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講義スライドNMR II - WEB PARK 2014

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現代物理学入門
磁性体における緩和率
ZrZn2 弱強磁性(Tc = 28 K)
β-Mn 反強磁性寸前の常磁性体
現代実験物理学I
超伝導状態におけるスペクトル
(a)
シールディング(軌道反磁性)によるシフト
λL:磁場侵入長
75
As
第2種超伝導のRedfieldパターン
Intensity (arb. units)
63
65
Cu
eiQ·r
e
2
2
1 + LHQ|| ab
H(r) = H0
7
Hanaguri et al., Q
Li
〃
PRB 85, 214505 (2012)
3
(b)
4
5
Intensity (arb. units)
e
6
7
LiFeAs単結晶NMR
µ0 H (T)
7
Li
6.6 T
8
(c)
15 K
4.2 K
(修正ロンドンモデル)
2Q
H || c
2Q
9
2
三角格子
Exp.
Calc.
ξ = 4.63 nm
λL = 110 nm
As
6.6 T
dKs = -0.025% 75
dKs = -0.035% 75
0
0.1 0.2 0.3
Knight shift (%)
0.4
(ロンドンモデル)
10
dKs = 0 As
4.25 T
-0.2 -0.1
ξ:コヒーレンス長
Cu
-0.2 -0.1
0
0.1
K4.2 K - K15 K (%)
0.2
saddle
c
ms
middle
core
H
現代実験物理学I
超伝導状態におけるスピン磁化率
NMRではスピン磁化率と軌道磁化率を分離できる(K-χプロット)
Kspin
Aspin spin
=
µB
spin
=
4µ2B
NS
0
f (E)
dE
E
準粒子状態密度
準粒子状態密度
ナイトシフト
スピン一重項
超伝導では減
少する
0.0
χorb
0
||ab
χorb
0.0000
||c
0.0005
0.0010
χab ( emu / mol )
現代実験物理学I
100
200
300
Temperature ( K )
スピン三重項超伝導状態におけるシフト
Fig. 2. Knight shift vs bulk susceptibility determined in the range
T ¼ 1:4{20 K for a weakly-correlated metal RuO2 , an electron-correlated
superconductor Sr2 RuO4 , and a correlated, nearly ferromagnetic metal
CaRuO3 . The hyperfine coupling constant due to the inner core polarization
Acp % !22:2 T/"B is estimated from a linear slope with the assumption that
Korb ¼ 1:59% is common among these ruthenates. The value of Acp is close
to Ahf % 30 T/"B in the FM state of SrRuO3 . A large negative K value of
Sr2 RuO4 indicates that its susceptibility is dominated by the d-electron spin
susceptibility (Fig. 5 in ref. 60).
Fig. 3. (Color online) Bulk susceptibility of Sr2 RuO4 (refs. 62 and 63)
illustrating the spin and orbital contributions (see text for details).
クーパー対の軌道部分に空間反転対称
ある
スピン部分
Kspin
↑↓ - ↓↑
ない
↑↑,↓↓,
↑↓ + ↓↑
nuclei as an internal thermometer (supplement
of ref. 47).
減らない
Third, based減少
on the systematic studies of K among different
ruthenates, i.e., RuO2 , Sr2 RuO4 , and (↑↑,↓↓)
CaRuO3 , it is clear that
a large negative value of K for Sr2 RuO4 originates from the
contribution from core polarization due to d-electrons
J. Phys. Soc. Jpn. 81 (2012) (Fig.
0110092).60) From such analysis,
S PECIAL
TOPICS
Y. MAENO et al.
the contribution
from the
orbital part is estimated as 1.0% while the spin contribution
is !4:5%
for K ksuperconductors.
c of Sr2 RuOHF:
anisotropy
of the NCS:
Table I. Selection of candidates
of spin-triplet
heavythe
fermion
superconductors,
FM:
99
4 . With
Fig.Noncentrosymmetric
4. Knight shift ofsuperconductors,
Ru nuclei of Sr
2 RuO4 with the applied field
61)
ferromagnetic superconductors,
!:
superconductivity
under
pressure.
spin part of only 2% as extracted from the !ab vs !c plot,
parallel to the basal ab plane (ref. 62). The spin part Ks ðT Þ, after subtracting
62,63)
orbital part Korb from the total shift KðT Þ, is
the dominance
of the spin part in the
is the temperature-independent
Materials
Classification
Spin susceptibility
evidence of triplet pairing
Properties
plotted.
illustrated
in
Fig.
