Dynamic scaling in the susceptibility of the spin-1/2 kagome lattice antiferromagnet herbertsmithite.
ABSTRACT The spin-1/2 kagome lattice antiferromagnet herbertsmithite, ZnCu(3)(OH)(6)Cl(2), is a candidate material for a quantum spin liquid ground state. We show that the magnetic response of this material displays an unusual scaling relation in both the bulk ac susceptibility and the low energy dynamic susceptibility as measured by inelastic neutron scattering. The quantity chiT(alpha) with alpha approximately 0.66 can be expressed as a universal function of H/T or omega/T. This scaling is discussed in relation to similar behavior seen in systems influenced by disorder or by the proximity to a quantum critical point.
arXiv:1002.1091v2 [cond-mat.str-el] 6 Apr 2010
Dynamic Scaling in the Susceptibility of the Spin-1
J.S. Helton1,3, K. Matan1a, M.P. Shores2b, E.A. Nytko2, B.M. Bartlett2c, Y. Qiu3,4, D.G. Nocera2, and Y.S. Lee1∗
1Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139
2Department of Chemistry, Massachusetts Institute of Technology, Cambridge, MA 02139
3NIST Center for Neutron Research, National Institute of Standards and Technology, Gaithersburg, MD 20899 and
4Department of Materials Science and Engineering,
University of Maryland, College Park, MD 20742
(Dated: April 7, 2010)
terial for a quantum spin liquid ground state. We show that the magnetic response of this material
displays an unusual scaling relation in both the bulk ac susceptibility and the low energy dynamic
susceptibility as measured by inelastic neutron scattering. The quantity χTαwith α ≃ 0.66 can be
expressed as a universal function of H/T or ω/T. This scaling is discussed in relation to similar
behavior seen in systems influenced by disorder or by the proximity to a quantum critical point.
2kagome lattice antiferromagnet herbertsmithite, ZnCu3(OH)6Cl2, is a candidate ma-
PACS numbers: 75.40.Gb, 75.50.Ee, 78.70.Nx
A continuing challenge in the field of frustrated mag-
netism is the search for candidate materials which dis-
play quantum disordered ground states in two dimen-
sions.In recent years, a great deal of attention has
been given to the spin-1
2nearest-neighbor Heisenberg an-
tiferromagnet on the kagome lattice, consisting of corner
sharing triangles. Given the high frustration of the lattice
and the strength of quantum fluctuations arising from
2moments, this system is a very promising can-
didate to display novel magnetic ground states, includ-
ing the “resonating valence bond” (RVB) state proposed
by Anderson. A theoretical and numerical consensus
has developed that the ground state of this system is
not magnetically ordered[2–8], although the exact ground
state is still a matter of some debate. Experimental stud-
ies of this system have long been hampered by a lack of
suitable materials displaying this motif.
The mineral herbertsmithite[9, 10], ZnCu3(OH)6Cl2,
is believed to be an excellent realization of a spin-1
kagome lattice antiferromagnet. The material consists
of kagome lattice planes of spin-1
perexchange interaction between nearest-neighbor spins
leads to an antiferromagnetic coupling of J = 17±1 meV.
Extensive measurements on powder samples of herbert-
smithite have found no evidence of long range mag-
netic order or spin freezing to temperatures of roughly
50 mK[11–13]. Magnetic excitations are effectively gap-
less, with a Curie-like susceptibility at low temperatures.
The magnetic kagome planes are separated by layers of
nonmagnetic Zn2+ions; however, it has been suggested
that there could be some site disorder between the Cu
and Zn ions[14, 15].This possible site disorder, with
≈ 5% of the magnetic Cu ions residing on out-of-plane
sites, as well as the presence of a Dzyaloshinskii-Moriya
(DM) interaction, would likely influence the low en-
ergy magnetic response.
2Cu2+ions. The su-
In this Letter we report a dynamic scaling analysis
of the susceptibility of herbertsmithite as measured in
both the bulk ac susceptibility and the low energy dy-
namic susceptibility measured by inelastic neutron scat-
tering. In particular, we find that the quantity χTαcan
be expressed as a universal function in which the en-
ergy or field scale is set only by the temperature. This
type of scaling behavior, when measured in quantum
antiferromagnets and heavy-fermion metals, has
long been associated with proximity to a quantum criti-
cal point (QCP). Power law signatures in the susceptibil-
ity have also been associated with random systems such
as Griffiths phase or random singlet phase sys-
tems. Such similarities could shed light on the relevant
low energy interactions in herbertsmithite.
