Magnetic susceptibility and specific heat of the spin-1/2 Heisenberg
  model on the kagome lattice and experimental data on ZnCu3(OH)6Cl2
EPJ manuscript No.
(will be inserted by the editor)
Magnetic susceptibility and specific heat of the spin-1
Heisenberg
model on the kagome lattice and experimental data on
ZnCu3(OH)6Cl2.
Grégoire Misguich1 and Philippe Sindzingre2
1 Service de Physique Théorique, CEA Saclay, 91191 Gif-sur-Yvette Cedex, France
2 Laboratoire de Physique Théorique de la Matière condensée, Univ. P. et M. Curie, 75252 Paris Cedex, France
the date of receipt and acceptance should be inserted later
Abstract. We compute the magnetic susceptibility and specific heat of the spin- 1
Heisenberg model on
the kagome lattice with high-temperature expansions and exact diagonalizations. We compare the results
with the experimental data on ZnCu3(OH)6Cl2 obtained by Helton et al. [Phys. Rev. Lett. 98, 107204
(2007)]. Down to kBT/J ' 0.2, our calculations reproduce accurately the experimental susceptibility,
with an exchange interaction J ' 190 K and a contribution of 3.7% of weakly interacting impurity spins.
The comparison between our calculations of the specific heat and the experiments indicate that the low-
temperature entropy (below ∼20 K) is smaller in ZnCu3(OH)6Cl2 than in the kagome Heisenberg model,
a likely signature of other interactions in the system.
PACS. 75.50.Ee Antiferromagnetics – 75.10.Jm Quantized spin models – 75.40.Cx Static properties
1 Introduction
After many years of theoretical investigations, the nature
of the ground-state of the spin- 1
Heisenberg model on the
kagome lattice is still not known. Although all numerical
studies have concluded to the absence of long-range mag-
netic (Néel) order [1,2,3,4,5,6,7,8], many basic questions
such as the existence of spontaneously broken symmetries,
or the existence of a finite gap to magnetic excitations re-
main open. In fact, many different states of matter have
been proposed for the kagome Heisenberg antiferromag-
net: Z2 gapped topological liquids [9,10], valence-bond
crystals [11,12,13,14], critical spin liquids with gapless
spinons [15,12].
Recently, a promising spin- 1
antiferromagnetic insula-
tor with an ideal kagome geometry, ZnCu3(OH)6Cl2, has
been synthesized and studied for its magnetic properties
[16,17,18,19,20]. Because these studies did not detect any
kind ordering (nor spin freezing) down to 50mK, it could
represent one of the first and most remarkable realization
of a 2D quantum spin liquid [21,22].
To extract some information about the low-energy physics
of the kagome Heisenberg model from the experiments on
ZnCu3(OH)6Cl2, it is important to first analyze in a quan-
titative way the possible role of magnetic defects (and
other “perturbations”to this model) in this compound.
In this paper we compare the experimental data for the
magnetic susceptibility χ and specific heat cv obtained by
Correspondence to: gregoire.misguich@cea.fr
Helton et al.[16] with calculations for the spin- 1
Heisen-
berg model on the kagome lattice based on exact diago-
nalization (ED) data (partial spectrum of a 36-site cluster
and full spectrum for 24-site and 18-sites clusters) and
high-temperature series expansion [23,24]. Down to tem-
perature kBT/J = 0.2, the experimental susceptibility
χexp(T ) can be very well fitted by that of the kagome lat-
tice Heisenberg model with J ' 190 K plus a contribution
of about 4% of impurity spins with weak mutual inter-
actions (modeled by a ferromagnetic Curie-Weiss temper-
ature of ' −6 K), likely mostly due to antisite disorder
(Cu substituted to Zn on sites between the kagome planes)
[20,25]. The low temperature specific heat is dominated
by impurities (and other perturbations) below 2 K and
by phonons above 15 K. In the intermediate range, the
calculated specific heat appears to be larger than in the
experiment. We comment on this feature at the end of the
paper.
2 Uniform static susceptibility
The spin- 1
Heisenberg model reads:
H = J
〈i,j〉
Si · Sj − gµBH
Szi (1)
where the sum runs over pairs of nearest neighbor sites on
the kagome lattice and H is an external magnetic field.
2 G. Misguich and P. Sindzingre: Magnetic susceptibility and specific heat of the spin- 1
Heisenberg model ...
To fix the notations, we define the (zero-field) uniform
susceptibility per site χ(T ) as χ(T ) = gµB
〈Szi 〉
where N is the total number of spins.
