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Dynamics and Scaling in a Quantum Spin Chain Material with Bond Randomness
T. Masuda,1 A. Zheludev,1 K. Uchinokura,2 J.-H. Chung,3,4 and S. Park3,4
1Condensed Matter Sciences Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831-6393, USA
2
The Institute of Physical and Chemical Research (RIKEN), Wako, Saitama 351-0198, Japan
3
NIST Center for Neutron Research, National Institute of Standards and Technology, Gaithersburg, Maryland 20899, USA
4Department of Materials Science and Engineering, University of Maryland, College Park, Maryland 20742, USA(Received 28 April 2004; published 12 August 2004)
Single crystal inelastic neutron scattering is used to study dynamic spin correlations in the quasione-dimensional quantum antiferromagnet BaCu2 Si0:5 Ge0:5 2 O7 , where the exchange constant fluctuates due to a random distribution of Si and Ge atoms. The measured low-energy spectrum isdominated by localized excitations and can be understood in the framework of the random singlet
model. The observed scaling relations for the frequency dependencies of the correlation length andstructure factor are in excellent agreement with recent theoretical predictions for the renormalizationgroup fixed point.
DOI: 10.1103/PhysRevLett.93.077206

PACS numbers: 75.10.Pq, 75.10.Nr, 75.40.Gb

The interplay between quenched disorder and quantum
correlations is at the heart of many exciting and diverse
phenomena in condensed matter. Among these are weak
localization, quantum Hall effect [1], percolation inquantum magnets [2], Griffith phases [3], etc. Certain
quantum systems that are critical in the absence of randomness are extremely susceptible to even weak disorder.
At sufficiently large length and time scales they behave as
if the magnitude of disorder was infinite. Such models
exhibit universal behavior and scaling that can often be
exactly predicted theoretically. Moreover, their realizations in real materials are very robust: the hard to characterize details of quenched disorder (e.g., defect size anddistribution) conveniently become irrelevant. Experiments on real-world compounds can thus be directlyinterpreted in the framework of purely hypothetical andesoteric ‘‘infinitely disordered’’ models.
The best known example is the one-dimensional (1D)
quantum S  1=2 Heisenberg antiferromagnet (AF) withrandom-bond strengths. In the absence of disorder themodel is a critical Luttinger spin liquid [4]. With disorder
it rapidly flows to the so-called random singlet (RS) phase[5,6], where randomly selected pairs of distant spins are
weakly bound into singlets. Renormalization group (RG)
theory was very successful in predicting universal thermodynamic [5–8] and spin transport [9,10] properties,
as well as universal scaling of the instantaneous [6]and dynamic [9,10] structure factors. Only a handful ofexperimental realizations of the random 1D Heisemberg
AF have been found to date [11,12]. Spin correlations were extensively studied in CD3 4 NMnx Cu1x Cl3[11,13], which realizes the random classical model, due
to the large value of spin involved (S  5=2 for Mn2 ).
However, recent theories for the dynamic scaling properties of the quantum RS phase [9,10] have not yet been
put to the test. In the present work we address this issue
in inelastic neutron scattering experiments on therandom S  1=2 quantum spin chain compound
BaCu2 Si0:5 Ge0:5 2 O7 .
Solid solutions with the general formula
BaCu2 Si1x Gex 2 O7 seem to be an almost ideal realization of the S  1=2 quantum random exchange model.
The x  0 system, BaCu2 Si2 O7 , is a prototypicalS  1=2 quasi-1D Heisenberg AF with an in-chain coupling constant J  24:1 meV [14 –16]. Weak interchaininteractions result in long-range AF ordering at TN 
9:2 K. Previous extensive neutron scattering studies
focused on the interplay between continuum and singleparticle spin excitations in this compound [17]. The
material with x  1 is very similar, but has a larger
exchange constant J  50 meV and orders at T  8:5 K[14,16]. Neither Si nor Ge atoms are directly involved in
the slightly zigzag —Cu2 —O2 —Cu2 —AF spin
chains that run along the c axis of the orthorhombic
structure, with the Cu2 sites carrying S  1=2. The
difference in J is due to the contrast in the covalent radiiof Si and Ge that indirectly affect the in-chain bondangles. In materials with intermediate Ge content J
may thus be expected to vary randomly due to random
fluctuations in the local Si=Ge environment of each chainsegment.
The bulk magnetic properties of BaCu2 Si1x Gex 2 O7
were investigated in Refs. [16] and interpreted in themean field spirit, ignoring any disorder effects. TheT curves were described in terms of an ‘‘effective’’
exchange constant Jeff that was deduced from the position of the Bonner-Fischer maximum and found to varylinearly with x. This approach entirely fails to explain an
important phenomenon observed in all 0 < x < 1 materials and absent in the two pure systems, namely, theapparent divergence of susceptibility at low temperatures.
The effect has nothing to do with three-dimensional
ordering, as T starts to increase dramatically well
above TN . In Ref. [16] such behavior was attributed to077206-1