Figure 1(a) shows the ac magnetic susceptibility of a
herbertsmithite powder sample as measured using a com-
mercial ac magnetometer (Quantum Design). An oscil-
lating field of 17 Oe, with a frequency of 100 Hz, was
applied along with a range of dc fields up to µ0H = 14 T.
The data were corrected for the diamagnetic contribution
by use of Pascal’s constants. These results, for data sets
with nonzero applied dc field, are plotted in Fig. 1(b)
with χ′Tα(with α = 0.66) on the y axis and the unitless
ratio µBH/kBT on the x axis. For this value of α, the
data collapse quite well onto a single curve for a range of
µBH/kBT spanning well over two decades. Scaling plots
with various exponent choices support α = 0.66 ± 0.02.
This scaling remains roughly valid up to moderate tem-
peratures, dependent upon the applied dc field. In the
data taken with µ0H = 0.5 T, shown in Fig. 1, the scal-
ing fails for temperatures greater than roughly T = 35 K;
under an applied field of µ0H = 5 T the scaling re-
mains valid to about T = 55 K. The functional form
of this collapse is qualitatively similar to the generalized
critical Curie-Weiss function seen in the heavy-fermion
compound CeCu5.9Au0.1, but with deviations demon-
strating that such a simple response function is not quite
FIG. 1: (color online) (a) The in-phase component of the ac
susceptibility, measured at 100 Hz with an oscillating field of
17 Oe. (b) A scaled plot of the ac susceptibility data measured
at nonzero applied field, plotted as χ′
the y axis and µBH/kBT on the x axis. Inset: A scaled plot
of the dc magnetization, showing MT−0.34vs µBH/kBT.
acTαwith α = 0.66 on
adequate. It should also be pointed out that in herbert-
smithite the entire bulk susceptibility obeys this scaling
relation, while in CeCu5.9Au0.1it is only the estimated lo-
cal contribution, χL(T) = [χ(T)−1−χ(T = 0)−1]−1, that
obeys scaling. A susceptibility of this form will imply a
similar scaling in the bulk dc magnetization of the sam-
ple, with MTα−1expressible as a function of H/T. As
a complementary measurement, such a scaling is shown
in the inset to Fig. 1(b).
measured up to µ0H = 14 T at temperatures ranging
from T = 1.8 K to 10 K, and is plotted as MT−0.34vs
The inelastic neutron scattering spectrum of herbert-
smithite was measured on the time-of-flight Disk Chop-
per Spectrometer (DCS) at the NIST Center for Neu-
tron Research.A deuterated powder sample of mass
7.5 g was measured using a dilution refrigerator with
an incident neutron wavelength of 5˚ A. Measurements
were taken at six different temperatures, with roughly
logarithmic spacing, ranging from 77 mK to 42 K. The
scattering data were integrated over a wide range of mo-
mentum transfers, 0.5 ≤ Q ≤ 1.9˚ A−1, to give a mea-
sure of the local response. The momentum integrated
dynamic scattering structure factor, S(ω), is shown in
Fig. 2(a). Similar to previous reports on the neutron scat-
tering spectrum of herbertsmithite, the data show a
The dc magnetization was
broad inelastic spectrum with no discernable spin gap
and only a weak temperature dependence for positive
energy transfer scattering. The negative energy trans-
fer scattering intensity is suppressed at low temperatures
due to detailed balance. The imaginary part of the dy-
namic susceptibility is related to the scattering struc-
ture factor through the fluctuation-dissipation theorem,
χ′′(ω) = S(ω)(1 − e−¯ hω/kBT). The dynamic suscepti-
bility can then be determined in a manner similar to
that used previously. For the two lowest tempera-
tures measured, detailed balance considerations will ef-
fectively suppress scattering at negative energy transfer
for values of |¯ hω| ≥ 0.15 meV. Thus these data sets are
averaged together and treated as background. This back-
ground is subtracted from the T = 42 K data, for which
the detailed balance suppression is not pronounced below
|¯ hω| = 2 meV. From this, χ′′(ω; T = 42 K) is calculated
for negative ω, and the values for positive ω are easily de-
termined from the fact that χ′′(ω) is an odd function of
ω. The dynamic susceptibility at the other temperatures
is calculated by determining the difference in scattering
intensity relative to the T = 42 K data set. It is rea-
sonably assumed that the elastic incoherent scattering
and any other background scattering are effectively tem-
perature independent. The calculated values of χ′′(ω)
at all measured temperatures are shown in Fig. 2(b).
