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Interplay of thermal and quantum spin fluctuations on the kagome lattice


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We present a Raman spectroscopic investigation of the Herbertsmithite ZnCu3(OH)6Cl2, the first realization of a Heisenberg s=1/2 antiferromagnet on a perfect kagome lattice. The magnetic excitation spectrum of this compound is dominated by two components, a high temperature quasi elastic signal and a low temperature, broad maximum. The latter has a linear low energy slope and extends to high energy. We have investigated the temperature dependence and symmetry properties of both signals. Our data agree with previous calculations and point to a spin liquid ground state.
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arXiv:1005.4831v2 [cond-mat.str-el] 10 Jun 2010
Interplay of thermal and quantum spin fluctuations on the kagome lattice
Dirk Wulferding,1Peter Lemmens,1Patric Scheib,1Jens Röder,2Philippe
Mendels,3Mark A. de Vries,4Shaoyan Chu,5Tianheng Han,6and Young S. Lee6
1Institute for Condensed Matter Physics, Technical University of Braunschweig, D-38106 Braunschweig, Germany
2Institute for Physical and Theoretical Chemistry,
Technical University of Braunschweig, D-38106 Braunschweig, Germany
3Laboratoire de Physique des Solides, Université Paris-Sud 11, UMR CNRS 8502, 91405 Orsay, France
4Department of Physics and Astronomy, University of Leeds, Leeds, LS2 9JT UK
5Center for Materials Science and Engineering, Massachusetts Institute of Technology,
77 Massachusetts Ave, Cambridge MA 02139, USA
6Department of Condensed Matter Physics, Massachusetts Institute of Technology,
77 Massachusetts Ave, Cambridge MA 02139, USA
(Dated: June 11, 2010)
We present a Raman spectroscopic investigation of the Herbertsmithite ZnCu3(OH)6Cl2, the first
realization of a Heisenberg s= 1/2antiferromagnet on a perfect kagome lattice. The magnetic
excitation spectrum of this compound is dominated by two components, a high temperature quasi
elastic signal and a low temperature, broad maximum. The latter has a linear low energy slope and
extends to high energy. We have investigated the temperature dependence and symmetry properties
of both signals. Our data agree with previous calculations and point to a spin liquid ground state.
Strongly frustrated spin systems are in the focus of
many recent works as the competition of exchange in-
teractions leads to interesting ground states and a rich
low energy excitation spectrum. Of these systems, the
s= 1/2Heisenberg antiferromagnet on the kagome lat-
tice has been of special interest, since it displays a unique
quantum disordered and highly degenerate ground state,
i.e. a spin liquid1. A kagome lattice consists of a
two-dimensional (2D) arrangement of corner-sharing tri-
angles. In systems with large spin so called weather-
vane modes, low energy collective rotations of spins on
a hexagon, destabilize any order. For s= 1/2a quan-
tum superposition of an infinitely dense spectrum of low
energy singlet and triplet excitations are supposed to ex-
ist. Defects on the kagome lattice may lead to a gapless
valence bond glass phase2with enhanced singlet bonds
opposite to the defects. Together with antisymmetric
Dzyaloshinskii-Moriya (DM) interactions a quantum crit-
ical point is induced3. Despite these and other4theoreti-
cal investigations, the understanding of the ground state
properties and the excitation spectrum of the quantum
kagome system is not completely satisfactory.
Earlier experimental investigations of compounds that
are close in realizing the kagome layer are hampered ei-
ther by interlayer interactions, a non-uniform, more com-
plex distribution of exchange coupling constants or spin
anisotropies. The first reported material with 2D kagome
layers was SrCr8Ga4O195with s= 3/2. It shows a spin-
glass transition for temperatures below T= 3.3K despite
the much larger interaction strength. In the s= 3/2com-
pound Y0.5Ca0.5BaCo4O76short ranged chiral spin cor-
relations develop at low temperature, while the s= 1/2
compound RbCu3SnF12 7experiences a spin gap that sep-
arates the singlet groundstate from low energy triplet ex-
Presently the compound ZnCu3(OH)6Cl2is the only
well characterized realization of the s= 1/2Heisen-
berg antiferromagnet on a perfect kagome lattice8,9. It
exists in two structurally slightly different forms: a)
Kapellasite10 and b) Herbertsmithite. Small single crys-
tals, however, are up to now only available as Herbert-
smithite. In this compound, the Cu2+ ions form the tri-
angular kagome lattice in the crystallographic ab-layer
and are coupled by oxygen atoms via superexchange
pathways, leading to strong antiferromagnetic correla-
tions of about J180 K. Despite this strong coupling,
susceptibility measurements reveal no magnetic ordering
down to 50 mK11.