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the Curie behavior of ‘‘chain moments,’’ though no explanation was given.
A more careful look reveals that the anomalous divergence of T is most certainly not Curie-like. In Fig. 1
we plot the magnetic susceptibility measured inBaCu2 Si0:5 Ge0:5 2 O7 and BaCu2 Si2 O7 powder samples
using a commercial SQUID magnetometer. The x  0material is disorder-free and shows conventional quasi1D behavior: a Bonner-Fischer maximum at T  163 K
is followed by 3D ordering at low temperature. In contrast, the random-bond x  0:5 compound does not ordermagnetically in the studied temperature range. Instead,following a weak Bonner-Fischer maximum at T 230 K, below T  50 K the system shows a huge increaseof susceptibility. From the log-log plot it is obvious that
this feature does not correspond to the 1=T free-spinCurie scaling, and must be related to the randomness ofexchange interactions. A key characteristic of a randombond spin chain are abundant low-energy excitations.
These correspond to transitions within the large-scale
singlets of loosely coupled spin pairs in the RS groundstate. The density of such excited states diverges as1=j!ln3 !j at ! ! 0 [5], giving rise to peculiar thermodynamic properties. In particular, TT / ln2 T=0 for T0 , where 0 is the microscopic cutoff that isroughly of the order of J. As shown by the solid line inFig. 1, our susceptibility data for BaCu2 Si0:5 Ge0:5 2 O7
measured up to T  50 K is reasonably well reproducedby this scaling relation. The fit yields 0  432 meV,which is consistent with the effective exchange constantfor x  0:5, estimated as Jeff  37 meV in Ref. [16].
The full impact of randomness on the spin correlations
in BaCu2 Si0:5 Ge0:5 2 O7 becomes apparent in inelastic
neutron scattering experiments. For these measurements

FIG. 1.
Temperature dependence of magnetic susceptibility measured in BaCu2 Si2 O7 (solid symbols) andBaCu2 Si0:5 Ge0:5 2 O7 (open symbols). The thin dashed line
indicates free-spin Curie behavior. The heavy solid and dashdotted lines are a fit to the data up to T  50 K using the exact
result for the RG fixed point. Arrows indicate the 3D ordering
temperature and the positions of the Bonner-Fischer maxima.