The high-temperature expansion of χ has been com-
puted up to order O
(included) by Elstner and
Young [23]:
χ(T ) =
χth(t = kBT/J)
χth(t) = (1/4)t
−1 − (1/4)t−2 + · · · (2)
where C0 = 0.25(gµB)2 is the Curie constant and t the re-
duced temperature. The truncated series χn(t) =
i=0 cit
at order n = 14 and n = 15 agree with a relative error
smaller than 10−2 for t > 1. Down to this temperature,
they already provide good approximations to χth(t). The
convergence of high-T series can be improved using Padé
approximants (PA). In the present case the PA provide a
reliable estimate of χth(t) at least down to t ∼ 0.5.1 One
representative PA (numerator of degree 8 and denomina-
tor of degree 7) is displayed Fig. 1.
On a small enough system, it is possible to obtain
by ED the full spectrum (2N energy levels for N spins).
We have done so for two 18-site and 24-site kagome clus-
ters (with periodic boundary conditions). Then thermo-
dynamic quantities can be computed exactly as a func-
tion of T . For bigger systems, where one can still compute
some eigenstates by ED, one may use the approximate
method described in reference [26] to compute thermody-
namic quantities. footnote In this method, one constructs
the density of states in each symmetry sector from the
exact low and high energy states obtained from ED and
approximating in between the unknown part of the spec-
trum with a smooth density of states. This smooth part
is constructed so that the first moments of the density of
states (Tr[Hn]), in each symmetry sector of the finite clus-
ter, are exact up to n = 5. This Ansatz guaranties that the
thermodynamics becomes exact at low T (when thermal
excitations only involve the eigenstates computed exactly)
as well as at high T (when a re-summed high-temperature
expansion up to T−5 is valid).
The results for the susceptibility χ(t) are shown in
Fig. 1. Above t = 0.2, the relative difference between the
(exact) N = 18 and N = 24 curves is smaller than 0.5%.
We therefore make the (rather safe) assumption that our
finite-size results are good approximations to the thermo-
dynamic limit down to tmin ' 0.2. This represents a small
gain over the coupled-cluster expansion of reference [27],
which is valid above t ∼ 0.3.
The χ(t) obtained with the approximate method for
N = 36 sites also agrees (with a relative error smaller than
2%) with the N = 24 results down to t ' 0.2. Slightly be-
low, the 36-site susceptibility is still increasing and might
be a better approximation to the infinite-size limit than
the 24-site curve. Still, it is not possible decide at which t
finite-size (and/or errors due to the approximation in the
1 Comparing the PA and the exact curves for 18 and 24-site
clusters shows that the PA is in fact correct down to T ∼ 0.4J ,
see Fig. 1.
density of states) will become too important. Safely, we
only use the theoretical results (noted χth) above t ' 0.2
to fit the experimental data χexp for the susceptibility.
We fit χexp to a sum of contributions from the kagome
spins χs and the impurities χimp in the following way:
χexp(T ) =
χs(kBT/J) + χimp(T ) (3)
with χimp(T ) =
T + θimp
where J is the (unknown) magnetic exchange in between
the spins in the kagome planes, x an impurity concentra-
tion, C0 = 0.25(gµB)2/kB the Curie constant and θimp
the Curie-Weiss temperature of the system of impurities.
Eq. 4 is the leading term in a high-temperature expan-
sion for the system of impurities, which provides a sim-
plified picture of their interactions. To be applicable, T
should therefore be large compared to θimp. We assume
that the system of impurities does not perturb the the
kagome spins.
We optimized numerically the parameters so that χs
fits the theoretical results χth for t ≥ 0.2. As can be seen
on Fig. 1, an almost perfect agreement can be obtained be-
tween χs = J4C0 (χexp(T )−χimp(T )) (squares) and the the-
oretical estimates for the kagome susceptibility χth with
the following parameters: C0 = 0.504 K cm3/(mol of Cu)
(equivalent to a gyromagnetic factor g = 2.32), J = 190.4 K,
x = 0.03655 and θimp = −6.1 K. We note that the value
of J is in rough agreement with the values reported in
reference [16] (17meV'200 K) and [27] (170 K). We also
check a posteriori that the lowest temperature of the fit
(t = 0.2 ' 38 K) is much bigger than |θimp|, so that a
Curie-Weiss approximation for the impurities is justified.
6 K is also approximatively the transition temperature re-
ported in ZnxCu4−x(OH)6Cl3 for x < 0.6 (replacing some
Zn by Cu between the kagome planes) [17], so this energy
scale may correspond to some couplings for spins located
between the planes, where magnetic impurities could sit.
We eventually notice that down to t = 0.15 (that is below
the lowest temperature used for the fit), χexp − χimp con-
tinues to increase and to follow the N = 36 curve. This
suggests that the maximum of the kagome susceptibility
could indeed be below t = 0.15.