The planes of these kagome structures are separated by
intermediate layers of Zn and Cl atoms and can therefore
ideally be considered as 2 dimensional layers with neg-
ligible magnetic interactions along the third dimension.
However, Zn/Cu antisite disorder of at about 6%12 is be-
lieved to be present in previously studied Herbertsmithite
samples. Hence, the presence of a small concentration of
impurities may affect the spin correlations in the kagome
Muon spin rotation (µSR) measurements on the parat-
acamite Zn0.5Cu3.5(OH)6Cl2showed a partly frozen
ground state13, which however disappears for higher
Zn/Cu ratio. For the Herbertsmithite, i.e. a Zn to Cu
ratio of 1/3, µSR did not reveal any ordering down to 50
mK. Electron spin resonance measurements revealed DM
interactions of about D= 0.08 ·Jmagnitude14 , which
causes a ferromagnetic-like increase of susceptibility at
low temperature. These and other experiments have been
performed either on natural (mineral) crystals or powder
samples9. Only recently, Herbertsmithite single crystals
were successfully grown via hydrothermal synthesis15.
Our approach of investigating the excitations in the
kagome lattice is using inelastic light scattering (Raman
scattering). This is a unique method in the sense that it
is sensitive to low energy topological, electronic, or mag-
netic excitations. Singlet modes (s= 0) can be observed,
which may otherwise only be probed in specific heat mea-
surements. In particular, it has recently been suggested16
that studying the polarization dependence of the mag-
netic Raman scattering contribution should allow to dis-
tinguish between different possible ground states such as
a valence bond crystal or a spin liquid. Of further inter-
est is the spectral distribution of magnetic scattering as
characteristic low energy slopes or sharp modes may be
observed as signs of the topological character of the spin
This study deals with two different types of samples:
(i) Natural crystallites from mineral sources, which show
a dark green color and are slightly transparent. A se-
lected crystallite from this source with the approximate
volume of about 0.5 mm3was studied via Buerger pre-
cession method19 to confirm the sample’s single domain
structure and to uncover a natural surface that is parallel
to the crystallographic ab-plane, in which the kagome lat-
tice is realized. Its stoichiometry, Zn0.8Cu3.2(OH)6Cl2,
is slightly shifted from optimum. (ii) A hydrothermally
synthesized single crystal with a size of about 1 x 0.2 x
0.2 mm3, which shows a very regular habitus, was stud-
ied in addition. It appears in light green color and high
transparency and has a stoichiometry very close to the
perfect kagome, i.e. ZnCu3(OH)6Cl2.
Raman spectroscopic studies were performed in quasi-
backscattering geometry with a solid state laser (λ= 532
nm) and 1 mW laser power. Room temperature measure-
ments were carried out under ambient conditions, while
low-temperature measurements were done in an evacu-
ated, closed-cycle cryostat. Investigations of the polar-
ization dependence were performed by rotating the sam-
ple within the ab plane. In the 0orientation, the a axis
was approximately perpendicular to both the polariza-
tion and the kvector of the incident laser light. The
spectra were collected via a triple spectrometer (Dilor-
XY-500) by a liquid nitrogen cooled CCD (HORIBA
Jobin Yvon, Spectrum One CCD-3000V). In a previous
experiment powder samples of ZnxCu4x(OH)6Cl2(with
x= 0.5,0.85,1.0,1.1,1.2and 1.4) were investigated to es-
timate the stoichiometry of the single crystals by compar-
ing their phonon frequencies20. These samples, however,
do not allow to successfully study the weaker magnetic
or topological Raman scattering in the Herbertsmithite.