The T = 42 K scattering data and χ′′(ω) were fit to
smooth functions for use in calculating the susceptibility
at other temperatures so that statistical errors would not
be propagated throughout the data; the smooth function
of χ′′(ω; T = 42 K) used in the calculation is also shown
in the figure.
The resulting values for χ′′(ω) follow a similar scaling
relation as the ac susceptibility, where the ratio ¯ hω/kBT
replaces µBH/kBT. In Fig. 3 we show χ′′(ω)T0.66on the
y axis and the unitless ratio ¯ hω/kBT on the x axis. The
scaled data collapse fairly well onto a single curve over
almost four decades of ¯ hω/kBT. Here we have used the
same exponent α = 0.66 that was observed in the scal-
ing of the ac susceptibility. However, the error bars on
the data allow for a wider range of exponents (α = 0.55
to 0.75) with reasonable scaling behavior. The collapse
of the χ′′(ω) data is again reminiscent of the behav-
ior observed in certain heavy-fermion metals, including
the shape of the functional form of the scaling function.
Let us assume that χ′′(ω)Tα
fermion metal CeCu5.9Au0.1 displays a scaling[21, 22]
that could be fit to the functional form F(ω/T) =
sin[αtan−1(ω/T)]/[(ω/T)2+ 1]α/2. A fit to this func-
tional form is shown as a dashed blue line in Fig. 3. This
simple form does not fit the herbertsmithite data well for
low values of ω/T. Other heavy-fermion metals[23, 24],
display a scaling relation that can be fit to the functional
form F(ω/T) = (T/ω)αtanh(ω/βT); this functional form
is similar to that used to fit the dynamic susceptibility in
La1.96Sr0.04CuO4. This functional form fits our data
∝ F(ω/T). The heavy-
FIG. 2: (color online) (a) Neutron scattering structure fac-
tor S(ω), measured using DCS integrated over wavevectors
0.5 ≤ Q ≤ 1.9˚ A−1. (b) The local dynamic susceptibility
χ′′(ω), determined as described in the text.
where indicated in this article are statistical in origin and
represent 1 standard deviation.
much better, shown (with fit parameter β = 1.66) as the
dark red line in Fig. 3. This function is somewhat un-
usual, in that for low values of ω/T it is proportional
to (ω/T)1−αrather than the expected ω/T; of course
such a dependence might be recovered at still smaller val-
ues of ω. For larger values of ω/T, this curve approaches
a power law dependence with χ′′(ω) ∝ ω−α. This is con-
sistent with the low temperature (T = 35 mK) behavior
of the dynamic susceptibility of herbertsmithite reported
Other works on kagome lattice systems have shown
evidence for similar behavior of the susceptibility. The
dynamic susceptibility in the kagome bilayer compound
SCGO has been shown to display power law behavior
and has been fit to a form identical to that shown as
the dark red line in Fig. 3 with α = 0.4. Both SCGO and
BSGZCO demonstrate anomalous power law behav-
ior in their bulk susceptibilities. Also, an early dynam-
ical mean-field theory study of a kagome RVB state
predicted such a scaling of the dynamic susceptibility.
A recent paper on herbertsmithite found that S(ω)
was roughly independent of both temperature and en-
ergy transfer for values of ω greater than 2 meV. This
simpler ω/T scaling is different from what we measure
here in the low energy susceptibility.
