Phase diagram and spin Hamiltonian of weakly-coupled anisotropic S=1/2 chains in CuCl2*2((CD3)2SO)

0.61 MB·English

by  Y. Chen

Phase diagram and spin Hamiltonian of weakly-coupled
anisotropic S = 21 chains in CuCl2 2((CD3)2SO)
·
Y. Chen1,2, M. B. Stone1,3, M. Kenzelmann1,4, C. D. Batista5, D. H. Reich1, and C. Broholm1,2
(1) Department of Physics and Astronomy, Johns Hopkins University, Baltimore, MD 21218
(2) NIST Center for Neutron Research, National Institute of Standards and Technology, Gaithersburg, MD 20899
(3) Neutron Scattering Science Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831 USA
(4) Laboratory for Solid State Physics, ETH Zurich, CH-8093 Zurich, Switzerland
7
0 (5) Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545
0 (Dated: February 6, 2008)
2
Field-dependent specific heat and neutron scattering measurements were used to explore the

n antiferromagnetic S=21 chain compound CuCl2·2((CD3)2SO). At zero field the system acquires
a magneticlong-rangeorderbelowTN =0.93Kwithanorderedmomentof0.44µB. Anexternalfield
J alongtheb-axisstrengthensthezero-fieldmagneticorder, whilefieldsalongthea-andc-axeslead

1 toacollapseoftheexchangestabilizedorderatµ0Hc =6Tandµ0Hc =3.5T,respectively(forT =
1 0.65K)andtheformation ofan energygap intheexcitation spectrum. Werelatethefield-induced
gap to the presence of a staggered g-tensor and Dzyaloshinskii-Moriya interactions, which lead to

] effective staggered fields for magnetic fields applied along the a- and c-axes. Competition between
l
e anisotropy, inter-chain interactions and staggered fields leads to a succession of three phases as a
r- functionoffieldappliedalongthec-axis. Forfieldsgreaterthanµ0Hc,wefindamagneticstructure
t thatreflectsthesymmetryofthestaggeredfields. Thecriticalexponent,β,ofthetemperaturedriven
s
phasetransitionsareindistinguishablefromthoseofthethree-dimensionalHeisenbergmagnet,while
.
t measurements for transitions driven by quantumfluctuationsproduce larger valuesof β.
a
m
PACSnumbers: 75.25.+z,75.10.Pq,74.72.-h
-
d
n
I. INTRODUCTION an incommensurate wave vector that moves across the
o
Brillouin zone with increasing field.5 This is expected
c
[ Weakly-coupled antiferromagnetic (AF) S = 1 chains to be a rather general feature of a partially magnetized
2
canbeclosetooneorseveralquantumcriticalpointsand quantum spin system without long range spin order at
2
v serve as interesting model systems in which to explore T = 0 K. When even and odd sites of the spin chain
8 strongly-correlatedquantum order. An isolated AF S = experience different effective fields, there is an entirely
2 1 Heisenberg chain, described by different ground state which is separated by a finite gap
2
2 from excited states. This has been observed in magnetic
1 materials with staggered crystal field environments and
=J S S , (1)
0 H n· n+1 Dzyaloshinskii-Moriya(DM) interactions. Most notably,
7 Xn the generation of an excitation energy gap was first ob-
0
is quantum critical at zero temperature. Due to the ab- servedin copper benzoate using neutron scattering6 and
/
t sence of a length scale at a critical point, the spin corre- the entire excitation spectrum of the S = 1 chain in the
a 2
m lations decay as a power law, and the fundamental exci- presence of an effective staggered field was mapped out
tationsarefractionalspinexcitationscalledspinonsthat in CuCl 2DMSO (CDC)7. These experiments showed
d- carry S = 1.1,2 Interactions that break the symmetry of thatthe2e·xcitationspectruminhighstaggeredfieldscon-
2
n the ground state can induce transitions to ground states sists of solitons and breathers that can be described as
o ofdistinctlydifferentsymmetry,suchasone-dimensional bound spinon states,and that the one-dimensionalchar-
c long-range order at zero temperature in the presence acter of the spinon potential leads to high-energy exci-
:
v of an Ising anisotropy or conventional three-dimensional tations that are not present in the absence of staggered
i long-rangeorder at finite temperature in the presence of fields.8
X
weak non-frustrating interchain interactions.3 In this paper,we explorethe strongly anisotropicfield
r
a Magnetic fields break spin rotational symmetry and dependence of long range magnetic order in CDC. Us-
allow tuning the spin Hamiltonian in a controlled way. ing neutron scattering, we have directly measured the
Whenitcouplesidenticallytoallsitesinaspinchain,an order parameters associated with the various competing
externalfielduniformlymagnetizesthechainandleadsto phases. We show that a magnetic field destroys the low-
novel gapless excitations that are incommensurate with field long-range order that is induced by super-exchange
the lattice. Experimentally this is observed with meth- interactions involving Cl− cations, and that an applied
odsthatcoupledirectlytothespincorrelationfunctions, field induces a first-order spin-flop transition followed
such as neutron scattering. Inelastic neutron scattering by a second-order quantum phase transition. Field-
experiments on model AF S = 1 chain systems such as dependent specific heat measurements provide evidence
2
copperpyrazinedinitrate4 showthatuponapplicationof for a gapless excitation spectrum in the low-field phase
a uniform magnetic field there are gapless excitations at and an energy gap for larger fields. These results com-
2
bined with neutron measurements enable a comprehen-
sivecharacterizationofthedominantspininteractionsin
thissystem,includingaquantitativedeterminationofthe
c
staggeredgyromagnetictensor and the DM interactions,
which give rise to the field-induced gap.