Rigol and Singh [27] analyzed the same experimen-
tal data with another high-temperature method (coupled
cluster expansion) and obtained a somewhat different con-
clusion. They argued that Dzyaloshinskii-Moriya (DM) in-
teractions provide a better description of the low-temperature
increase of the susceptibility than impurities. Although
we agree that DM interactions are certainly present in
ZnCu3(OH)6Cl2 and that they should affect the physics
of the system (at least at low-temperatures), it is also
clear that a few percent of impurities must be present too
and should have a visible effect on the susceptibility, even
at rather high temperatures. According to reference [27],
free impurities cannot explain the sharp increase of χexp
below 60 K. In our analysis, this issue is solved by al-
lowing for a small ferromagnetic Curie-Weiss temperature
G. Misguich and P. Sindzingre: Magnetic susceptibility and specific heat of the spin- 1
Heisenberg model ... 3
Fig. 1. (color online) Magnetic susceptibility per spin as a
function of temperature. Dashed (magenta) curve : experimen-
tal data (Helton et al.), multiplied by J/(4C0) as a function
of kBT/J with J ' 190 K and C0 = 0.504 K cm3/(mol of
Cu). Black squares: χs =
(χexp(T ) − χimp(T )), obtained
from the experimental data χexp by subtracting the contribu-
tion χimp of a concentration x = 0.03655 of impurity spins
with a Curie-Weiss temperature θimp = −6.1 K. Red (resp.
cyan) curve : Exact χth for a N = 24 (resp. 18) site kagome
cluster with periodic boundary conditions. Blue curve: Results
for N = 36 spins obtained with the (approximate) method of
reference [26]. Green curve: Padé approximant from the high-
temperature expansion at order t−15. The Padé approximant
is not converged below t ' 0.4 whereas the finite-size curves
are practically converged to the thermodynamic limit down to
t ' 0.2. Below t = 0.2J , the later curves are only indicative.
θimp ' −6 K for the impurities.2 One can indeed see that,
once χimp has been subtracted, the experimental results
show a saturation of χ around 20 K, in rough agreement
with the measurements of reference [19]. The location of
the maximum we obtain is however quite sensitive to the
value x of the impurity concentration.
3 Specific heat
The experimental data for the specific heat cv(T ) are only
available at very low temperature where the size effects
on the ED results are large. We therefore also applied
the high-temperature entropy method [28]. It combines
three pieces of information about the system: 1) The high
2 The Curie-Weiss temperature is an average of the differ-
ent exchange constants. However, due to the complexity of the
interactions between impurities (disorder), the ferromagnetic
sign of θimp does not necessarily imply that they behave ferro-
magnetically at low temperatures.
temperature series expansion of cv(T ), up to T−17 [23,
24]. 2) The ground-state energy per site e0 of the Hamil-
tonian. Here we use the following estimate e0 = 2〈0|Si ·
Sj |0〉 = −0.44 [24]. 3) The exponent α describing the low-
temperature behavior of the specific heat: cv(t→ 0) ∼ tα.
The method then provides a set of cv(t) curves (for dif-
ferent Padé approximants) which all satisfy exactly the
following properties: i) cv(T → 0) ∼ Tα, ii) cv(T →∞) ∼
the series expansion, iii)
cv(T )dT = −Ne0, and iv)∫∞
cv(T )/TdT = NkB ln(2). When the value of e0 and α
are both correct, one usually gets a large number of very
similar curves but if either is incorrect, only a few and scat-
tered curves will be obtained (more details in Refs. [28,
24]). In the case of the kagome antiferromagnet, the value
of α is not known and the entropy method gives a reason-
able convergence for α = 1 and α = 2 [24]. Motivated by
the experimental observation [16] of a low-temperature cv
with an exponent smaller than one, we also include here
a calculation with α = 0.5.
The results for some valid3 PA from orders 14 to 17 are
displayed in Fig. 2 together with the exact result obtained
by ED of the full spectrum of a 24-site kagome cluster. For
t > 0.3, these results are practically exact.
In the two scenarios α = 1 and 2 there is a significant
dispersion of the results for t < 0.3 from one PA to another
[24].4 In both case, cv show a low-t peak or shoulder as
found from ED of finite-size systems. The choice α = 0.5
leads to a smoother cv(t) and improves significantly the
convergence. It is however not clear if this improved con-
vergence for small values of α (0.5 . α . 1) indicates that
α is actually smaller than one or an artifact of the present
entropy method, which might be “slower” to stabilize a
low-t peak (as with α = 2, see Fig. 2), than a smooth
curve (as with α = 0.5). In any case, this is clearly re-
lated to the unusually large low-temperature entropy of
the kagome system.