Figure 1 a) shows Raman spectra of the natural and
of the synthesized single crystal, measured at room tem-
perature (T= 295 K) and in xx polarization. Rotating
the samples within the crystallographic ab plane (with
mineral crystal
FIG. 1. a) Raman spectrum of natural and synthesized single
crystal at room temperature in xx polarization. b) left: inten-
sity of the quasi elastic signal at RT as function of the angle
between the light scattering polarization and a fixed direction
within the ab plane as described in the text. The rotational
anisotropy of this scattering intensity is consistent with that
of A1gphonon modes; right: intensity of the Egmode at 363
cm1plotted in a similar fashion. c) Comparison of phonon
modes in ZnCu3(OY)6Cl2assigned to H/D vibrations com-
paring a hydrated (Y=H) and a deuterated (Y=D) powder
the axis of rotation along the kvector of the incident
light) allows to distinguish between modes of A1gand Eg
symmetry representation. Furthermore, the previously
discussed rotational anisotropy of the magnetic modes
can be probed.
We observe 7 sharp modes at low to intermediate ener-
gies (123, 148, 365, 402, 501, 697-702 and 943 cm1) and
five modes at high energies. We attribute these lorentzian
lines to phonons. The factor group analysis yields 12 Ra-
man active phonon modes according to the R3mspace
group: ΓRaman = 5 ·A1g+ 7 ·Eg. In addition, we will
later consider the A2gsymmetry representations that can
be induced by electronic resonances or DM interaction17 .
The corresponding tensors are:
a0 0
0 0 b
, Eg=
c0 0
0c d
, A2g=
e0 0
0 0 0
By rotating the samples within the ab plane (xy
plane) the A1gmodes are expected to give a rotation
invariant contribution in parallel (xx) polarization and
disappear in crossed (xy) polarization of incident and
scattered light. The Egmodes should show a four-fold
symmetry in both polarizations. Due to the layered crys-
tal structure and strongly anisotropic electronic polariz-
ability perpendicular to the ab plane slight but inevitable
misalignments of the crystals will show up as an addi-
tional two-fold modulation of the scattering intensity.
The 5 high energies phonons at about 3500 cm1(2 ·
A1g+ 3 ·Eg) are due to vibrations of hydrogen atoms. In
a deuterated powder sample they shift to lower energies
with a renormalization factor of 0.74, which is in good
accordance to a simple estimate (ωpmH/mD=
0.71). This is depicted in Fig. 1 c).
We assign the modes at 365, 402 and 501 cm1to
Egsymmetry, as their intensity clearly displays a four-
fold rotational symmetry (see Fig. 1 b). The phonon
modes at 123, 148 and 702 (697) cm1are distinctly
different as they show a two-fold rotational symmetry.
We assign these modes to A1gsymmetry and the two-
fold modulation of intensity to a small deviation of the k
vector from the c axis, i.e. a slight canting of the sample
In a previous investigation of a series of
ZnxCu4x(OH)6Cl2powder samples we stated a
linear dependence of the phonon mode at 702 (697)
cm1on composition. This allows us to estimate x= 0.8
for the natural and x= 1.0for the synthesized single
crystal, respectively. The synthesized single crystal is
therefore in the regime of Herbertsmithite stoichiometry.
Still, the natural single crystal is reasonably close to this
state in the sense that thermodynamic and spectroscopic
experiments did not observe evidence for long range
ordering for x= 0.813 . Studying the two different
samples should therefore allow determining the influence
of spin defects and static doping on the spin dynamics.
A less intense, asymmetric mode with a fano lineshape
is observed at 230 cm1. This mode shows an even
smaller intensity in the powder samples. Its lineshape
may originate from the coupling of the corresponding
phonon to a continuum of states. As its energy range
is within the energy range of spin-fluctuations21 and the
compound is an insulator the corresponding continuum
is attributed to magnetic fluctuations. The mode itself
is probably induced by crystallographic disorder as the
larger intensity modes already exhaust the total number
of Raman active modes.
At low frequency and for high temperatures there is a
pronounced quasi elastic scattering contribution (QES)
visible with a two-fold rotational anisotropy (Fig. 1 b)
that is similar to the phonon modes at 123, 148 and 702
(697) cm1, identified as A1gphonons. Therefore we as-
sign the continuum to the same A1gcomponent. At low
temperatures (below T= 50 K) the quasi elastic scatter-
ing is completely depressed as demonstrated in Fig. 2 for
the case of the natural sample.
