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m wavelength data. Blue, green, bright green, red, and g
Zn, Cu, Cl, O, and H atoms, respectively. (a) Local c
Downloaded
www.sciencemag.org
on Junein
7, a
2009
ment
of thefrom
intersite
is highlighted
projection perpe
Thermal ellipsoids are drawn at the 70% probability le
of the kagomé layer is highlighted, in a projection
c-axis. All atoms outside the kagomé plane have been
(c) Photograph of a crystal from the batch used for
interatomic distances (Å) and angles (deg): Zn-O
1.9821(3); Cu-Cl, 2.7676(2); Zn · · · Cu, 3.0560;
Japanese name for the weave pattern of a basket (see the second figure, top panel). The
structure consists of corner-sharing triangles
and is even more frustrated than the triangular
lattice considered by Anderson. Last year saw
the synthesis of such a solid-state Kagome
system: ZnCu3(OH)6Cl2, where a single electron spin resides on the Cu (10) (see the second figure, bottom panel). Although the
exchange energy is ~200 K, this material does
not show any magnetic ordering down to millikelvin temperatures. The magnetic excitations are apparently gapless, but unlike the
organic compound, the large specific heat at
low temperatures is sensitive to magnetic
field, which suggests that the low-temperature
properties may be dominated by a few percent
of local moment defects.
Spin liquids are not limited to twoA R T Isystems.
C L E S A newly synthesized
dimensional
material, Na4Ir3O8, has Ir ions that form a
three-dimensional network of corner-sharing triangles, termed a hyper-Kagome structure (11). Despite an exchange energy of
~300 K, no magnetic order was found down
to 1 K and below.
It is an exciting time in the history of antiferromagnetism. After decades of searching,
three examples of the defeat of Néel order by
quantum fluctuations have been discovered
in quick succession. There are good reasons
to believe that fermionic spinons will
emerge as the low-energy excitations, but
more work will be needed to confirm this.
Figure 1. Crystal structure of herbertsmithite obtain
gredients are dimers of an organic molecule,
T [bis(ethylenedithio) - tetrathiafulvalene]. A
ngle electron is localized on each dimer,
hich forms layers of approximately triangular
ttices. Despite an exchange energy of ~250 K,
o magnetic order was detected down to 30
K. This material is an insulator but becomes a
uperconductor (critical temperature Tc = 3.5 K)
nd then a metal under pressure. It is believed
at the proximity to an insulator-to-metal trantion implies that the spins interact with a more
omplicated Hamiltonian than the Heisenberg
odel and allows the spin liquid state to form
, 9). Remarkably, the spin susceptibility goes
a constant at low temperatures and the spefic heat is linear in temperature (7). These
operties are normally associated with metals,
eing consequences of the electron Fermi surce. The linear specific heat is particularly
nusual for an insulator that is relatively defectee. Furthermore, the ratio of magnetic suseptibility to the linear temperature coefficient
the specific heat is close to that of free fermi-
ARTICLES
Spinons in the
S=1/2 Heisenberg
antiferromagnet
on kagome
Tuesday, April 23, 13
Oleg Tchernyshyov
Johns Hopkins University
Papers
• Zhihao Hao and Oleg Tchernyshyov:
Tuesday, April 23, 13
•
Spin-1/2 Heisenberg antiferromanget on kagome: a Z2 spin
liquid with fermionic spinons. arxiv:1301.3261
•
"Structure factor of low-energy spin excitations in a S=1/2
kagome antiferromagnet," Phys. Rev. B 81, 214445 (2010).
•
"Fermionic spin excitations in two and three-dimensional
antiferromagnets," Phys. Rev. Lett. 103, 187203 (2009).
Long story
1-body physics
Spinons on kagome resemble strongly spinons of the Heisenberg
model on the Delta/sawtooth chain. Both come in two flavors
and have similar kinematics.
Tuesday, April 23, 13
Long story
2-body physics
Spinons exhibit Fermi statistics: upon exchange of two spinons,
the 2-spinon wavefunction changes sign.
They feel strong attraction in the S=0 channel mediated by the
exchange interaction. This attraction binds two spinons into a
small pair with a binding energy of 0.06J.