II. CRYSTAL STRUCTURE
CDC crystallizes in the orthorhombic space group
Pnma (No. 62)with room-temperaturelattice constants
a = 8.054˚A, b = 11.546 ˚A, and c = 11.367 ˚A .9 Based
on the temperature dependence of the magnetic suscep- b
tibility, CDC was proposed as an AF S=1 chain with
2
exchange constant J = 1.43 meV.10 Inspection of the
a
crystal structure (the positions of the Cu2+ sublattice
aregiveninAppendixA)suggestsstrongsuper-exchange
interactions between Cu2+ ions via Cl− cations, so that
theCu2+ ionsformspinchainsalongthea-axis,asshown FIG. 1: Position of the Cu2+ (red), Cl− (green) and O−
in Fig. 1. Inelastic neutron scattering measurements de- (blue)inthecrystalstructureofCDC.Thedominantsuperex-
scribed below yield a consistent value of J =1.46meV. change interactions are via Cu2+-Cl−- Cu2+ paths forming
The local symmetry axis of the Cl− environment lies linear chains along the a-axis. The axes of the quasi-planar
entirely in the ac-plane, but alternates its orientation CuCl2O2 groups alternates by 22◦ along the chain direction,
leading to a staggered gyromagnetic tensor g.
alongthe a-axis. This alternationofthe localcrystalline
environment leads to an alternating g tensor in CDC of
the following form:
2.31 0 ( 1)ngs III. EXPERIMENTAL TECHNIQUES
g = 0 2.02 − 0 =gu+( 1)ngs,
n
( 1)ngs 0 2.14  −
Single crystals of CDC were obtained by slow cool-
−
  (2) ing from T = 50oC to 20oC of saturated methanol so-
where n denotes the n-th spin in the chains along the lutions of anhydrous CuCl and dimethyl sulphoxide in
2
a-direction10. gu is the uniform and gs is the staggered a 1:2 molar ratio. Emerald-green crystals grow as large
part of the g factor. tabular plates with well developed (001) faces. Specific
The crystal symmetry also allows the presence of DM heatmeasurementswereperformedonsmall,protonated
interactions11,12 crystalswithatypicalmassof15mg ina dilutionrefrig-
erator,usingrelaxationcalorimetryinmagneticfieldsup
DM = Dn (Sn−1 Sn). (3) to 9T applied along the three principal crystallographic
H · ×
Xn directions. Measurements were performed in six differ-
The DM vector D = ( 1)nDbˆ alternates from one ent magnetic fields applied along the a- and b-axes, and
n − eight different fields applied along the c-axis.
chain site to the next as can be seen from the space
Neutron diffraction measurements were performed on
group symmetry: First, the structure is invariant under
deuterated single crystals of mass 1g using the SPINS
a translation along the a-axis by two Cu sites, meaning
spectrometer at the NIST Center for Neutron Research.
that there can be at most two distinct DM vectors. Sec-
Inelastic neutron scattering experiments were performed
ond, the ac plane is a mirror plane, implying that the
onmultipleco-alignedsinglecrystalswithatotalmassof
DM vectors point along the b-axis. Third, the crystal
7.76g using the DCS spectrometer atNIST. The SPINS
structure is invariant under the combined operation of
spectrometer was configured with horizontal beam colli-
translationbyoneCusitealongthechain,andreflection
mations45′/k (58Niguide) 80′ 80′ 300′. Apyrolytic
in the ab-plane. i
− − −
graphite (PG) monochromator was used to select inci-
It was shownthat alternatingDM interactions canre-
dent neutron energies of either E = 3 meV, 5 meV (us-
sult in an effective staggered field H upon application i
st
ing the (0,0,2) reflection) or 20 meV (using the (0,0,4)
of a uniform external field H (Ref. 13). Taking into ac-
reflection). For E = 3 meV and 5 meV a cooled Be
countalsothe staggeredg tensor,the effective staggered i
filter was used before the sample to eliminate contam-
field that is generated by a uniform field can be written
ination from higher-order monochromator Bragg reflec-
as
tions. The DCS measurements were performed with an
guH = 1 D guH+gsH. (4) incident energy Ei=4.64 meV and an angle of 60o be-
st 2J × tween the incident beam and the a-axis, as detailed in
3
0.8 (a) H || a 1 (a)
H = 0 T
H = 5 T
H = 7 T
0.4 0.1
H = 9 T
H || a
Lattice ) H = 6 T
0 K0.01 H = 7 T
(b) H || b ol/ H = 9 T Lattice
H = 0 T m (b)
) 1.2 H = 2 T J/ 1
K H = 7 T (
ol/ H = 9 T C
m 0.1
J/ 0.8 H || c
( H = 4 T
C 0.01 H = 7 T
0.4
H = 9 T
0.5 1.0 1.5 2.0 2.5
0
(c) 1/T (K-1)
0.8 H || c
H = 0 T
H = 2 T
H = 3.5 T FIG. 3: Semilogarithmic plot of the specific heat C(H,T)
0.4
H = 7 T in the high-field gapped phase as a function of the inverse
H = 9 T
temperature 1/T for different magnetic fields applied along
0 the(a)aand(b)c-axes,respectively. Thesolid(dashed)line
0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
correspondstothelow-temperaturespecificheatcalculatedin
T (K)
the framework of the sine-Gordon (boson) model and given
in Eq. 10 (Eq. 6). Dotted line plotted in (a) corresponds to
FIG.2: SpecificheatofCDCasafunctionoftemperaturefor aT3 lattice contribution to thespecific heat.
magnetic field applied along the a, b and c-axis. The peaks
in the temperature dependence correspond to the onset of
long-rangemagneticorder. Thesolid(andthemostlyhidden
dashed)linecorrespondstothelow-temperaturespecificheat Ne´eltemperatureincreaseswithincreasingfieldstrength,
calculatedintheframeworkofthesine-Gordon(boson)model as shown in Fig. 2(b).
and described in Eq. 10 (Eq. 6). Dotted line plotted in (a) We analyze the specific heat for H > H to deter-
c
corresponds to aT3 lattice contribution to thespecific heat. mine the field dependence of the spin gap. The low-
temperature specific heat is first compared to a simple
model(whichwerefertoastheboson model)ofn˜ species
ofone-dimensionalnon-interactinggappedbosonswitha
Ref. 7. dispersion relation
¯hω(q˜)= ∆2+[v(q˜ q˜ )]2. (5)
0
−
IV. SPECIFIC HEAT MEASUREMENTS Here∆isthespingap,pv isthespin-wavevelocity,which
dependsonboththestrengthandorientationofthemag-
The temperature dependence of the specific heat at netic field,q˜is the chainwave-vectorq˜=πh andq˜ =π.