An analysis of the QES (black dots) at 295 K is given
in Figure 3 a) together with a Lorentzian (solid red line)
and a Gaussian (dashed green line) fit. The data is in
better agreement with the Lorentzian line shape. Figure
3 b) shows the temperature development of the QES in-
tensity in the synthesized (black dots) and the natural
crystal (blue triangles). The decrease in intensity with
decreasing temperature can be fitted with power laws
(red and green curves). The observed exponent varies
from I(ω0, T )T3/2to I(ω0, T )T5for the
synthesized and the natural sample, respectively.
With decreasing temperatures we observe a moderate
reduction of linewidth of all phonon modes, see Fig. 2 a.
We attributed this to the decrease of anharmonic phonon
scattering processes or other fluctuations. The integrated
intensity of the asymmetric Fano line shape at 230 cm1
is increasing strongly; the intensity at 10 K is about 5x
mineral crystal
FIG. 2. a) Temperature dependence of Raman spectra of
the natural sample in the frequency regime 30 to 725 cm1
and in parallel xx polarization. Spectra are shifted for clar-
ity. b) and c) show the Bose corrected spectra for the nat-
ural crystal and the synthesized sample, respectively, with
phonon modes subtracted (dots) together with a fit to the
background. The dashed black lines in b) and c) correspond
to the fitted background in crossed xy polarization at 5 K.
The spectra at T= 295 K in b) and c) are shifted in intensity
for clarity.
stronger compared to 120 K. In the synthesized sample
this increase is only half that strong. At the same time,
the linewidth of this mode is decreasing steadily before
saturating for temperatures below approximately 70 K.
With decreasing temperatures also the A1gphonons at
123 and 148 cm1gain a Fano line shape.
At low temperatures a broad, finite energy maximum is
observed with increasing scattering intensity. This con-
tinuum is observed for both the natural as well as the
synthesized single crystal, as shown in Fig. 2 b) and
c). For the synthesized sample the maximum position
is higher (Emax = 265 cm12.1J) compared to the
natural one (215 cm11.7J). In addition, the former
sample also shows a more pronounced decrease of scat-
tering intensity towards higher energies. We attribute
the broad continuum in ZnCu3(OH)6Cl2to scattering on
magnetic correlations as it resembles two-magnon scat-
tering of a quantum antiferromagnet in the paramagnetic
state22. Further below we will discuss and compare its
properties in detail with existing theories. The dashed
lines in Fig. 2 b) and c) show the corresponding re-
sults in crossed polarization. There are similar maxima
and frequency shifts comparing the two sample qualities.
However the onset of the scattering is shifted to higher
energy for the natural sample.
We observe a larger spectral weight of the continuum
in the synthesized sample, especially in xx polarization.
For both samples there is less spectral weight at low
energies in xy polarization. This suggests the possibil-
ity of decomposing the spectral weight into a low and a
high energy part, to which the allowed symmetry chan-
nels A1g, A2gand Egcontribute to varying extend. To
estimate their magnitudes we investigate the rotational
anisotropies of these signals in Fig. 5. It is important to
FIG. 3. a) Fit of Lorentzian (solid red line) vs. Gaussian
(dashed green line) line shapes to the data (black dots) of
the natural sample at low frequency and T= 295 K. The
phonon modes at 123 cm1and 148 cm1are subtracted.
b) Temperature dependence of the intensity of the QES in
the synthesized crystal (black circles) and the natural sample
(blue triangles). The dotted (dashed) line is a fit to the data
using a power law according to IT3/2(IT5) for the
synthesized (natural) sample as described in the text.
note that we observe no further signal that could be in-
terpreted as of magnetic origin. In particular, there are
no sharp modes at low energies. This statement is as-
sured down to energy scales of approximately 5-10 cm1
as no upturn is visible at the edge of the experimentally
accessible energy window.
Figure 4 a) zooms into the low frequency regime (up
to 100 cm1) at low temperature (T= 5 K). There is a
linear frequency dependence towards the maximum and
we can use its slope as a measurand of the integrated
intensity, less sensitive to fluorescence backgrounds. In
Fig. 4 b) the slope of the low energy scattering is plotted
as function of temperature. It increases approximately
linearly with decreasing temperature (T < 50 K). Both
samples show this linear temperature dependence. How-
ever at higher temperatures the synthesized sample has
again the more pronounced Tdependence as shown in
Fig. 4 c). Such a linear increase of scattering intensity
mineral crystal
FIG. 4. a) Linear fit to the low frequency scattering in xx
polarization of the natural crystal at T= 5 K. b) Temperature
development of its slope including a linear fit. c) Temperature
development of the intensity of the continuum in both the
natural and synthesized sample with guides to the eye.