3.
Such
properties
of
Sr
RuO
ensure
that
3
2
4 NMR etc.7)
He
Superfluid
magnetization,
p-wave, A phase is chiral
its NMR Knight shift serves as a reliable high-sensitivity
Oxide
NMR, polarized neutron
2D analogue of 3 He-A
Sr2 RuO4
probe for the Cooper pair spin susceptibility.
Chiral p-wave
Ishida and coworkers have measured Knight shifts
KðT Þ polarization effect, where "B is the Bohr magneton. This
18)
HF
NMR
f -wave
UPt3
at the Ru and O sites in high-quality single13)crystals. means that the Ru Knight shift can detect much smaller
HF
NMR
UBe13 , URu2 Si2 , UNi2 Al3
Measurements were performed using a 3 He–4 He dilution changes of the Ru-4d electronic
state when superconducUGe2 ! , URhGe, UCoGe
FM, HF
Indirect
Anomalous Hc2 19–22)
refrigerator with the sample crystals directly immersed in tivity sets in. The temperature dependence of the 99 Ru shift
UIr!
NCS, FM, HF
Indirect
liquid helium to ensure good thermal contact to the23)bath. As was measured in the fields of 0.68, 0.9, and 1.05 T parallel
!
NCS, HF
NMR
CeIrSi3
for the 17 O Knight-shift measurements, they assigned
NMR to the RuO2 plane and the spin part of the Knight shift
Li2 Pt3 B
NCS
NMR24)
signals arising from
the different crystal sites
(planer and [99 Kspin ¼ KðT Þ ! Korb ] obtained at 0.9 T is shown in Fig. 4.
CePt3 Si
99
apical O sites),NCS,
andHF
estimated the spin part Indirect
of the Knight
Kspin does not change on
passing through Tc % 1:0 K at
!
NCS, HF
Indirect
Anomalous Hc2 25)
CeRhSi3
shift above Tc at each site. They investigated temperature
"0 H % 0:9 T. If a spin-singlet d-wave state with a line-node
S/FM/S
Junctions
Indirect (Ic )26–29)
even-parity, s-wave
dependence of the Knight shift at the planer O site, where gap wereOdd-freq.,
realized,
the T dependence of 99 Kspin would
JPSJ
81,dashed
011009
the spin density is 6 times larger than that in the apical OMaeno
behaveetasal.,
drawn
by the
curve(2012).
in Fig. 4. Here we
パウリリミット
存在
無し(↑↑,↓↓)
現代実験物理学I
超伝導状態における緩和率
4
1
=
T1
n Ahf
2
dEk dEk
1+
2
Ek Ek
NS (Ek )NS (Ek )f (Ek ){1
コヒーレンス因子
f (Ek )} (Ek
Ek )
コヒーレンスピーク
(Hebel-Slichterピーク)
現代実験物理学I
REVIEW ARTICLE FOCUS
量子液体
reside on the vertices of square lattices, w
other in the three-dimensional crystal. T
layers are negligible (although this small
obtaining a non-zero magnetic ordering
need only consider the S = 1/2 spins S
a square lattice, i. The spins are couple
a superexchange interaction, and so w
hamiltonian
X
H0 =
J ij Si · Sj
Localized spins
Neél order
120° order
hiji
Here, hiji represents nearest-neighbour
exchange interaction. The antiferromagn
the positive value of J ij , which prefers an
the spins. We will initially consider, in Se
the sites of a square lattice, and all J ij = J
in the subsequent sections to other lattice
between the spins.
Our strategy here will be to begin by
of H0 , and its perturbations, in some sim
then describe the low-energy excitations o
field theories. By extending the quantum
parameter regimes, we will see relations
discover new phases and critical points
global perspective of the phase diagram.