0
low fields has a well-defined peak for all field direc- The specific heat of a one-dimensional boson gas in this
tions, as shown in Fig. 2. The peak position varies with boson model is given by15
field strength and direction, all of which suggests that
3
the peak corresponds to the onset of long-range mag- n˜R ∆ 2 ∆
netic order. At zero-field, the transition temperature is Cmag(T)= exp( ∆/kBT). (6)
T =0.93 K, in agreement with prior zero-field heat ca- √2π (cid:18)kBT(cid:19) v −
N
pacity measurements.14 When the field is applied in the Termsproportionalto H2 andtoT3 wereincludedinthe
mirror plane, either along the a or the c-axis, the Ne´el T2
fittotakeintoaccountthe smallnuclearspinandlattice
temperatureT decreaseswithincreasingfield,asshown
N contributions, respectively. The total function fitted to
in Fig. 2(a) and Fig. 2(c). The peak becomes unobserv-
the specific heat was
able above µ H =5.7 and µ H =3.9T for fields applied
0 c 0 c
along the a- and c-axis, respectively. At higher fields, C(T,H)=C +aT3+b(H/T)2. (7)
mag
the specificheatisexponentiallyactivated,withspectral
weight shifting to higher temperatures with increasing All data for T < ∆ and H > H were fit simulta-
c
field. This is evidence for a spin gap for magnetic fields neously to Eq. 7, yielding a = 0.027(1) J/molK4 and
along the a- and c-axes. In contrast, a magnetic field b=8(2) 10−6 JmolK/T2. The averagespin-wave veloc-
·
along the b-axis enhances the magnetic order and the ity is found to be v/n˜ = 0.65(1) and 0.58(2)meV/mode
4
for fields along the a- and c-axis, respectively. From whereM M. Weusethisexpressionforanadditional
0
≡
our previous measurements of the field-induced incom- comparison to the experimentally-observed temperature
mensurate mode8 we found that the spin-wave velocity andfielddependenceofthespecificheatC(T,H)forT <
is v = 1.84(4)meV. This yields n˜ = 2.8 for the number M and H >H , in order to determine the soliton mass
1 c
ofgappedlow-energymodeswhichcontributetothespe- as a function of field strength and orientation. These
cific heat, inexcellentagreementwiththe expectationof results are shown as solid lines in Figs. 2 and 3. The
one longitudinal and two transverse modes. The result- adjustedsolitonandbreathersmassesareshowninFig.4,
ing field and orientation dependent spin gaps are shown in comparison with the results from the boson model.
inFig.4,andthecalculatedC(T,H)forthebosonmodel The adjusted boson mass is larger than the breather
are plotted as dashed lines in Figs. 2 and 3. mass,butsmallerthanthesolitonmass,asexpectedfrom
amodelwhichisonlysensitivetoanaveragegapenergy.
The soliton energy is different for fields applied along
0.4 boson spin gap
the a- and c-axis as a result of different staggered fields
soliton spin gap
V) 0.3 breather spin gap arisingfromthe staggeredg-factorandDMinteractions.
e
m The field dependence of the gap for two different field
( 0.2
directions allows an estimate of the strength of g and
wh s
0.1 D/J. The bestestimate isobtainedforhighfields where
CDC specific heat H || a-axis the effect ofthe low-fieldorderedphase is small. Assum-
0.0
ingH =cH andfitting Eq.8tothefielddependenceof
st
0.5 boson spin gap thegap,weobtainc=0.045(2)forfieldsalongthea-axis
soliton spin gap
V) 0.4 breather spin gap and c = 0.090(4) for fields along the c-axis. Comparing
me 0.3 these results with Eq. 4 we obtain
(
wh 0.2
gs gaaD/2J =0.045
0.1 H || c-axis −
gs+gccD/2J =0.09. (11)
0.0
0 2 4 6 8 10
H (T) With gaa = 2.28 and gcc = 2.12 (Ref. 10), this leads
to gs = 0.068(3) and D/J = 0.0102(5), establishing the
FIG.4: Fielddependenceoftheenergygapsforthebosonand presence of both sizeable DM interactions and a stag-
sine-Gordonmodels,derivedfromfitstospecificheatdataas
gered g factor in CDC.
shown in Fig. 3. The error bars are smaller than the symbol
size.
We also analyze the specific heat in the framework of H || b H=0T T=0.25K
the sine-Gordon model, to which the Hamiltonian can n.) 10000 Q=(h,0,0) HH==07T.2 5TT= 2TK=0.25K
be mapped in the long wave-length limit13. The sine- mi
Gordon model is exactly solvable, with solutions that per
include massive soliton and breather excitations. The nts
soliton mass, M, is given by16,17 ou
C
M J(gµBHst)(1+ξ)/2 y (
≈ J sit 1000
{B(gµJBH)(2π−β2)/4π(2− βπ2)1/4}−(1+ξ)/2. (8) Inten
Here ξ = β2/(8π β2) 1/3 for µ H 0, B =
0
− → →
0.422169,and µ0Hst is the effective staggered field. The 0.8 0.9 1.0 1.1 1.2
mass of the breathers, M , is given by
n h (r.l.u.)