FIG. 5. Angular dependence at T= 5 K of a) the low energy
slope in the natural crystal and b) its continuum intensity.
The data follow the rotational anisotropy expected for Egand
A1gsymmetry components. Note that measurements were
performed only in the range from 0to 180.
is anomalous and not compatible with the expected Bose
factor. However, it is frequently observed for magnetic
Raman scattering in low dimensional spin systems that
are close to quantum critical.
Finally, in Figure 5 we probe the rotational anisotropy
of the low energy slope in xx and xy polarization (shown
in a) and the rotational anisotropy of the intensity of
the broad continuum (shown in b) in the natural sample
at T= 5 K. The low energy slope in a) shows a four
fold Eg-like symmetry in parallel polarization, while in
crossed polarization the rotational anisotropy is rather
two-fold or of mixed character. The intensity of the broad
continuum in b) shows slight anisotropies in both xx and
xy polarization.
In the following we will first discuss the QES observed
at high temperatures followed by the implications of the
low temperature data at intermediate and high energies.
As mentioned above, the most characteristic feature
of the QES is its strong depression of intensity with de-
creasing temperatures. Such a scattering is frequently
observed in compounds that realize low dimensional spin
systems, as spin chains23 or frustrated dimer systems24 .
It is related to energy density fluctuations that couple via
spin-phonon coupling to the lattice and can be mapped
on the specific heat Cmof the spin system, I(ω0, T )
CmT2. Its lineshape and symmetry should be Lorentzian
and of A1gsymmetry, respectively22. The fit to the data
given in Fig. 3 follows all these predictions very well.
Again, the rotational invariance is distorted with two-
fold symmetry but within experimental error identical to
the one of the phonon modes identified as belonging to
the A1gsymmetry representation.
From the temperature dependence of the quasi elastic
scattering intensity I(ω0, T ) we can derive informa-
tion on the evolution of low energy excitations. In, e.g.,
the Shastry-Sutherland compound SrCu2(BO3)2the ex-
ponential drop of the respective intensity has been used
to derive a spin gap24 . The power law-like tempera-
ture dependence in ZnCu3(OH)6Cl2differs clearly from
that and implies a spin liquid with more gradually evolv-
ing correlations25. We attribute the softer power law
I(ω0, T )T3/2in the synthesized single crystal to
the reduced number of defects leading to an even more
gradual evolution of spin correlations.
We will now discuss the three most recent theoreti-
cal investigations (using different approaches and tech-
niques) of finite energy, magnetic Raman scattering in a
kagome system and their implications. As was suggested
by Cépas et al.16, the formation of a valence bond crys-
tal leads to a weakly broken translation symmetry of the
spin system. Thereby the magnetic Raman response of
the kagome plane should depend in a decisive way on the
polarization of the incoming / scattered light, i.e. there
should be characteristic oscillations of magnetic modes in
the energy range of Jwhen the polarizations are rotated
within the plane. In contrast, the “true” spin liquid ex-
hibiting no broken symmetry should show a polarization
independent magnetic Raman response. Since experi-
mental investigations using different techniques have up
to now not observed a spin gap, the Raman experiment
is highlighted to be potentially sensitive to even weakly
broken symmetries in kagome plane. It should be noted
that in the second order perturbational treatment used
by Cépas et al. magnetic scattering is only observed in
Egsymmetry, as the A1gcomponent commutes with the
used exchange scattering Hamiltonian.
Recently, calculations based on a modified Shastry-
Shraiman model describing a U(1) Dirac spin liquid have
been performed by Ko, et al.17 . In this approximation
also higher order terms, induced e.g. by electronic reso-
nances of the phonon energy with correlated states of the
system, are considered. Here, the included chiral ~s ×~s
component leads to additional scattering contributions in
A1g, A2gand Egsymmetry components. From the Ra-
man tensors given above we notice that the now allowed
A2gcomponent is only observed in xy polarization as a
rotational invariant contribution. It does not contribute
to xx polarization at all. The spectral distribution of
the Raman scattering cross section derived by Ko, et al.
shows a broad continuum that extends up to about 600
Finally, Läuchli et al.18 calculated dynamical singlet
fluctuations on the kagome lattice using large-scale exact
diagonalizations on a 36 site lattice. The results show
a broad continuum extending over a range of 2 – 3J.