Similar scaling has been reported in other quantum
antiferromagnets, many of which are believed to be close
to a quantum phase transition. The neutron scat-
FIG. 3: (color online) The quantity χ′′(ω)Tαwith α = 0.66
plotted against ¯ hω/kBT on a log-log scale. The data collapse
onto a single curve. The lines are fits as described in the text.
tering results on the spin-glass La1.96Sr0.04CuO4 show
a scaling that at small values of ω worked best with
α = 0.41 ± 0.05, while at higher energy transfers the
data followed a pure ω/T scaling with α = 0. Further
comparisons can be made to neutron results on various
heavy-fermion metals with doping levels that place them
near a transition to a spin-glass or antiferromagnetically
ordered state[22–24, 31]. In addition, a scaling of the
ac susceptibility similar to that reported here was seen
in CeCu5.9Au0.1 and Ce(Ru0.5Rh0.5)2Si2. Recent
exact diagonalization work has suggested that the
ground state of the spin-1
2kagome lattice antiferromag-
net with a Dzyaloshinskii-Moriya interaction will be a
quantum disordered state for the Heisenberg Hamilto-
nian, but a N´ eel ordered state when the component of
the Dzyaloshinskii-Moriya vector perpendicular to the
kagome lattice plane exceeds ≈ J/10. The presence of
a nearby QCP is also possible in models without a DM
interaction. Futhermore, several of the theoretically
proposed ground states for the spin-1
antiferromagnet[35, 36] are critical or algebraic spin liq-
uid states. These proposed quantum ground states would
possibly display excitations that are similar to fluctua-
tions near a QCP. Thus, the observed low energy scaling
behavior in herbertsmithite might signify quantum crit-
ical behavior or a critical spin liquid ground state.
In many doped heavy-fermion metals, the observed
non-Fermi liquid behavior is likely related to disor-
der. In herbertsmithite, the low temperature suscepti-
bility roughly resembles a Curie tail, and it has been
suggested[14, 15] that this is attributable to S = 1/2 im-
purities (consisting of ≈ 5% of all magnetic ions) with
weak couplings to the rest of the system. We find that
the scaling behavior seen in herbertsmithite does have
features in common with the disordered heavy-fermion
metals, such as χ′′(ω) proportional to (ω/T)1−αat low
values of ω/T rather than linear in ω/T. The divergence
of the low temperature susceptibility may also be indica-
tive of a random magnetic system, such as a Griffiths
phase[19, 38] or random singlet phase. A collection
of impurity spins subject to a broad distribution of cou-
plings, P(Jimp) may result in a power law susceptibility
at sufficiently low temperatures; however, the scaling
presented here describes the entire measured susceptibil-
ity rather than the response of a small impurity frac-
tion. In this scenario, the scaling of the ac susceptibility,
with χ′T0.66∝ F(H/T), would be useful in determining
the distribution of couplings experienced by the impurity
spins in herbertsmithite. The disordered heavy-fermion
metal Ce(Ru0.5Rh0.5)2Si2 displays a scaling of the ac
susceptibility that is remarkably similar to that in
herbertsmithite, except with a considerably larger effec-
tive moment [a much smaller field will suppress the low
temperature susceptibility of Ce(Ru0.5Rh0.5)2Si2]. That
scaling was attributed to a broad distribution of coupling
strengths that likewise diverges at low coupling, quite
likely with a power law distribution: P(Jimp) ∝ J−α
In terms of our data, this would imply a distribution of
impurity couplings which extend to several meV, which
is surprisingly large for the assumed out-of-plane impu-
rity ions. A further prediction in this scenario is that the
distribution of local susceptibilities would diverge at low
temperatures such that the width of the Knight shift,
δK, as measured by NMR would go asδK
λ ≃ 0.17. The observed NMR signal certainly broadens
at low temperatures, and it would be most interest-
ing to see if it follows this specific power law.
mal transport measurements would be important to help
differentiate between scenarios where the scale-invariant
spin excitations are localized near impurities or extended
(as in the aforementioned criticality scenarios).
In conclusion, we have shown that the low energy dy-
namic susceptibility of the spin-1
ferromagnet herbertsmithite displays an unusual scaling
relation such that χTαwith α ≃ 0.66 depends only on
the thermal energy scale kBT over a wide range of tem-
perature, energy, and applied magnetic field. This be-
havior is remarkably similar to the data seen in certain
quantum antiferromagnets and heavy-fermion metals as
a signature of proximity to a quantum critical point. In
addition to scenarios based on impurities, the results may
indicate that the spin-1
2kagome lattice antiferromagnet
is near a QCP, or that the ground state of herbertsmithite
may behave like a critical spin liquid.
We thank J.W. Lynn, C. Payen and T. Senthil for
helpful discussions.JSH acknowledges support from
the NRC/NIST Postdoctoral Associateship Program.
The work at MIT was supported by the Department of
Energy (DOE) under Grant No. DE-FG02-07ER46134.