M =2Msin(nπξ/2), (9)
n
FIG. 5: (h,0,0) scans through Q = (1,0,0) magnetic Bragg
wplhieedrefi1el≤d18n.≤Th[1e/tξe]mwpitehraξtudreepdenepdeinngdeenncteireolfythonestpheecaifipc- apneadkTin=C2DKCatazterTofi=el0d..25TKh,eµba0Hrin=th7e.2fi5guTr|e|bre,pTres=en0ts.2t5hKe
heat due to breathers and solitons is given by calculated peak width of Q = (1,0,0) based on the known
instrumental resolution. The peak at h=1.09 appears to be
[1/ξ](1+δ )M k T 3 k T 2 fieldandtemperatureindependentindicatingthatitdoesnot
α0 α B B
C 1+ +
arise from magnetic scattering.
∼ α=0 √2πv " Mα 4(cid:18) Mα (cid:19) #
X
3
M 2 M
α α
exp , (10)
· k T −k T
(cid:18) B (cid:19) (cid:18) B (cid:19)
5
V. NEUTRON DIFFRACTION
16000 H || b H=7.25T
e) H || c H=3T
Neutron diffraction measurements were performed in nut 14000 H = 0T
the (h,0,l) and (h,k,0) reciprocal lattice planes. Fig. 5 mi 12000
showsh-scansthroughthe (1,0,0)reflectionfordifferent er
temperatures and fields. At zero field and T = 0.3 K a s p 10000
resolution-limited peak is observed at h = 1. The peak unt 8000
o
is absent for temperatures higher than T = 0.93 K. C
No incommensurate order was observed iNn any of our y ( 6000
experiments. The reflections we found can be indexed nsit 4000
e
by (2·n,m,0) in the (h,k,0) plane, whereas magnetic Int 2000 Q=(1,0,0)
diffractionwasfoundonlyatthe(1,0,0)andthe(3,0,0)
0
reciprocal lattice points in the (h,0,l) plane for Q < 0.2 0.4 0.6 0.8 1.0 1.2
2.6˚A−1. | |
T (K)
14000 FIG. 7: Measured temperature dependence of the Q =
e) 12000 Q=(1,0,0) (1,0,0) magnetic peak intensity for zero field, µ0H =7.25T
ut appliedparalleltotheb-axisandµ0H =3Tappliedparallel
n
mi 10000 H || c T=0.25K to the c-axis. The solid lines represent the fits explained in
er H || b T=1K Sect. VC.
p
s 8000 H || b T=0.25K
nt
u
o 6000
C
sity ( 4000 (1,F0i,g0.)7masghnoewtisc Bthreaggtepmepaekrafotrurfieelddsepaepnpdlieendcealoonfg tthhee
n
e b-andc-axis. Forfieldsalongtheb-axis,theBraggpeak
nt 2000
I has higher intensity atlow T andthe enhancedintensity
0 survives to higher temperatures than in the zero field
0 2 4 6
measurement. For fields along the c-axis, the intensity
H(T) is reduced and the transition temperature is lower than
at zero field. The phase transition remains continuous,
FIG. 6: Measured field dependence of the Q=(1,0,0) mag- regardless of the direction of the field.
neticpeakintensityatT =0.25Kwiththefieldappliedalong
the c-axis and at T = 0.25K and at T = 1 K with the field
10
applied along the b-axis. The solid lines represent the fits
H || a
explained in Sect. VC.
8 H || b
H || c
a) 6
sl
e
T
H ( 4
A. HT-Phase Diagram
2
The phase diagram of CDC was further explored with
neutron diffraction for fields up to 7.25T, applied along
0
the b and c-axis. Figure 6 summarizes the magnetic 0.0 0.2 0.4 0.6 0.8 1.0 1.2
field dependence of elastic scattering measurements at T (K)
the (1,0,0) wave-vector. Applying a field along the b-
axis increases the intensity of the (1,0,0) reflection and FIG.8: Field-temperaturephasediagramofCDCdetermined
we associate this with an increase in the staggeredmag- from heat capacity measurements (filled symbols) and from
netization. In contrast, a field along the c-axis decreases neutron scattering (open symbols). The squares indicate the
the intensity of the (1,0,0) Bragg peak, first in a sharp spin-floptransition observed for afield along thec-axis. The
drop at 0.3 T suggestive of a spin-flop transition, and solid and dashed lines are a guide tothe eye.
then in a continuous decrease towards a second phase
transition at 3.9 T. The transition at 3.9 T is evidence
that exchange-inducedlong-rangeorderis suppressedby The phase diagram in Fig. 8 shows a synopsis of the
the application of fields along the c-axis. The value of results obtained from specific heat and neutron diffrac-
thecriticalfield,3.9T,isinagreementwithspecificheat tion measurements. CDC adopts long-range magnetic
measurements. order below T . This order can be strengthened by the
N
6
applicationofamagneticfieldalongtheb-axis,whichen- ble I and Fig. 10a-b) with the experiment is obtained
hances the ordered moment and increases the transition for a magnetic structure belonging to representation Γ2
temperature to long-range order. In contrast, applying as defined in Appendix B with χ2 = 3.1 and R = 0.43.
fieldsalongthea-andc-axescompeteswiththezero-field The magnetic structure is collinear with a moment of
long-rangeorder,decreasesthecriticaltemperaturelead- 0.44(5)µ along the c-axis as shown in Fig. 9(a). The
B
ing eventually to the collapse of the low-field long-range alignment of the moment along the c-axis suggests an
order at 5.9 and 3.9 T respectively, as extrapolated to easy-axisanisotropyalongthisdirection,whichisnotin-
zero temperature. cluded in the zero-field Hamiltonian defined by Eq. 1.