Fig. 2 also displays the experimental results (black
squares) obtained by Helton et al. [16]. The only parame-
ter here is the exchange constant, taken to be J ' 190 K
from the fits of the susceptibility data. Above 15 K, the
phonon contribution is dominant and we have to focus on
the lower temperatures to analyze the magnetic contribu-
tion. Below 15 K the order of magnitude agrees with our
calculation but there is no quantitative agreement. Several
“perturbations” such as weakly interacting magnetic im-
purities or magnetic anisotropies should indeed contribute
to the specific heat at such low temperatures.
Results for the integrated entropy S(T ) =
cv(x)/xdx
are plotted in Fig. 3, where one sees that the choice of the
3 By the entropy method, the entropy s(e) is obtained as the
power α/(α + 1) of a rational fraction (PA) of the energy per
site e. The specific heat curve is then obtained parametrically
through T (e) = 1/s′(e) and cv(e) = −s′(e)2/s′′(e). Only the
PA which satisfy s(e) > 0, s′(e) > 0 and s′′(e) < 0 in the range
]e0, 0[ are physically “valid”.
4 This is due to the finite order in the high-temperature
expansion. Still, for a given value of α, we believe that this
method gives a qualitatively correct picture for cv(T ), even at
low T.
4 G. Misguich and P. Sindzingre: Magnetic susceptibility and specific heat of the spin- 1
Heisenberg model ...
Fig. 2. (color online) Specific heat cv vs temperature T .
Black squares: experimental data from Helton et al. [16], as-
suming J ' 190 K. Red triangles: exact specific heat of a 24-
site kagome cluster. Green (full), blue (dashed) and magenta
(dotted) curves: cv calculated by the entropy method for three
different values of the low-temperature exponent (α = 2, 1 and
0.5). All valid PA from order 14 to 17 with numerator and de-
nominator of degrees ≥ 3 are shown. For a given α and at each
temperature, their dispersion provides a rough estimate of the
error bars.
Fig. 3. (color online) Entropy S vs temperature. Square, tri-
angles and circles: experimental data from Helton et al. [16]
with magnetic field B= 0, 5 and 14 Teslas, plotted as a func-
tion of kBT/J with J = 190 K. Green (full), blue (dashed)
and magenta (dotted) curves: entropy calculated by the en-
tropy method for α = 2, α = 1 and α = 0.5 (same PA as in
Fig. 2).
low-T exponent α of cv has practically no influence on the
theoretical S(t = kBT/J) above t ' 0.06 and that, around
this temperature, the experimental value is significantly
lower than in our calculations. Of course, subtracting the
contribution of the phonons (hard to estimate quantita-
tively) and from the impurities would make the discrep-
ancy even larger. Concerning the impurities, one sees in
Fig. 3 that an applied magnetic field of 5 and 14 Teslas
is enough to significantly reduce S(t) for t . 0.06. Such
fields are low in comparison to J but of the order of the
estimated coupling between the impurities. The difference
between the curves at 0 and 5 (or 14) Teslas may thus
provide a rough estimate of the contribution of the impu-
rities. Note that S(t) at 5 Teslas become closer to the S(t)
computed by the entropy method with α = 2. The experi-
mental value α < 1 could be due to the impurities and the
actual value of α for the kagome spins could be larger than
1. In any case, at t = 0.06(' 11 K), the experimental en-
tropy (' 0.2 ln 2) is thus at least 7% of ln 2 below the the-
oretical estimates (0.27 ln 2). This seems a rather robust
indication that some additional interactions play some role
in this energy range, by freezing some degrees of freedom
of the spins in the kagome planes and pushing the corre-
sponding entropy to higher energies. We may mention in
particular DM interactions [27], non-magnetic impurities
in the kagome planes (“dilution”) [20] and interactions
between impurities and the kagome spins. We conclude
that 15∼20 K is a minimal temperature for a kagome lat-
tice Heisenberg model description of ZnCu3(OH)6Cl2 to
be valid.
Acknowledgments
We are grateful to C. Lhuillier for many discussions and
comments about the manuscript. We also thank P. Mendels
and F. Bert for useful discussions. GM also thanks Y. S. Lee
and J. Helton for discussions and for providing their data
as well as P.A. Lee, Y. Ran, T. Senthil, X.-G. Wen for
discussions on related topics.
Note added : After the first submission of this manuscript,
two preprints [20,25] (neutron scattering) confirmed the
importance of magnetic impurities (from 6% to 10%) in
ZnCu3(OH)6Cl2. The smaller value found here could be
due to our simplified model to fit the magnetic suscepti-
bility.
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	Introduction
	Uniform static susceptibility
	Specific heat