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we used two coaligned single crystal samples of total
mass 0.5 g and mosaic spread 0:6 . The lattice parameters(space group Pnma) are a  6:92 A, b  13:28 A, andc  9:94 A. Neutron data were taken in the energy range
3–15 meV (note: h!
!
Jeff ) using the HB1 3-axis spectrometer installed at the High Flux Isotope Reactor atORNL. Neutrons with a fixed final energy Ef  13:5 meVwere used in conjunction with a pyrolitic graphite (PG)
monochromator and analyzer and a PG higher-order filter
after the sample. Somewhat limited data were collectedin the range 0:5 < h!
! < 4 meV using the SPINS cold
3-axis spectrometer at the NIST Center for Neutron
Research. In this experiment we employed Ef  5 meV
fixed final energy neutrons with a PG monochromator,
horizontally focusing the multiblade PG analyzer andBe filter.
Typical raw constant-E scans measured at T  12 K
near the 1D AF zone center q0  3d , d  a=2 [18], areshown in solid symbols in the insets of Fig. 2 . Subtracting
a sloping background yields the scans plotted in open
symbols in the main panels. As a clear indicator ofdominant AF correlations, each scan shows a peak centered at the AF zone center. However, the shape of thisFIG. 2. Symbols:
constant-EscanscollectedinBaCu2 Si0:5 Ge0:5 2 O7 at T  12 K near the 1D AF zone centerl  3. Insets: raw data. Main panels: after subtracting a sloping
background. The hatched areas in the right indicate strongspurious scattering from Al in the sample holder. Lines andshaded and hatched peaks are as described in the text.

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feature is highly unusual. In a classical magnet with norandomness the excitations are sharp spin waves with alinear dispersion relation h!
! q  vjq  q0 jd at low energies. Additional multimagnon excitations occur at higherenergies. In the more relevant model of a disorder-freeS  1=2 quantum spin chain, the spectrum is a multispinon continuum with a sharp lower bound that follows
the same ‘‘dispersion relation’’ [19,20]. The crucial pointis that in both cases the dynamic spin correlations areconfined to a triangular ‘‘wedge’’ h!
! > h!
! q , which is asignature of coherent quasiparticle propagation. This, indeed, is the case in BaCu2 Si2 O7 [17]. In contrast, inBaCu2 Si0:5 Ge0:5 2 O7 there is clearly no sharp lower
bound to the dynamic structure factor, and the observed
peak has extended ‘‘tails’’ that stretch most of theBrillouin zone. To illustrate this fact, we have simulatedscans for the classical and quantum S  1=2 disorderfree-spin chains using the effective value J  37 meV.
The spinon continuum in the quantum model was approximated using the Müller ansatz function [20]. The
simulation takes into account experimental resolution
effects, and the results are plotted as hatched and shaded
areas in Fig. 2. The measured peaks are considerably
broader than the simulated ones. Unlike the simulations,they lack any internal structure and can be very wellapproximated by Lorentzian profiles convoluted withthe calculated Gaussian q resolution. The corresponding
Voigt fits are shown in heavy solid lines in Fig. 2. The
obtained intrinsic Lorentzian half-width " is the inverse
correlation length and is plotted as a function of energy inFig. 3.
A comparison with expectations for disorder-freechains also reveals an anomalous behavior of the intensityof scattering measured in BaCu2 Si0:5 Ge0:5 2 O7 : when
integrated over q, it increases with energy. For comparison, q-integrated intensity S! for classical spin waves
decreases with energy transfer as 1=!, and is practicallyenergy independent for the spinon continuum in theFIG. 3. Energy dependence of the reciprocal correlation
length, as measured in BaCu2 Si0:5 Ge0:5 2 O7 . Open and shaded
symbols represent cold- and thermal-neutron data, respectively.
The line is a fit using the universal scaling function of Ref. [9].

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quantum model. The area of the Lorentzian feature measured in BaCu2 Si0:5 Ge0:5 2 O7 is plotted against energy
transfer in Fig. 4, where it has been scaled by ! for thebenefit of the discussion below.
All the unusual scaling properties observed inBaCu2 Si0:5 Ge0:5 2 O7 are related to bond disorder andcan be understood within the existing theories for theRG fixed point. Here the excitations are not coherently
propagating quasiparticles as in the pure system. Instead,they are localized and diffusive in nature, and hence lackthe characteristic dispersion wedge. Their size diverges asa power of ln! at ! ! 0, as does the typical distance
between them. The dynamic structure factor was predicted to have a universal scaling form that in our casecan be written as [9,10]

1J 2Sq; ! /lnjd
=!
;Fjq

q00
J!ln3 0 =!
(1)
where J is the variance of the exchange constants in thechains, and F is a universal function. Equation (1) implies the following scaling laws:!S! / ln5 0 =!;(2)