This work used facilities supported in part by the NSF
under Agreement No. DMR-0454672.
2kagome lattice anti-
∗ email: firstname.lastname@example.org
a) Department of Physics, Mahidol University, Thailand
b) Department of Chemistry, Colorado State University
c) Department of Chemistry, University of Michigan
 P. W. Anderson, Science 235, 1196 (1987).
 C. Zeng and V. Elser, Phys. Rev. B 42, 8436 (1990).
 J. Marston and C. Zeng, J. Appl. Phys. 69, 5962 (1991).
 R. R. P. Singh and D. A. Huse, Phys. Rev. Lett. 68, 1766
 S. Sachdev, Phys. Rev. B 45, 12377 (1992).
 P. Lecheminant, et al., Phys. Rev. B 56, 2521 (1997).
 C. Waldtmann, et al., Eur. Phys. J. B 2, 501 (1998).
 F. Mila, Phys. Rev. Lett. 81, 2356 (1998).
 M. P. Shores, E. A. Nytko, B. M. Bartlett, and D. G.
Nocera, J. Am. Chem Soc. 127, 13462 (2005).
 R. S. W. Braithwaite, K. Mereiter, W. H. Paar, and A. M.
Clark, Mineral. Mag. 68, 527 (2004).
 J. S. Helton, et al., Phys. Rev. Lett. 98, 107204 (2007).
 P. Mendels, et al., Phys. Rev. Lett. 98, 077204 (2007).
 O. Ofer, et al. (2006), arXiv:cond-mat/0610540v2.
 G. Misguich and P. Sindzingre, Eur. Phys. J. B 59, 305
 M. A. de Vries, et al., Phys. Rev. Lett. 100, 157205
 M. Rigol and R. R. P. Singh, Phys. Rev. B 76, 184403
 S. Sachdev and J. Ye, Phys. Rev. Lett. 69, 2411 (1992).
 P. Coleman and C. P´ epin, Physica B 312-313, 383
 A. H. Castro Neto, G. Castilla, and B. A. Jones, Phys.
Rev. Lett. 81, 3531 (1998).
 R. N. Bhatt and P. A. Lee, Phys. Rev. Lett. 48, 344
 A. Schr¨ oder, et al., Nature 407, 351 (2000).
 A. Schr¨ oder, et al., Phys. Rev. Lett. 80, 5623 (1998).
 J.-G. Park, D. T. Androja, K. A. McEwan, and A. P.
Murani, J. Phys.: Condens. Matter 14, 3865 (2002).
 M. C. Aronson, et al., Phys. Rev. Lett. 75, 725 (1995).
 B. Keimer, et al., Phys. Rev. Lett. 67, 1930 (1991).
 C. Broholm, G. Aeppli, G. P. Espinosa, and A. S. Cooper,
Phys. Rev. Lett. 65, 3173 (1990).
 C. Mondelli, et al., Physica B 284-288, 1371 (2000).
 P. Bonnet, et al., J. Phys.: Condens. Matter 16, S835
 A. Georges, R. Siddharthan, and S. Florens, Phys. Rev.
Lett. 87, 277203 (2001).
 M. A. de Vries, et al., Phys. Rev. Lett. 103, 237201
 S. D. Wilson, et al., Phys. Rev. Lett. 94, 056402 (2005).
 Y. Tabata, et al., Phys. Rev. B 70, 144415 (2004).
 O. C´ epas, C. M. Fong, P. W. Leung, and C. Lhuillier,
Phys. Rev. B 78, 140405(R) (2008).
 D. Poilblanc, M. Mambrini, and D. Schwandt (2009),
 Y. Ran, M. Hermele, P. A. Lee, and X.-G. Wen, Phys.
Rev. Lett. 98, 117205 (2007).
 S. Ryu, et al., Phys. Rev. B 75, 184406 (2007).
 S. Sachdev, Nature Physics 4, 173 (2008).
 R. B. Griffiths, Phys. Rev. Lett. 23, 17 (1969).
 S.-K. Ma, C. Dasgupta, and C.-K. Hu, Phys. Rev. Lett.
43, 1434 (1979).
 C.-Y. Liu, et al., Phys. Rev. B 61, 432 (2000).
 T. Imai, et al., Phys. Rev. Lett. 100, 077203 (2008).