Tuesday, April 23, 13
Long story
Many-body physics
We find a finite concentration of spinons in the ground state,
one spinon per unit cell (3 sites). Their ground state is not a
Fermi sea (as in large-N inspired approaches), for two reasons.
First, strong attraction in the S=0 channel (see 2-body physics).
Second, strong interaction with an emergent U(1) gauge field.
Spinons carry the U(1) charge +1. They also have a small electric
charge and couple to the real electric field.
Spin pairs are bosons with charge +2. Their condensation may
create a Z2 spin liquid with vison excitations.
Tuesday, April 23, 13
Spinons on kagome:
The gory details
Tuesday, April 23, 13
Kagome lattice
lattice of corner-sharing triangles
Test
H=J
⇥ij⇤
Tuesday, April 23, 13
J
Si · Sj =
2
S + const.
2
Single triangle of s=1/2
hw02, problem 4
E= -3/4 J is the energy of a spin
singlet: the lowest energy state of the
single triangle is made of a singlet
bond and a free unpaired spin
OS addition
Tuesday, April 23, 13
Construct a ground state?
Place a valence bond on every triangle.
Every triangle minimizes its exchange
energy and the entire lattice is in a
state of lowest possible energy.
Tuesday, April 23, 13
Construct a ground state?
Place a valence bond on every triangle.
Every triangle minimizes its exchange
energy and the entire lattice is in a
state of lowest possible energy.
Tuesday, April 23, 13
Construct a ground state?
Place a valence bond on every triangle.
Every triangle minimizes its exchange
energy and the entire lattice is in a
state of lowest possible energy.
Tuesday, April 23, 13
Construct a ground state?
Place a valence bond on every triangle.
Every triangle minimizes its exchange
energy and the entire lattice is in a
state of lowest possible energy.
No matter how hard we try, we always end up with
a few defect triangles containing no valence bond.
Tuesday, April 23, 13
Defects are inevitable!
• N unit cells.
• 2N triangles (up and down).
• 3N sites.
• 3N/2 valence bonds.
• 3N/2 have valence bonds.
• N/2 triangles are empty.
• That is one in four triangles.
V. Elser, PRL 62, 2405 (1989).
Tuesday, April 23, 13
Focus on defect triangles
Defect triangles are sources of
quantum fluctuations. Without them,
the system would have valence bonds
frozen in place.
Tuesday, April 23, 13
Focus on defect triangles
Defect triangles are sources of
quantum fluctuations. Without them,
the system would have valence bonds
frozen in place.
Tuesday, April 23, 13
Focus on defect triangles
Defect triangles are sources of
quantum fluctuations. Without them,
the system would have valence bonds
frozen in place.
Tuesday, April 23, 13
Focus on defect triangles
Defect triangles are sources of
quantum fluctuations. Without them,
the system would have valence bonds
frozen in place.
Tuesday, April 23, 13
Focus on defect triangles
Defect triangles are sources of
quantum fluctuations. Without them,
the system would have valence bonds
frozen in place.
Quantum fluctuations of valence bonds
near a defect triangle. No fluctuations
away from the defects.
Tuesday, April 23, 13
Look at defect triangles
Can’t study isolated defects on kagome. Instead, use
the Husimi cactus: same local connectivity, but no
loops apart from triangles.
Tuesday, April 23, 13
Look at defect triangles
Can’t study isolated defects on kagome. Instead, use
the Husimi cactus: same local connectivity, but no
loops apart from triangles.
The dual lattice is a tree
Tuesday, April 23, 13
Husimi cactus
(kagome in a hyperbolic plane)
V. Elser and C. Zeng (1993).
P. Chandra and B. Douçot (1994).
Tuesday, April 23, 13
Husimi cactus
(kagome in a hyperbolic plane)
A ground state.
V. Elser and C. Zeng (1993).
P. Chandra and B. Douçot (1994).
Tuesday, April 23, 13
A single defect triangle
The triangle without a bond is not in a stationary state.
Nearby bonds fluctuate, those farther away remain static.