There is also a pronounced intensity shooting up at lowest
energies, ω/J 0.225 cm1, caused by a high density
of low energy singlet and triplet excitations.
We can decompose the experimentally observed scat-
tering continuum into two contributions both attributed
to specific spin liquid excitations. The rotation depen-
dence of the low energy slope (Fig. 5 a) shows in xx
polarization a clear Eg-like contribution, while in the xy
configuration it is oval shaped, most probably a mix of
both Egand A2gsymmetry. The dominance of the Eg
channel at frequencies below 100 cm1is in contrast to
theory17. The rotation dependence of the overall con-
tinuum’s intensity (Fig. 5 b) shows a weak asymmetry
in both xx and xy configurations for the natural and
the synthesized sample. This anisotropy has no domi-
nant four-fold contribution. However, a superposition of
a four-fold and a two-fold contribution leads to a good
description of the scattering intensity. We conclude that
the A1g(for xx) and the A2g(for xy) symmetry channels
are major contributions for the continuum at an energy
of 2J. These components are proposed to be due to
higher order perturbation theory17. Comparing our ex-
perimental data in crossed polarization to the data in
parallel polarization, no clear separation of the respec-
tive spectral weight is evident.
This behavior is in clear contrast to the phonon Ra-
man scattering, where modes of A1gand Egshow a pro-
nounced selection rule and different in-plane anisotropy.
The observed Fano-like line shape of phonons is also only
observed for modes with A1gsymmetry. On the other
hand phonon and magnetic scattering differ within two
respects that in the theoretical literature on the Her-
bertsmithite are jointly discussed3. One is related to
defects, which for the local spin excitations play a dif-
ferent role than for q0optical phonons with large
coherence length. The second aspect is DM interaction,
which mixes singlet and triplet excitations. DM interac-
tion and structural defects as intersite mixing will allow,
further enhance and possibly mix A1gand A2gcontribu-
tions, while spin defects in the plane reduce the scattering
intensity. Therefore, a clear assignment of contributions
from the A1g, A2gand Egchannels to the background
is not straight forward. Still, the energy range and gen-
eral shape of the observed background are in very good
agreement with the above mentioned calculations of Ko,
et al.17.
The effects of impurities in our Herbertsmithite sam-
ples should be considered, and these effects should more
pronounced for the natural sample (x= 0.8). While
keeping the kagome plane intact, Cu atoms in the in-
termediate layer may induce lattice deformations due
to Jahn-Teller distortion, as well as additional magnetic
coupling to the kagome Cu atoms. The effect of impu-
rities in quantum spin systems with dedicated low en-
ergy excitations can be exemplified using the frustrated
dimer system SrCu2(BO3)2, where 1-2% of Zn doping
on Cu broadens the low energy excitations and consider-
ably redistributes their spectral weight22. This may be
the reason that in contrast to theory, our experimental
data do not show sharp magnetic modes or a pronounced
low energy maximum. In NMR experiments on Herbert-
smithite it has been shown that the local susceptibility
in the proximity to a defect is different and anomalous
compared to the regular behavior26. With respect to Ra-
man scattering it can be assumed that the excitations in
the continuum that consist mainly of two-pair states17)
should be reduced in intensity, comparing the natural
and the synthesized sample. Such a reduction of contin-
uum scattering intensity is indeed observed, as well as a
softening of about 20%.
Our Raman data give further evidence that the alge-
braic spin liquid is the ground state of the Heisenberg
s= 1/2kagome lattice antiferromagnet. This is based
on the observation of a power law depletion of thermally
induced fluctuations of the magnetic energy density with
decreasing temperature and scattering continua in the
low- and mid energy range. We give estimates of the con-
tributions to these scattering continua to different sym-
metry components and conclude that except for the low
energy continuum in xx configuration, no strong Egcon-
tribution has been observed.
Our data leads to the conclusion that the kagome plane
has short range and continuous correlations that form a
maximum (at 2J) with no clear cutoff but a moderate
suppression at higher energy scales (at 6J). The over-
all energy range and shape of the continua agrees well
with existing calculations for a spin liquid state.
