
in NTENP vanishes in relatively modest applied fields
of H & 4 T. This is borne out in the constant-q scans
shown in the two lower panels of Fig. 2. Thus, even
well below H the excitation spectra of NTENP and
c
NDMAP are qualitatively different. It is interesting that
while the intensities of excitations in NTENP behave
anomalously, the field dependencies of the correspond-
ing gap energies at H < H are very similar to those in
c
NDMAP. These field dependencies measuredin NTENP
are plotted in Fig. 1(b). As in the case of NDMAP,
below H these data are well described by a simple-
c FIG.2: Inelasticenergyscansnearq=(1,0,1.5)at100mKand
minded perturbation theory calculation [24]: ∆ (H) =
± H=0,2 and 4 T.
∆z+2∆y ± [(∆z−2∆y)2 + (gµBH)2]1/2 and ∆0(H) = ∆x,
where ∆ , ∆ and ∆ are the gaps at H =0 for excita-
x y z 8
tionspolarizedalongx,yandz,respectively. Thebestfit
7
isobtainedwith∆ =2.0±0.1meV,∆ =0.97±0.02meV,
z x
∆ =1.13±0.02and g=2.09. 6
y
The second crucial finding of this work is that at 5
)
H > H there is only one low-energy mode at q = π V
c c e 4 Continuum
icnonNstTanEtN-qPs.caTnhsiscoclalenctbedebseeleonwf(r8o.m5TF)i,ga.ro4unthda(t11s.h5oTw)s w (m 3 meV)
adnadtaaabroevien(1s4ta.5rkTc)otnhteracsrtitwiciatlhfiseilmdil(aHrcsc∼an1s1.m3eTa)s.uTrehdesine h 2 wh (21
1 0 h (r.l.u.) 0.5
NDMAP (Fig. 1 of Ref.16), where three distinct modes
0
are visible at H > Hc. As discussed above, the upper 0.0 0.1 0.2 0.3 0.4 0.5
mode in NTENP vanishes well below Hc. In addition, h (r.l.u.)
in NTENP the lowest-energy mode that goes soft at H
c
does not reappear at higher fields, as it does in NDMAP. FIG.3: Measureddispersionrelationofmagneticexcitations
The data for NTENP at 14.5 T show only a statistically in NTENP (open circles). Solid symbols show the positions
insignificant hint of a peak at 0.4 meV. Scans in several of peaks in the in-plane (squares) and out-of-plane (circles)
Brilloinzonesconfirmedthat the low-energymode is ab- dynamic structure obtained by Lanczos calculations. Solid
andbrokenlinesaretheresultsoffittingtotheform(~ω)2=
sent at H >H in NTENP in all polarization channels.
c ∆2+ωm2axsin2πh. Inset: excitationcontinuum(shadedarea),
ComparingthebehaviorsofNTENPandNDMAP,we
as calculated for NTENP. The lower bound (thin solid line)
conclude that the spin dynamics of anisotropic Haldane coincides with the observed position of the observed z-axis
anddimerizedS=1chainsarequalitativelydifferentboth mode(thehighest excitation mode). Thein-planemodes are
below andabove the critical field. This fact is confirmed shown bya heavy solid line.
bya recentLanczosfirst-principlesstudy ofthe dynamic
structure factors in NTENP and NDMAP, based on ac-
tual measured exchange and anisotropy parameters [25]. the neutron results. Finally, the numerical calculations
The calculated dispersion relations for NTENP (solid reproduce the key features at H > Hc: three modes for
symbolsinFig.3)matchourexperimentaldataremark- NDMAP and only one for NTENP.
ably well. The solid squares and circles correspond to An understanding of the underlying physical mecha-
peaks in the dynamic structure factors for excitations nism emerged from a numerical size-dependence analy-
polarizedparalleltotheeasyplane(Sxx andSyy)orper- sis [26] that can distinguish between long-lived single-
pendiculartoit(Szz),respectively. Thecalculationsalso particleexcitationsandpeaksin multi-particlecontinua.
predict that at H = 0 the intensity of the out-of-plane Thekeyliesinthe violation of translational symmetryin
Szz mode is only 20%ofthat forSxx, inagreementwith a dimerized chain. This symmetry breaking makes the
4
particle state. Since at any H >0 both the highest- and
lowest-energymodeshavemixedpolarizationinthe(y,z)
plane, in the high-field phase they both become subject
to decay into multi-particle states. At H > H only the
c
middle mode survives in NTENP, being polarized along
x. For this excitation all decay channels remain closed.
In conclusion,our experimental results clearly demon-
strate fundamental quantum-mechanical differences be-
tween the two exotic spin-liquid phases of uniform and
bond-alternating integral spin chains.
Acknowledgments.— This work was in part supported
by the Molecular Ensemble research program from
RIKEN and the Grant-in-Aid for Scientific Research on
Priority Areas(B): Field-Induced New Quantum Phe-
nomena in Magnetic Systems (No.13130203) from the
Japanese Ministry of Education, Culture, Sports, Sci-
ence and Technology. Work at ORNL was carried out
under DOE Contract No. DE-AC05-00OR22725. Ex-
FIG. 4: Inelastic energy scans at q=(1.1,0,1.5) and 100 mK
for H=8.5 (<Hc), 11.5 (∼Hc) and 14.5 T (>Hc). periments at NIST were supported by the NSF through
DMR-0086210 and DMR-9986442. The high-field mag-
netatNIST wasfunded byNSFthroughDMR-9704257.
wave vectors q = π and q = 0 equivalent. As a result,
k k
in NTENP around q=π there is a low-lying continuum
that consists of paired q = 0 and q = π magnons. In
k k
the presence of sufficiently strong anisotropy the upper
∗ Present address: KYOKUGEN, Osaka University, 1-3
mode at q = π is pushed up in energy and can en-
k Machikaneyama, Toyonaka560-8531, Japan
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