Tuesday, April 23, 13
Ground state without defects
V. Elser and C. Zeng (1993).
P. Chandra and B. Douçot (1994).
Tuesday, April 23, 13
Ground state without defects
A ground state.
V. Elser and C. Zeng (1993).
P. Chandra and B. Douçot (1994).
Tuesday, April 23, 13
S=1 excitations
Tuesday, April 23, 13
S=1 excitations
H
Tuesday, April 23, 13
= 5/4
1/2
1/2
Displaced bonds are shown in blue.
Tuesday, April 23, 13
Displaced bonds are shown in blue.
Tuesday, April 23, 13
Displaced bonds are shown in blue.
kink
antikink
The free spin can freely travel along a particular line.
It shifts bonds along its path (cf. the sawtooth chain).
Tuesday, April 23, 13
It’s a Δ/sawtooth chain embedded in the Husimi cactus.
kink
antikink
The rest of the lattice remains frozen in time.
Tuesday, April 23, 13
Spin excitations on the cactus
• are spinons: quasiparticles with S=1/2;
• come in two flavors:
• kinks are completely localized,
• antikinks are able to move along a line,
rearranging bonds along the way;
• resemble closely spinons on the Δ/sawtooth
chain.
Tuesday, April 23, 13
Defect triangle revisited
The triangle without a bond is not in a stationary state.
Tuesday, April 23, 13
State with a 3rd-neighbor singlet bond.
Tuesday, April 23, 13
Long-range bond = spins ● and ● with total S=0.
Tuesday, April 23, 13
Long-range bond = spins ● and ● with total S=0.
Tuesday, April 23, 13
He II at low temperatu
α
⌅S1 · S2 ⇥ S3⇧ =
⇤ 0.
a a = 2S.
†
β
S=a
†
⇥
2
Long-range bond = spins ● and ● with total S=0.
Consider a general state of the two spins, |
Tuesday, April 23, 13
• ⇥• ⇧
a⇥ .
Spin ● is able to move around.
Tuesday, April 23, 13
Spin ● is able to move around.
Tuesday, April 23, 13
So is spin ●.
Tuesday, April 23, 13
The spinons are antikinks traveling along three branches
of the cactus.
Tuesday, April 23, 13
We can now probe their exchange statistics.
Tuesday, April 23, 13
We can now probe their exchange statistics.
Tuesday, April 23, 13
An important technical point
Arrows indicate a sign convention for singlet bonds.
He II at low temperatures
| ⇥• ⇤ ⌅ | ⇤• ⇥ ⌅
⌃
=
2
Tuesday, April 23, 13
An important technical point
Arrows indicate a sign convention for singlet bonds.
He II at low temperatures
| ⇥• ⇤ ⌅ | ⇤• ⇥ ⌅
⌃
He II at low temperatures=
2
| ⇥• ⇤ ⌅ | ⇤• ⇥ ⌅
⌃
=
2
Tuesday, April 23, 13
| ⇥ ⇤• ⌅ | ⇤ ⇥• ⌅
⌃
=
2
An important technical point
Arrows indicate a sign convention for singlet bonds.
He II at low temperatures
| ⇥• ⇤ ⌅ | ⇤• ⇥ ⌅
⌃
He II at low temperatures=
2
| ⇥• ⇤ ⌅ | ⇤• ⇥ ⌅
⌃
=
2
Tuesday, April 23, 13
| ⇥ ⇤• ⌅ | ⇤ ⇥• ⌅
⌃
= −=
2
Tuesday, April 23, 13
Tuesday, April 23, 13
Tuesday, April 23, 13
Tuesday, April 23, 13
Tuesday, April 23, 13
Tuesday, April 23, 13
Initial state
Tuesday, April 23, 13
State with spinons exchanged
Initial state
Tuesday, April 23, 13
State with spinons exchanged
Initial state
State with spinons exchanged
Spinon exchange flips one valence bond.
That produces a minus sign.
Tuesday, April 23, 13
Forget about the singlet signs and treat valence bonds
as passive dimers. Then the extra minus sign must be assigned
to the spinons themselves.
Tuesday, April 23, 13
Forget about the singlet signs and treat valence bonds
as passive dimers. Then the extra minus sign must be assigned
to the spinons themselves.