The ordered magnetic moment is significantly less than
thefree-ionmoment,asexpectedforasystemofweakly-
coupledS = 1 chainsclosetoaquantumcriticalpoint19.
B. Magnetic Structures 2
To understand the field-induced phase transitions, it (h,k,l) Iexp Ical
is important to determine the symmetry of the mag- ( -1, 7, 0) -4 (2) 1.38
netic structures below and above the transitions. The ( -1, 8, 0) 0 (2) 3.43
intensities of 40 magnetic Bragg peaks were measured ( 0, 3, 0) 0 (2) 0
∼
at 0.1 K at zero field, at µ H = 0.5 T and at 10 T
0 ( 0, 7, 0) 0 (2) 0
applied along the c-axis. The ordering wave-vector for
( 3, 4, 0) 1 (2) 0.722
both structures is k = (0,0,0). We use group theory
( 3, 2, 0) 3 (2) 1.12
to identify the spin configurations consistent with the
( 5, 1, 0) 3 (2) 4.17
wave-vector k for the given crystal symmetry, and com-
pare thestructure factor of possible spin configurations ( 3, 5, 0) 4 (2) 9.11
with the observed magnetic Bragg peak intensities. The ( 5, 2, 0) 5 (2) 2.21
group theoretical analysis which leads to the determina- ( 3, 7, 0) 5 (2) 3.7
tion of the symmetry-allowed basis vectors is presented ( 3, 6, 0) 5 (2) 0.344
in Appendix B. ( -1, 5, 0) 5 (2) 3.32
At zero field and T = 0.1 K, best agreement (see Ta-
( 5, 3, 0) 5 (2) 3.09
( 1, 8, 0) 5 (2) 3.43
( 1, 7, 0) 5 (2) 1.38
c c
( 1, 5, 0) 5 (2) 3.32
+
( 1, 3, 0) 6 (2) 6.55
3 (a) 3 (b)
( 5, -1, 0) 7 (2) 4.17
-
J’ 4 J’ 4 ( -1, 6, 0) 7 (2) 9.76
+
( -1, 3, 0) 8 (2) 6.55
1 ( 1, 6, 0) 8 (2) 9.76
1J 2 J -2
( 5, 0, 0) 11 (2) 2.57
a a
TABLE I: Measured magnetic Bragg peak intensities at
c c µ0H = 0 T, compared with the best fit with χ2 = 3.1 and
R = 0.43. The found magnetic structure belongs to Γ2 and
3 (c) 3 (d) H the moment is aligned along the z-axis. The next best fit
yieldsamagneticstructurethatbelongs toΓ5 with themag-
J’ 4 J’ 4 netic moments aligned along the y-axis, with χ2 = 5.2 and
H R=0.67.
st
1 2 1
J J 2
At µ H = 0.75 T after the system undergoes a spin-
0
a a flop transition, the magnetic order can be described by
the representation Γ5 with χ2 = 1.75 and R = 0.49 (
Table II and Fig. 10c) . Slightly lower values for χ2 are
FIG. 9: Magnetic structures at zero field (a), 0.75T (b) and
obtained if a small antiferromagnetic moment belonging
10T (c),with thefield H applied along thec-axis. Thesolid
to Γ8 is also allowed for. As shown in Fig. 9(b), the
arrows indicate the spin directions, + and - indicate spins
pointingintooppositedirections alongtheb-axis. TheCu2+ spin structure is collinear with a magnetic moment of
positions are defined in Appendix A. Because the ordering 0.3(1)µB per Cu2+ along the b-axis (and perpendicular
wave-vector is k = (0,0,0), the spin arrangement in other to the field as expected for a spin-flop transition).
unit cells is identical. (d) Directions of the staggered fields At µ H = 10 T, the magnetic order can be described
0
Hst upon application of an external field H along the c-axis. by representation Γ8 with χ2 =2.41 and R=0.21 ( Ta-
ble III and Fig. 10) . The spin structure is a collinear
7
(h,k,l) Iexp Ical (h,k,l) Iexp Ical
( 1, 5, 0) -1 (1) 0.321 ( 5, 1, 0) -5 (2) 0.0479
( 5, 1, 0) 0 (1) 5.05 ( 5, 0, 0) -3 (2) 0
( 1, 7, 0) 0 (1) 0.0705 ( 5, -1, 0) -2 (2) 0.0479
( 1, 6, 0) 0 (1) 0.671 ( 3, 5, 0) -1 (2) 0.308
( 1, 3, 0) 0 (1) 1.54 ( 3, 7, 0) -1 (2) 0.159
( 0, 3, 0) 0 (2) 0 ( 5, 3, 0) -1 (2) 0.278
( 0, 7, 0) 0 (2) 0 ( 5, 2, 0) 0 (2) 0.273
( 3, 4, 0) 0 (1) 0.485 ( 0, 3, 0) 0 (2) 0
( 3, 2, 0) 0 (1) 1.15 ( 0, 7, 0) 0 (2) 0
( 3, 7, 0) 1 (2) 1.28 ( -1, 7, 0) 0 (2) 6.05
( 1, 8, 0) 1 (1) 0.136 ( -1, 8, 0) 1 (2) 0.792
( 3, 6, 0) 2 (2) 0.147 ( 1, 8, 0) 2 (2) 0.792
( 3, 5, 0) 2 (2) 4.87 ( 1, 6, 0) 2 (2) 22
( 5, 2, 0) 2 (2) 2.54 ( -1, 6, 0) 2 (2) 22
( 5, 3, 0) 4 (2) 3.26 ( 3, 2, 0) 4 (2) 3.57
( 5, 0, 0) 4 (2) 3.17 ( 3, 6, 0) 6 (2) 4.13
( 3, 4, 0) 7 (2) 6.04
TABLE II: Measured magnetic Bragg peak intensities at
( 1, 7, 0) 8 (2) 6.05
µ0H = 0.75 T, compared with the best fit with χ2 = 1.75
and R = 0.49. The found magnetic structure belongs to Γ5 ( -1, 5, 0) 14 (2) 14
and the moment is aligned along the y-axis. The next best ( 1, 5, 0) 20 (2) 14
fit yields a magnetic structure that belongs to Γ4 with the ( -1, 3, 0) 21 (2) 24.3
magnetic moments aligned along the x-axis, with χ2 = 5.3
( 1, 3, 0) 23 (2) 24.3
and R=0.96.