Presently the role of impurities in the single crystal, i.e.
antisite disorder and local, defect-like interlayer interac-
tions, is difficult to evaluate. Comparing the cleaner syn-
thesized with the natural, natural samples a weak sup-
pression of the continuum intensity with a frequency shift
is evident. As no sharp, pronounced magnetic peaks are
observed their relation to disorder remains an open issue.
This work was supported by DFG and the ESF pro-
gram Highly Frustrated Magnetism. We like to thank
W. Brenig, O. Cépas, P. Lee, C. Lhuillier, E. Sherman,
R. Singh and O. Tchernyshyov for important and helpful
discussions. D.W. acknowledges support from B-IGSM of
the TU-BS. The work at MIT was supported by the De-
partment of Energy (DOE) under Grant No. DE-FG02-
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... However, heat capacity is found not to be sensitive to an antiferromagnetic transition, see for example [22], thus further studies such as NMR are necessary to identify the temperature of magnetic transition in κ-Hg-Cl. On the other hand, to clarify magnetic low-temperature state of the studied materials much information can be received from magnetic Raman scattering Raman scattering was previously successfully used to study spinon excitations in both inorganic kagome-lattice [23] and BEDT-TTF-based triangular lattice [24,25] spin liquid candidates. ...
Full-text available
Quantum dipoles go liquid Quantum spin liquids do not achieve an ordered magnetic state, even at the lowest temperatures. Hassan et al. studied an organic compound that may be both a spin liquid and a dipole liquid (see the Perspective by Powell). In the layered material κ-(BEDT-TTF) 2 Hg(SCN) 2 Br, molecules form charged dimers whose sites are arranged on a triangular lattice. The extra charge associated with each dimer can “live” on one of the two molecules in the dimer, resulting in a nonzero electric dipole moment for the dimer. Raman spectroscopy and heat capacity measurements revealed that, like spins in a quantum spin liquid, these dimers remained disordered down to the lowest temperatures. Science , this issue p. 1101 ; see also p. 1073
This work is devoted to the study, the development, and the application of a new systematic method yielding the dominant correlations that govern a quantum many-body state in an unbiased way. The dominant correlations between any two disjoint blocks of a system are extracted by performing a singular value decomposition of the correlation density matrix (CDM) between those blocks. We determine several mathematical properties and features of this method, in particular the consequences of the lattice symmetries or the symmetries intrinsic to the studied state on the singular values spectrum. We investigate the relation between the norm of the CDM – providing a natural measure of the total correlation between the two blocks – and the so-called mutual information, a quantity originally introduced in quantum information theory. This novel tool is utilized for sheding new light on the zero temperature physics of the spin-1/2 frustrated ferromagnetic J1–J2 Heisenberg chain in a magnetic field as well as on the low-energy physics of the spin-1/2 antiferromagnetic Heisenberg model on the two-dimensional kagomé lattice. The states are computed using exact diagonalization and the density matrix renormalization group procedure in the first case, and exact diagonalization only in the second case. This work is introduced in Chapter 1. The first model is then presented in Chapter 2. Chapter 3 introduces the CDM method, and Chapter 4 is devoted to the study of the kagomé antiferromagnet. In the J1–J2 chain, we reveal a vector chiral phase at low magnetic field and a sequence of multipolar Luttinger liquid phases at high field. We explicitly show that these multipolar phases result from the destabilization – driven by a locking mechanism – of the classical spiral ground state in the absence of magnetic field. This point of view is completely new: multipolar phases were known to be a possible destabilization of ferromagnetic phases, but they have never been reported as a destabilization of spiral states yet. Regarding the kagomé antiferromagnet, we address for the first time the question of the nature of the singlet states forming its quite dense singlet spectrum above the ground state. We show that some of these low-lying singlet states have large dimer correlations which do not seem to significantly decrease with the distance, moreover our CDM studies confirm that the dominant correlations in those singlet states are of the dimer-dimer type. Studies of Von Neumann block entropies reveal a very short correlation length on the one hand, and entropies that are roughly independent on the energy of the state under consideration on the other hand. The scenario of a valence bond crystal phase is investigated and the relevance of different kinds of crystals (from the literature or ad hoc) for reproducing the dimer correlations in the 36-site sample is probed.
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