In this sense, spinons are fermions.
Tuesday, April 23, 13
On a more technical level
• Obtain an orthogonal basis (similar to the
sawtooth chain).
• Find the ground state of two spinons with a given
total spin S and check the spatial wavefunction.
Fermi statistics requires that
• Ψ(r , r ) = +Ψ(r , r ) for total S=0. ✔
• Ψ(r , r ) = −Ψ(r , r ) for total S=1. ✔
●
●
●
●
●
●
●
●
• Spinons are fermions.
Tuesday, April 23, 13
Finite-size scaling of the
ground-state energies
S=1: no bound state.
E1 = bottom of the
2-spinon continuum
E1 + π2/2(R+2)2
E
0.65
0.6
S=0: bound state with
binding energy Δ = 0.06J
and radius ξ = 1.4 spacings
of the dual lattice.
Tuesday, April 23, 13
S=1
0.55
0.5
0.45
0.4
Δ
S=0
0
5
10
15
20
R
What about kagome?
• Antikinks are still fermions.
• 1 in 4 triangles contains a spinon pair,
• That translates to 1 spinon per unit cell.
Tuesday, April 23, 13
Low-energy spin excitations
S=0
Tuesday, April 23, 13
S=1
Low-energy spin excitations
S=0
Tuesday, April 23, 13
S=1
Low-energy spin excitations
S=0
S=1
Away from the pair:
S=0 bond → S=1 bond → kink + antikink. E ≈ 0.25J.
Tuesday, April 23, 13
Low-energy spin excitations
S=0
S=1
Away from the pair:
S=0 bond → S=1 bond → kink + antikink. E ≈ 0.25J.
Near the pair:
S=0 bond → S=1 bond → 2 antikinks. E ≈ 0.06J.
Tuesday, April 23, 13
Inelastic neutron scattering just
above the spin-gap threshold
N=5
Single dimer
2
Ke
1
K
0.4
Ke
0.3
K
0 qy 0.2
-1
-2
-2 -1
0
qx
1
2
0.1
0
-2 -1
0
qx
1
2
Left: I(Q) of inelastic neutron scattering just above the spin gap.
Right: same for isolated dimers of three different orientations.
Z. Hao and O.T., PRB 81, 214454 (2010); arXiv:1004.2293.
R. R. P. Singh, Phys. Rev. Lett. 104, 177203 (2010).
A. Läuchli and C. Lhuillier, arXiv:0901.1065.
L. Messio, O. Cépas, and C. Lhuillier, Phys. Rev. B 81, 064428 (2010).
Tuesday, April 23, 13
1.4
2
1.2
1
1
0.8
0 qy
0.6
-1
0.4
0.2
-2
0
LETTER Experiment Nature v.492, p.406 (2012)
doi:10.1038/nature11659
Fractionalized excitations in the spin-liquid state of a
kagome-lattice antiferromagnet
Tian-Heng Han1, Joel S. Helton2, Shaoyan Chu3, Daniel G. Nocera4, Jose A. Rodriguez-Rivera2,5, Collin Broholm2,6 & Young S. Lee1
LETTER RESEARCH
The experimental realization of quantum spin liquids is a long- Cu21 ions (,5% of the total) substituting for Zn21 ions in the intersought goal in physics, as they represent new states of matter. Quan- layer sites, the kagome planes contain only Cu21 ions10. Measurements
0
5
10
20 011–13
0.15
0.3
0.45
tum spin liquids cannot be described
by the
broken symmetries
on15 powder samples
indicate strong
antiferromagnetic
superassociated with conventional ground
states.