TABLE III: Measured magnetic Bragg peak intensities at
µ0H = 10 T, compared with the best fit with χ2 = 2.4 and
magnetic structure where the magnetic moments point R = 0.21. The found magnetic structure belongs to Γ8 and
along the a-axis, perpendicular to the external field and the moment is aligned along the x-axis. The next best fit
along the staggered fields in the material, as shown in yieldsamagneticstructurethatbelongs toΓ6 with themag-
Fig. 9(c). The antiferromagnetically ordered moment netic moments aligned along the z-axis, with χ2 = 8.2 and
along the a-axis is 0.49(5)µ per Cu2+. Our numerical R=0.34.
B
calculations8usingatheLanczosmethodforfinitechains
of 24 sites yield a orderedstaggeredmoment 0.55µ per
B ready perpendicular to that field direction. At higher
site, in good agreement with the experiment.
fields, however,the competition between magnetic order
The magneticstructuresrevealwhyfields alongthe a-
arising from either spin exchange or staggered fields be-
andc-axis lead to the collapse oflong-rangeorder,while
comes important, leading eventually to the phase transi-
a field along the b-axis strengthens magnetic order. A
tionatµ H =5.7Tintoagappedphasethatisdescribed
0
field along the c-axis leads to staggeredfields due to the
by the sine-Gordon model.
alternating g tensor and DM interactions, as illustrated
in Fig. 9(d). These staggered fields compete with inter-
chain interactions which favor a different antiferromag- C. Critical Exponents
netic arrangement of spins. For example, inter-chain ex-
changefavorsAF alignmentofthe neighboringmagnetic
The neutron diffraction measurements allow determi-
moments on sites 1 and 4, but the staggered fields favor
nation of the order parameter critical exponent β for
ferromagneticalignmentalongthea-axis. Thismustlead
the phase transition to long-range order. Fig. 11 shows
to an increase in quantum fluctuations which leads to a
the magnetic intensity of the (1,0,0) reflection close to
collapse of magnetic order, identifying the transition as
the critical region. The data were corrected for a non-
a quantum phase transition. This situation was recently
magnetic, constant background.
analyzed in great detail by Sato and Oshikawa.20.
The temperature scans are shown as a function of re-
Application ofa field along the b-axisdoes notinduce duced temperature (1 T/T ) for zero field, µ H =
c 0
staggeredfields,andsomerelyquenchesquantumfluctu- 7.25 T applied along th−e b-axis and µ H = 3 T applied
0
ationsforsmallfieldstrengths,therebyleadingto anen- along the c-axis. Power law fits were performed using
hancement of the antiferromagnetically orderedmoment
as observed in the experiment. Application of a field I(H) T T 2β. (12)
N
alongthea-axisshouldnotleadtoaspin-floptransition, ∝| − |
because the moments in the zero-field structure are al- It was found that β = 0.30(1) for zero field, β =0.35(1)
8
transitions discussedhere, because here the field induces
calculated intensity calculated intensity an ordered state starting from a paramagnetic state.
The field scans are shown as a function of reduced field
0 2 4 6 8 10 12 0 2 4 6 8 10 12
14
exp. Int. (arb. units) 11022468 (a) HTG 2==00.T1K (b) HTG 5==y00.T1K 24681102 exp. Int. (arb. units) if|1nou−onrdHdet/hrHattco|β.dP=etoew0r.em4r0Iin0l(a(Tew1)t)hfi∝feotsr|cHTwriet−=riceHa0ml.c2e|a52xdβpKeouansneinndtgβs a=nd0.4it8(1w1(a33s))
z
0 0
for T = 1 K. Fits were performed over the range 0.2 in
6 30
b. units) 45 (c) HTG 5==y00..17K5T (d) HTG 8==x01.01TK 2205 b. units) rpeodFnuoecrnettdshβfieetfleadml.lpinertahteurreandgreiv0e.n30tran0s.i3ti6ond,epthenedcirnigticoanltehxe-
exp. Int. (ar 123 51105 exp. Int. (ar dexirpeocntieonntsooffththeraeep-pdliimedenfiseilodn,acloI∼nssinisgteunntivweirtshaltihtyecclraistsic,2a1l
towhichCDCbelongsatzerofield. Forthefield-induced
0 0
0 1 2 3 4 5 6 0 5 10 15 20 25 30 transitions, the Ising character of the spin symmetry is
calculated intensity calculated intensity
enhancedduetothepresenceofauniformandperpendic-
ular staggered field. The critical exponent for the field-
FIG. 10: (a-d) Observed magnetic intensities plotted as a
induced transitions, β = 0.400(1) and β = 0.481(3), are
function of calculated intensities, showing the quality of our
fits. (a-b) compares the best fit for µ0H = 0 T with the however higher than the critical exponent β = 0.325 of
secondbestfit,while(c-d)showthebestfitsforµ0H =0.75T the three-dimensional Ising universality class.
and µ0H =10T, respectively. For an Ising quantum phase transition (T = 0), the
effective dimension of the quantum critical point is D =
d+z where d = 3 is the spatial dimension and z = 1 is
the dynamical exponent. Since D = 4 is the upper crit-
for µ H = 7.25 T and β = 0.36(1) for µ H = 3 T. Fits
0 0 ical dimension, the quantum critical point is Gaussian
were performed over a reduced temperature of 0.2.