REVIEW ARTICLE FOCUS
Quantum Liquid
Fermi Liquid
3He
hiji
(Superfluid)4He
Here, hiji represents nearest-neighbour pairs and J ij > 0 is the
exchange interaction. The antiferromagnetism is a consequence of
the positive value of J ij , which prefers an antiparallel alignment of
the spins. We will initially consider, in Section IIA the case of i on
the sites of a square lattice, and all J ij = J equal, but will generalize
in the subsequent sections to other lattices and further interactions
between the spins.
Our strategy here will be to begin by guessing possible phases
Figure
states.
a,b, Ground
statessimple
of the limiting
S = 1/2cases.
antiferromagnet
H0 with all
of H10 , Néel
and its
perturbations,
in some
We will
then
describe
the low-energy
excitations
of these states
by quantum
J ij =
J on
the square
(a) and triangular
(b) lattices.
The spin
polarization is (a)
field theories.
By extending the quantum field theories to other
collinear
and (b) coplanar.
parameter regimes, we will see relationships between the phases,
discover new phases and critical points, and generally obtain a
global perspective of the phase diagram.
Sachdev, Nature Phys. 4, 173 (2008).
Itinerant
electrons
e-
reside on the vertices of square lattices, which are layered atop each
other in the three-dimensional crystal. The couplings between the
layers are negligible (although this small coupling is important in
obtaining a non-zero magnetic ordering temperature), and so we
need only consider the S = 1/2 spins Si , residing on the sites of
a square lattice, i. The spins are coupled to each other through
a superexchange interaction, and so we will consider here the
hamiltonian
X
H0 =
J ij Si · Sj .
Surpress
detectable.
Indeed, the quantum critical point can serve as the best
A. NEEL ORDERED STATES
point
entire
phase diagram in
Withof
alldeparture
J ij = J , there for
is nounderstanding
parameter, g , that the
can tune
the ground
H0 ; J sets the overall energy scale, but does not modify
thestate
g , Tof plane.
the
wavefunction
of any
of the eigenstates.
So there is only
We
will begin
Section
II by describing
thea single
rich variety of
quantum phase to consider.
quantum
phases
that
appear
insulators,
whose
For the
square
lattice,
thereinistwo-dimensional
convincing numerical
and
excitations
=ground
1/2 electronic
spins
on the sites of a
experimental
evidence2are
that Sthe
state of H0 has
Néel order,
Figure 1 Néel states. a,b, Ground states of the S = 1/2 antiferromagnet H0 with all primary
as
shown
in
Fig.
1a.
This
state
spontaneously
breaks
the
spinJ ij = J on the square (a) and triangular (b) lattices. The spin polarization is (a)
lattice. Interest in such two-dimensional spin systems was initially
rotation symmetry
of H0 , and isofcharacterized
by the expectation
collinear and (b) coplanar.
stimulated
by the discovery
high-temperature
superconductivity
value
in the cuprate compounds. However,
since then, such quantum
j
(1)
h
S
j i = ( 1 ) 8.
detectable. Indeed, the quantum critical point can serve as the bestspin systems have found experimental applications in a wide variety
systems,
we willthedescribe
below. The
critical
quantum
phases
point of departure for understanding the entire phase diagram inof Here,
( 1) jas
represents
opposite orientations
of the
spins on
the
the g , T plane.
andtwo
points
of such
materials
be discussed
in Section
sublattices
as shown
in Fig.will
1a. The
vector 8 represents
the III. Section
We will begin Section II by describing the rich variety of orientation and magnitude of the Néel order. We have |8| < 1/2,
IV will extend our discussion to include charge fluctuations,
quantum phases that appear in two-dimensional insulators, whose implying that the ground-state wavefunction is not simply the
will describe some recent proposals for novel metallic and
primary excitations are S = 1/2 electronic spins on the sites of aand
classical state shown in Fig. 1a, but has quantum fluctuations about
states,
and associated
quantum
critical
lattice. Interest in such two-dimensional spin systems was initiallysuperconducting
it. These fluctuations
will entangle
the spins with
each other,
but points.
stimulated by the discovery of high-temperature superconductivity qualitatively
character
of thethese
state isissues
captured
by therelated ideas
Althoughtheweessential
will not
discuss
here,
in the cuprate compounds. However, since then, such quantum pattern in Fig. 1a. Note that this pattern implies a long-range,
also apply in other spatial dimensions. Similarly, quantum states
spin systems have found experimental applications in a wide variety and classical, correlation between the spins, but not a significant
inofthree
dimensions,
critical points
of systems, as we will describe below. The critical quantum phasesappear
amount
entanglement
because although
there are no the
EPR quantum
eVects between
and points of such materials will be discussed in Section III. Sectionoften
haveofawell-separated
distinct character.