In
fact,
the
interacting
exchange
(J
<
17
meV,
where
J
is
the
exchange
coupling
that
appears
a
3
magnetic moments in these systems
do not order, but are highly in the nearest-neighbour Heisenberg Hamiltonian) and the absence
d3
entangled with one another over long ranges1. Spin liquids have a of long-range magnetic
order or spin freezing down to temperaprominent role in theories describing
high-transition-temperature tures of T 5 0.05 K. The bulk magnetic properties reveal a small
2
superconductors2,3, and the topological properties of these states Dzyaloshinskii–Moriya
interaction and an easy-axis exchange
2
may have applications in quantum information4. A key feature of anisotropy14,15, both of order J/10. Despite these small imperfections,
1 exotic spin excitations carrying the nearest-neighbour Heisenberg model on a kagome lattice is still an
spin liquids is that they support
1
fractional quantum numbers. However, detailed measurements of excellent approximation
of the spin Hamiltonian for herbertsmithite.
these ‘fractionalized excitations’ 0have been lacking. Here we report This is especially important, because recent calculations on record
neutron scattering measurements
on single-crystal samples of the lattice sizes indicate that the ground state of this model is in fact a
b
spin-1/2 kagome-lattice antiferromagnet ZnCu3(OD)6Cl2 (also called quantum spin liquid016. Thus, experiments to probe the spin correla2
herbertsmithite), which provide striking evidence for this characte- tions in herbertsmithite are all the more urgent.
ristic feature of spin liquids. At low temperatures, we find that the
To this end, we recently succeeded in developing a technique for the
e3
spin excitations form a continuum,
single crystals of herbertsmithite17, and
1 in contrast to the conventional growth of large, high-quality
spin waves expected in ordered antiferromagnets. The observation of small pieces have been used in studies involving local probes18,19,
such a continuum is noteworthy because, so far, this signature of anomalous X-ray diffraction10, susceptibility15 and Raman scattering20.
0
2 inelastic neutron scattering measurements on
fractional spin excitations has been
c 2 observed only in one-dimensional In this Letter, we report
systems. The results also serve as a hallmark of the quantum spin- a large, deuterated,
single-crystal sample of herbertsmithite. The neu20
liquid state in herbertsmithite.
tron scattering cross-section is directly proportional to the dynamic
1
1
In a spin liquid, the atomic magnetic
moments are strongly corre- structure10
factor Stot(Q, v) (where Q and v stand for the momentum
lated but do not order or freeze even in the limit as the temperature, T, and energy transferred to the sample, respectively), which includes
goes to zero. Although many types0 of quantum spin-liquid states exist both the nuclear
0
and0 magnetic signals. The magnetic part, Smag(Q, v),
in theory, a feature that is expected –2
to be common
is the
correlation
–1to all is the presence
0
1 Fourier transform
2
–2 (in time and–1space) of the spin–spin
0
1
2
of deconfined spinons as an elementary excitation from the
ground
function and can be obtained by subtracting the
nuclear
scattering as
(H,
H,
0)
(H,
H,
0)
state1. Spinons are spin-half (S 5 1/2) quantum excitations into which described in the Supplementary Information. After calibration with
conventional
spin-wave
excitations
with
S 5 1 fractionalize.
In one
respectplotted
to a vanadium
the9 meV.
measured
structure factors
are
1 # Bv #
e, Calculation
of the equal-time
structure factor,
Figure
1 | Inelastic
neutron
scattering
from the spin
excitations,
in overstandard,
dimension,
this phenomenon
well established
the atS T
55
1/21.6 Kexpressed
in absoluteSunits.
reciprocal
space. a–c,isMeasurements
werefor
made
on a singlemag(Q), for a model of uncorrelated nearest-neighbour dimers. The intensity
Heisenberg
antiferromagnetic
chain,
where
may bestructure
thought factor,Contour
plotsis of Scorresponds
T 5 1.6sum
K and
The dynamic
Stot(Q, v),
to 1/8 in
of Fig.
the 1a–c
total for
moment
rule S(S 1 1) for the spins on the
crystal
sample of ZnCu
tot(Q, v) are shown
3(OD)
6Cl2.spinons
of as magnetic
domain
thatand
disrupt
Néel
order(b)
and
are Efree
three
energy
transfers
BvThe
(B data
denotes
Planck’s
constant
meVdifferent
and
plotted
for Bvboundaries
5 6 meV (a)
Bv 5
2 meV
with
kagome
lattice.
presented
in a–c
are expressed in barn sr21 eV21 per
f 5 5.1
to propagate
from
each
the3.0
one-dimensional
compound
divided by
2p).