(with logarithmic corrections) and β = 0.5. Therefore,
by measuring the field dependence of the order param-
eter at different temperatures, we expect to observe a
10000
H=0T crossover between the classical (finite T and β = 0.325)
ute) H=7.25T andthe quantumphase transitions (T =0 and β =0.5).
n H=3T
mi T=1K Thisseemstobethenaturalexplanationfortheobserved
er T=0.25K values of β close to 0.5. A similar increase in the appar-
p
s entβcriticalexponentwasobservedinthesingletground
unt 1000 state system PHCC.22
o
C
y (
sit
n VI. INELASTIC NEUTRON SCATTERING
e
Int Q=(1,0,0)
100 A. Exchange Interactions
0.01 0.1
1-T/T; |1-H/H|
c c Tocharacterizethe Hamiltonianandaccuratelydeter-
minethestrengthofthespininteractionalongthechain,
FIG. 11: Logarithmic plot of the background corrected Q= we performed an energy scan for wave-vector transfer
(1,0,0) magnetic intensity plotted as a function of reduced Q a = π along the chain. At this wave-vector the two-
temperature (open symbols) or versus reduced field (filled spi·non spectrum characteristic of the AF S = 1 chain
symbols). Dashed lines corresponds to the power law fits de- 2
should be concentrated over a small energy window and
scribedinthetext. Forthetemperaturescans,µ0H =7.25T form a well-defined peak located at h¯ω = πJ/2. Fig. 12
was applied along the b-axis, and µ0H = 3 T was applied showsthatthisisindeedthecase,andthatforCDCthis
alongthec-axis. Forthefieldscansatconstanttemperature,
well-defined excitation is located at h¯ω = 2.43(2) meV.
the field was applied along the b-axis for T =1K and along
thec-axis for T =0.25K. A fit to the exact two-spinon cross-section23 convolved
with the experimental resolution yields an excellent de-
scription of the observed spectrum, as shown in Fig. 12
Figure11alsoshowstwofieldscans: onescanwiththe for a value of intra-chain exchange J = 1.46(1) meV.
field applied along the c-axis and performed at 0.25 K, This value is very similar to that obtained from the
and a scan with the field applied along the b-axis, per- temperature dependence of the magnetic susceptibility,
formed at 1 K. The latter transition is unlike all other which was 1.43meV.10
9
1.00
200 SPINS CDC T=0.1K CDC T = 40mK
Ef=5meV Q=(0.5,k,0)
H = 0T
9 foc. blades 0.75
H = 11T
min) 150 eV)
2 m
ounts ( 100 w<> (h 0.50
C
0.25
50
0.00
0.0 0.2 0.4 0.6 0.8 1.0
1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2
h (r.l.u.)
h w (meV)
FIG. 13: First moment hω˜i = ¯h2 dωωI(h,ω) of the well-
FIG. 12: Neutron scattering intensity as a function of en- h
defined excitations only as a function of wave-vector h along
Terhgeymtreaanssuferermfoernztewroa-sfipeledrffoorrmhed=w0i.t5hm9eaansuarlyedzeursbinlagdSePchINanS-. the chain, for zero field and µ0H=1R1 T. The raw scattering
datawerecorrectedforatime-independentbackgroundmea-
nels set to reflect E = 5 meV to the center of the detec-
f sured at negative energy transfer, for monitor efficiency, and
tor. A Be filter rejected neutrons with energies higher than for the Cu2+ magnetic form factor, folded into the first Bril-
5 meV from the detection system. The measurements were
louin zone, and put onto an absolute scale using the elastic
performed with the strongly dispersive direction along the
incoherent scattering from CDC.
scattered neutron direction, so that the measurements inte-
grate neutron scattering over wave-vectors along a weakly-
dispersive direction. The solid line represents the exact two-
spinon cross-section23 for J = 1.46(1) meV, convolved with
trum, thus excluding any contributions from continuum
theexperimental resolution.
scattering. Itwascalculatedby fitting a Gaussianto the
experimentally observed peak in the energy spectra and
by integrating the area under the Gaussian curve. ω˜
h
h i
B. Ground state energy at zero field and at µ H = 11 T is generally about 3 times smaller than
0
µ0H =11T ω˜ h at zero field. This is evidence for the presence of
h i
substantial magnetic continuum scattering because both
Inelastic neutron scattering not only allows determi- numerical and analyticalcalculations show that the first
nation of exchange constants, but also a measurement moments in an applied magnetic field should be larger
of the ground state energy. The expectation value of than this measured value. This is based upon (A) nu-
the Hamiltonian, < >, is proportional to the first mericalcalculations8 usingtheLanczosmethodforfinite
moment of the dynaHmic structure factor, defined as chainsoflengthbetween12and24spinswhichshowthat
ω˜ Q = h¯2 dωωS(Q,ω). Here the normalization of thegroundstateenergyatµ0H =11TforHst =0.075H
h i
S(Q,ω) is chosen such that the total moment sum rule is< >= 0.34J,and(B)analyticalcalculationsofthe
reads: h¯ SR(Q,ω)dω = S(S +1)/3. So if the dynamic fieldHdepend−ence of the first moment which show that
structure factor, S(Q,ω), is measured in absolute units, Eq. 14 acquires merely a small constant value with the
neutron sRcattering yields a measurement of < > as a application of uniform and staggered fields. Points (A)
H
function of field or temperature. and(B)thusshowthattheoretically,onlyaslightdepres-
For a Heisenberg AF chain, the wave-vector depen- sion of the first moment is expected upon application of
dence of ω˜ is given by magnetic fields.