" / ln2 0 =!:(3)

These relations fit our experimental data on
BaCu2 Si0:5 Ge0:5 2 O7 remarkably well, as shown by solidlines in Figs. 3 and 4. The energy scales obtained fromthese fits are 0  16425 meV and 0  464 meVfor Eqs. (2) and (3), respectively, reassuringly of the sameorder of magnitude as the average exchange constant in
the chains. The fact that the two values differ by a factorof 3 is not necessarily a reason for concern: the two
expressions contain different powers of the logarithmicenergy variable, and therefore involve different effective
cutoff frequencies.

FIG. 4. Energy dependence of the q-integrated intensity ofthe peak seen in constant-E scans. Open and shaded symbols
represent cold- and thermal-neutron data, respectively. The
heavy solid line is a fit using the universal scaling functionof Ref. [9].

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The actual form of the universal function F should beconsidered separately from the scaling laws. Its analyticalform was derived in Refs. [9,10] and is plotted in Fig. 3 ofRef. [10]. It can be described as a peaklike ‘‘feature’’
centered at the AF zone center q0 resting on top of an‘‘incoherent’’ flat component. The peak value is exactly
twice the incoherent level. In practice, a flat magnetic
signal is hard to separate from the background originatingfrom the spectrometer, multiphonon scattering in thesample, etc. For this reason, our data analysis procedure(fits using Voigt profiles with a sloping background) is
designed to measure only the properties of the feature.
Knowing the ‘‘dark’’ background generated by the instrument (dashed line in the insets of Fig. 2), we can put an
upper limit on the flat component of F . In all measured
scans it is considerably smaller than the amplitude of thepeak. In this our data are at variance with the analyticalpredictions for the scaling function.
An additional discrepancy is in the actual shape of thepeak feature. Theory predicts a weakly oscillatory form,
so that the central peak in F is surrounded by a pair ofshallow minima [9]. No hint of such minima are observedin our data, which instead follows a plain Lorenzian
shape. A likely source for the discrepancies is thatBaCu2 Si0:5 Ge0:5 2 O7 is, after all, not an infinitely disordered system. The authors of Ref. [10] have specifically
cautioned that higher-order renormalizations of the spinoperators, not included in their calculations, can affect
the q dependence of the structure factor of a realistic
system with finite disorder. In particular, they pointed outthat the feature in F may actually become rescaled by afactor of the order of unity, and the relative strength of theincoherent contribution may change as well.
The characteristic width of the peak in Sq; ! is notexpected to be strongly affected by higher-order renormalizations of spin operators. Making only a small leapof faith, we can thus estimate J=J by comparing the coefficient obtained in fitting Eq. (3) to the data shown inFig. 3 with the full width at half maximum of the universal scaling function of Ref. [9]. This comparison yields
J=J  0:031 for BaCu2 Si0:5 Ge0:5 2 O7 . Unfortunately,
this quantity is difficult to independently estimate basedon structural considerations. It largely depends on theradius of effect  (number of Cu—O—Cu bonds influenced) that each Ge atom substituted on a Si site has on
the nearby spin chains. The width of the distribution of
exchange constants decreases very rapidly with increasing , and the estimated J=J is compatible with 
spanning of a couple of lattice repeats.
In summary, we have found an interesting experimental realization of the quantum S  1=2 spin chain withrandom bonds. The new material allowed us to directly
measure the universal scaling laws for the dynamic spincorrelations in the random singlet spin chain. Future
experiments will tackle two important issues: the effectof finite temperature and the evolution of dynamic proper077206-4

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ties with the continuously changing disorder strength x inBaCu2 Si1x Gex 2 O7 systems. For either problem, very
little theoretical guidance is currently available.
We thank I. Tsukada for his participation in an earlystage of this project. Work at ORNL was carried out underContract No. DE-AC05-00OR22725, U.S. Department ofEnergy. Experiments at NIST were partially supported bythe NSF through DMR-0086210 and DMR-9986442.

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