1a shows
for Bv
6 meV.
Surprisingly,
Bv away
5 0.75
meV
(c)other.
with EInf 5
formula
unit,data
as shown
by5the
left colour
bars. The data presented in parts d and
meV. The background
was measured
with
anFigure
KCuF3, a empty
continuum
of spinon
has been
characterized
the scattered
intensity
is exceedingly
diffuse,
spanning
e are
dimensionless,
with
the scalea large
givenfraction
by the right colour bar. The Brillouin
sample
holder excitations
and subtracted.
Thewell
diffuse
scattering is mostly
magnetic
5
using neutron
scattering
. Inthe
twophonon
dimensions,
the nature
thesignal
spinon
of the
hexagonal
zone. A similar
pattern
zone boundaries
are drawn
in of
thediffuse
figurescatterfor clarity; they correspond to the
in origin,
because
contribution
toofthe
is small
(except
near Brillouin
excitations
is less
First,positions,
the existence
of two-dimensional
ingare
is strong).
observed for Bv
5 2 meV (Fig.
The diffuse
nature
ofbthe
conventional
unit 1b).
cell with
parameters
a5
5 6.83 Å, c 5 14.05 Å,
the
(2, 2,clear.
0)-type
where
the fundamentalmagnets
Bragg peaks
with a quantum
spin-liquid
ground
state
is
still
a
matter
of
great
debate.
scattering
at
a
temperature
that
is
two
orders
of
magnitude
below
the
a 5 b 5 90u and c 5 120u.
d, The magnetic part of the dynamic structure factor, Smag(Q, v), integrated
Second, the various spin-liquid states which are proposed in theory exchange energy scale, J, is in strong contrast to observations in nongive rise to a variety of spinon excitation spectra, which may be either frustrated quantum magnets. The S 5 1/221
antiferromagsites, which
are believed to affect the low-energy
The observed Q dependence of the scattered intensity provides the interlayer Zn square-lattice
gapped or gapless.
net La2CuO4 develops substantial
antiferromagnetic
correlations
for
25
.
scattering
important
information
on
the
ground-state
spin
correlations.
The
The S 5 1/2 kagome-lattice Heisenberg antiferromagnet has long T , J/2 (ref. 21), temperatures at which the low-energy scattering is
scattering
in reciprocal
hastothe
shape
of broadened
hexagonal
The
scattering
pattern’s space.
overall
insensitivity to energy transfer is
been recognized
as a promising
systemspace
in which
search
for quantum
strongly
peaked near the
(p, p)
point in reciprocal
In herbertrings
centred
(0,kagome
0, 0)- and
(2, 0,
positions.
of thethe
scans
another
remarkable
feature
of theatdata.
spin-wave excitaspin-liquid
states,
becauseatthe
network
of0)-type
corner-sharing
tri- All
smithite,
scattered
intensity
is not strongly
peaked
any Conventional
speangles frustrates
magnetic from
order6–8
. We
syn-show
cificsimilar
point, patand thistions
remains
for allofenergies
measuredoffrom
that welong-range
have performed
Bv
5 have
1.5 todevised
11 meV
taketrue
the form
sharp surfaces
dispersion in Q–v space. Such
thetic methods
produce
(ZnCu3(OH)
whichenergy-integrated
Bv 5 0.25 to 11 meV.
This behaviour
is also
markedly
6Cl2) inThe
terns to
for
the herbertsmithite
scattered magnetic
intensity.
spin-wave
excitations
were
indeeddifferent
observed in the S 5 5/2 kagome
21
the S 5 1/2
Cu
moments
are
arranged
on
a
structurally
perfect
from
that
observed
in
the
larger,
S
5
5/2
kagome
antiferromagnet
dynamic
structure
factor
over
the
integration
range
1
#
Bv
#
9
meV
(OH)
(SO
)
(ref. 26). In herbertsmithite, no
antiferromagnet
KFe
3
6
4
2
21
kagome lattice9 and nonmagnetic Zn ions separate the lattice planes. KFe3(OH)6(SO4)2 which becomes magnetically ordered at low temis
plotted
in
Fig.
1d.