h
h i BecauseourdatapresentedinFig.13includeonlyres-
2
ω˜ = < >(1 cos(πh)). (14) onantmodesthis resultindicatesthatthere aresubstan-
h
h i −3 H − tial contributions to the first moment from continuum
We use this relation to describe the measured first mo- scattering- perhaps not a surprisegiventhat continuum
ment, ω˜ . Figure 13 shows that at zero field the ex- scattering is the sole source of the first moment at zero
h
h i
perimental ω˜ follows this expression with < >= field.
h
h i H
0.4(1)J - in excellent agreement with Bethe’s result of
−
< > 0.44J (Ref. 24). ω˜ was obtained by fitting
h
theHexa≃ct−two-spinon cross-hsecition S (Q,ω) as calcu- C. Comparison with the chains-MF model
2sp
lated by Bougourzi et al.23 to the data and computing
ω˜ from the analytical expression for S (Q,ω). The magnetic order of weakly-coupled chains can be
h 2sp
h i
Fig.13showsthefirstmoment ω˜ atµ H =11Tin- described by a chain mean-field theory developed by
h 0
cludingcontributionsonlyfromshharipmodesinthespec- Schulz19. Itgivesthefollowingrelationsbetweenordered
10
magnetic momentm , T andthe averageof inter-chain was characterized by an apparent critical exponent β,
0 N
′
interactions J : which is higher than expected for a classical phase tran-
| |
sitionina3Dantiferromagnet. Theinterpretationofthis
T
J′ = N (15) result is that the experiment probes a cross over regime
| | 1.28 ln(5.8J/T ) affected by the T =0 quantum critical transition, which
N
is above its upper critical dimension and therefore mean
p
field like.
m =1.017 J′ /J. (16)
0
| |
The first relation allows anpestimate for J′ . In our
| | VIII. ACKNOWLEDGEMENTS
case, where T = 0.93 K and J = 1.5 meV, this gives
N
′ ′
J = 0.03 meV. The real value for J may be some-
| | | | We thank C. Landee, J. Copley, I. Affleck, and F.
what higher because quantum fluctuations are not fully
Essler for helpful discussions, and J. H. Chung for help
accounted for in the chain mean-field theory.
during one of the experiments. Work at JHU was sup-
The second relation predicts the ordered moment for
′ ported by the NSF through DMR-9801742 and DMR-
a Heisenberg chain magnet m = 0.14 µ for J =
0 B
| | 0306940. DCS and the high-field magnet at NIST were
0.03meV, whichis considerablylowerthanthe observed
supportedinpartbytheNSFthroughDMR-0086210and
ordered moment even if one considers a higher value
′ DMR-9704257. ORNL is managed for the US DOE by
for J due to quantum fluctuations. The large dis-
| | UT-Battelle Inc. under contract DE-AC05-00OR2272.
crepancy may be due to the Ising-type spin anisotropy
This workwasalso supportedby the Swiss NationalSci-
which fixes the zero-field ordered moment along the c-
ence Foundation under Contract No. PP002-102831.
axis, thereby quenching quantum fluctuations and en-
hancing long-range order at low temperatures. Indeed,
thisterm,nomatterhowsmall,induceslong-rangeorder
APPENDIX A: DEFINITION OF THE
even in the purely 1D model at T =0K.
MAGNETIC SUB-LATTICE IN CDC
VII. CONCLUSIONS The unit cell contains four Cu2+ sites, which occupy
the 4c position in the Pnma space group and are given
by:
CDC is a rich model system in which to study quan-
tum magnetism because an applied magnetic field leads r =(0.3209,0.25,0.3614)
1
first to a first-order phase transition and then to a novel
r =(0.8209,0.25,0.1386)
high-field phase with soliton excitations via a quantum 2
r =(0.1791,0.75,0.8614)
critical phase transition. In this paper, we character- 3
ized the material and its proximity to quantum critical- r =(0.6791,0.75,0.6386) (A1)
4
ity using specific heat and neutron scattering measure-
ThisnumberingoftheCu2+ positionsisusedthroughout
ments. We determined the phase diagram as a function
the paper.
of temperature and field applied along all three crystal-
lographic directions. The Braggpeak intensities and the
temperaturedependenceofthespecificheatdemonstrate
APPENDIX B: MAGNETIC GROUP THEORY
that when applying a field along the a- and c-axes, the
ANALYSIS
spin system undergoes a quantum phase transition due
to competing magnetic order parameters. Beyond the
Magneticstructureswereinferredfromdiffractiondata
critical field, specific heat measurements show activated
by considering only spin structures ”allowed” by the
behavior typical for a gapped excitation spectrum. The-
space group and the given ordering wave-vector k =
oretical analysis indicates that the spin gap results from
(0,0,0). Landau theory of continuous phase transitions
spinon binding due to a staggered gyromagnetic factor
implies that a spin structure transforms according to an
and DM interactions. The gap energy was extracted
irreducible representation of the little group G of sym-
from specific heat data using both a non-interacting bo- k
metryoperationsthatleavethewave-vectorskinvariant.
son model and an expression derived from a mapping of
The eigenvectors φλ of the λ-th irreducible representa-
the Hamiltonian to the sine-Gordon model. We deter-
tions Γλ were determined using the projector method.25
mined the strength of the staggered gyromagnetic fac-
They are given by
tor and the DM interactions by analyzing the different
field dependence of the gap for fields along the a- and
φλ = χλ(g)g(φ), (B1)
c-axes. A magnetic field along the c-axis leads to two
g
phase transitions: a first-orderspin-floptransition and a X
continuous phase transition at higher field to magnetic whereg isanelementofthelittlegroupandφisanyvec-
order described by only one irreducible representation. tor of the order parameter space. χλ(g) is the character
The continuous field driven quantum phase transition of symmetry element g in representation Γλ.