This
quantity
serves
as
an
approximation
of
the
surfaces
of
dispersion
are
observable
in the low-temperature data.
A depiction of the crystal structure is shown in Supplementary Fig. 1. peratures and has magnetic peaks at q 5 0 wavevectors above the
equal-time
structure
factor.
For
comparison,
a
calculation
of
the
equal(Q,
v)
on
Bv
and
Q is plotted in Fig. 2 for
The
dependence
of
S
22
tot
Whereas herbertsmithite typically contains a small percentage of excess ordering temperature .
(–K, K, 0)
ARTICLES
Figure 1. Crystal structure of herbertsmithite obtained OS
withaddition
0.41 × 10-10
time structure factor for a collection of uncorrelated nearest-neighbour
singlets on a kagome lattice is shown in Fig. 1e. To a first approximation, the observed magnetic signal resembles this calculation.
Therefore,
the ground-state wavefunction of herbertsmithite has a
Tuesday, April 23,
13
1
m wavelength data. Blue, green, bright green, red, and gray spheres represent
Zn, Cu, Cl, O, and H atoms, respectively. (a) Local coordination environment of the intersite is highlighted in a projection perpendicular to the c-axis.
Thermal ellipsoids are drawn at the 70% probability level. (b) The geometry
of
the kagomé layer is highlighted, in a projection perpendicular to the
two high-symmetry directions in reciprocal space: the (H, 0, 0) direction (Fig. 2a) All
and theatoms
(H, H, 0) direction
(Fig. 2b).
Thesekagomé
directions are plane have been removed for clarity.
c-axis.
outside
the
indicated by thick black lines in Fig. 2d. These plots show that the spin
(c)
Photograph
of abandcrystal
from
the batch used for these data. Selected
excitations
form a broad, continuous
(or a continuum),
extending
Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA. 2NIST Center for Neutron Research, National Institute of Standards and Technology, Gaithersburg,
Maryland 20899, USA. 3Center for Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA. 4Department of Chemistry, Massachusetts Institute
of Technology, Cambridge, Massachusetts 02139, USA. 5Department of Materials Science and Engineering, University of Maryland, College Park, Maryland 20742, USA. 6Institute for Quantum Matter and
Department of Physics and Astronomy, The Johns Hopkins University, Baltimore, Maryland 21218, USA.
excitation continuum is observed over the entire range measured. The colour
bar shows the magnitude of Stot(Q, v) in barn sr21 eV21 per formula unit.
c, Energy dependence of Stot(Q, v) measured at high-symmetry reciprocal
normalized to have units of eV21 per formula unit, consistent with the
magnetic structure factor defined in Supplementary Information. d, The
integrated areas in reciprocal space referred to a–c.
the free-Cu21 magnetic form factor. Here the measured data indicate
longer-range correlations than the nearest-neighbour singlet model.
Figure 3c depicts a line scan of the dynamic structure factor (integrated
over 1 # Bv # 7 meV) along the (0, K, 0) direction. The nearestneighbour singlet model does not account for the observed scattering
intensity at the (0, 2, 0)-type positions.
30
2 meV
6 meV
10 meV
0.2
20
10
0
0.3
0.1
Integrated over 1 to 7 meV
Dimer calculation
0.0
0
Integrated over 1 to 11 meV
Dimer calculation
1
2
(0, K, 0)
d
3
0.2
(–K, K, 0)
Integrated Smag(Q, ω)
b
c
Integrated Smag(Q, ω)
Smag(Q, ω) (eV–1 form. unit–1)
a
0.1
0.0
–1
0
(–2, 1 + K, 0)
Figure 3 | The measured dynamic structure factor along specific directions
in reciprocal space with comparison to the nearest-neighbour singlet model.
a, Smag(Q, v) along the (22, 1 1 K, 0) direction, indicated by the thick red line
on the reciprocal space map in d. Three energy transfers, Bv 5 2, 6 and 10 meV,
are shown. b, Smag(Q, v) along the (22, 1 1 K, 0) direction integrated over
1
2
ut
)c
,0
,
(–2
1
0
–2
K
1+
–1
0
(H, H, 0)
ut
)c
,0
K
(0,
1
2
1 # Bv # 11 meV. c, Smag(Q, v) along the (0, K, 0) direction, indicated by the
thick orange line on the reciprocal space map in d, integrated over
1 # Bv # 7 meV. The solid lines in b and c are the calculated equal-time
structure factors for uncorrelated nearest-neighbour singlets multiplied by
| F(Q) | 2. d, The trajectories in reciprocal space referred to in a–c. Error bars, 1 s.d.
4 0 8 | N AT U R E | VO L 4 9 2 | 2 0 / 2 7 D E C E M B E R 2 0 1 2
©2012 Macmillan Publishers Limited. All rights reserved
OS addition
Tuesday, April 23, 13
Spin gap
• Our calculation: 0.06 to 0.10.
• less than 0.1, exact diagonalization*.
• 0.06 to 0.10, perturbation series .
• 0.05 to 0.06, DMRG (Sheng).
• Steve White’s DMRG: 0.14 or so?
†
*C. Waldtmann et al., Eur. Phys. J. B 2, 501 (1998).
†R.R.P. Singh and D. A. Huse, arXiv:0801.2735.
H. C. Jiang, Z.Y. Weng, and D. N. Sheng, Phys. Rev. Lett. 101, 117203 (2008).
Tuesday, April 23, 13
Spinons
in d=1:
Test
Δ/sawtooth chain
H=J
⇥ij⇤
J
Si · Sj =
2
S2 + const.
Ground state: every triangle has total spin 1/2.
T. Nakamura and K. Kubo, Phys. Rev. B 53, 6393 (1996).
D. Sen, B. S. Shastry, R. E. Walstedt, and R. Cava, ibid., 6401 (1996).
Tuesday, April 23, 13
Spinons
in d=1:
Test
Δ/sawtooth chain
H=J
⇥ij⇤
J
Si · Sj =
2
S2 + const.
Ground state: every triangle has total spin 1/2.
T. Nakamura and K. Kubo, Phys. Rev. B 53, 6393 (1996).
D. Sen, B. S. Shastry, R. E. Walstedt, and R. Cava, ibid., 6401 (1996).
Tuesday, April 23, 13
Spinons
in d=1:
Test
Δ/sawtooth chain
H=J
⇥ij⇤
J
Si · Sj =
2
S2 + const.
Ground state: every triangle has total spin 1/2.
T. Nakamura and K. Kubo, Phys. Rev. B 53, 6393 (1996).
D. Sen, B. S. Shastry, R. E. Walstedt, and R. Cava, ibid., 6401 (1996).
Tuesday, April 23, 13
Spin excitations
Tuesday, April 23, 13
Spin excitations
Tuesday, April 23, 13
Spin excitations
Tuesday, April 23, 13
Spin excitations
Tuesday, April 23, 13
Spin excitations
Tuesday, April 23, 13
Spinons
kink
antikink
• Serve as domain walls between the 2 vacua.
• Carry spin S=1/2.
• Kinks are localized in this model.
• Antikinks are mobile.
• Their quantum statistics is undefined:
can’t exchange particles in 1 spatial dimension.
Tuesday, April 23, 13
Test
Energetics
J
H=J
S ·S =
i
j
⇤ij⌅
H=J
⇤ij⌅
J
Si · Sj =
2
⇥
kink
2
S⇥
2
⇥
2
S⇥
+ const.
E (k) = 0.
antikink
E (k) = 0.
3 1
E+(k) = +
4 2
1 k2
cos k ⇤ + .
4
2
1
= min(E + E+) = .
4
2
1 k
cos k ⇤ + .
4
2
– Typeset by FoilTEX –
3 1
E+(k) = +
4 2
–
Tuesday, April 23, 13
+ const.
1
Virtual excitations
+1
(b)
Tuesday, April 23, 13
T
T
Virtual excitations
+1
T
(b)
T
1 band
2 bands
∆ = 0.250 in 1-band approximation.
∆ = 0.219 in 2-band approximation.
∆ = 0.215 exact diag (Kubo 1996).
Fast convergence.
Not much room for improvement.
2J
J
0
/2
Tuesday, April 23, 13
0
/2
k