A fairly complete understanding of
any pair
spins.
IV will extend our discussion to include charge fluctuations,one-dimensional
Having described
the ground
state,has
we been
now consider
thebuilding on
correlated
states
achieved,
and will describe some recent proposals for novel metallic and excitations. This is most conveniently done by writing the form
methods.
superconducting states, and associated quantum critical points. ‘bosonization’
of the Feynman path integral for the trajectories of all the spins
classical orders
somehow
eUT Low-Temp. Center
HP
Geometrical frustration
J < 0
Although we will not discuss these issues here, related ideas in imaginary time, ⌧ . After taking the long-wavelength limit to
also apply in other spatial dimensions. Similarly, quantum statesII. PHASES
OF INSULATING
QUANTUM
MAGNETS
a continuous
two-dimensional
space, this
path integral defines
appear in three dimensions, although the quantum critical points a quantum field theory in 2 + 1 dimensional space-time with
often have a distinct character. A fairly complete understanding of coordinates (r, ⌧). The quantum field theory is for a field 8(r, ⌧),
superconductivity appears when insulators such
one-dimensional correlated states has been achieved, building onHigh-temperature
which is the value of the Néel order when averaged over the
‘bosonization’ methods.
as square-lattice
La2 CuO4 are
doped
with mobile
holes or
electrons.
spins,
Si , located
on sites within
some
averaging La2 CuO4 is
of r . Ain
quick
derivation
of the eVective
actionof
forfreedom are
an neighbourhood
antiferromagnet,
which
the magnetic
degrees
II. PHASES OF INSULATING QUANTUM MAGNETS
this
quantum
field
theory
is
provided
by
writing
all
terms
in
powers
S = 1/2 unpaired electrons, one on each Cu atom.
The Cu atoms
and gradients of 8 that are invariant under all the symmetries of the
3,4
High-temperature superconductivity appears when insulators such hamiltonian. In this manner, we obtain the action
Z
as La2 CuO4 are doped with mobile holes or electrons. La2 CuO4 is174
an antiferromagnet, in which the magnetic degrees of freedom are
S8 = d2 r d⌧ (@ ⌧ 8)2 + v 2 (rx 8)2 + s82 + u(82 )2 . (2)
S = 1/2 unpaired electrons, one on each Cu atom. The Cu atoms
triangular, Kagomé,
(AF)
honerycomb, pyrochlore,…
174
?
A. NEEL ORDERED STATES
With all J ij = J , there is no parameter, g ,
state of H0 ; J sets the overall energy sc
the wavefunction of any of the eigenstate
quantum phase to consider.
For the square lattice, there is co
experimental evidence2 that the ground s
as shown in Fig. 1a. This state sponta
rotation symmetry of H0 , and is charact
value
hSj i = ( 1) j 8.
Here, ( 1) j represents the opposite orien
two sublattices as shown in Fig. 1a. The
orientation and magnitude of the Néel o
implying that the ground-state wavefun
classical state shown in Fig. 1a, but has qu
it. These fluctuations will entangle the s
qualitatively the essential character of th
pattern in Fig. 1a. Note that this patte
and classical, correlation between the sp
amount of entanglement because there ar
any pair of well-separated spins.
Having described the ground state
excitations. This is most conveniently d
of the Feynman path integral for the tr
in imaginary time, ⌧ . After taking the
a continuous two-dimensional space, t
a quantum field theory in 2 + 1 dime
coordinates (r, ⌧). The quantum field the
which is the value of the Néel order
square-lattice spins, Si , located on sites
neighbourhood of r . A quick derivation
this quantum field theory is provided by w
and gradients of 8 that are invariant unde
hamiltonian. In this manner, we obtain th
Z
S8 = d2 r d⌧ (@ ⌧ 8)2 + v 2 (rx 8)
nature physics VOL 4 MARCH 2008 www.nature.com/naturephysics
Ising triangular AF
nature physics VOL 4 MARCH 20
現代実験物理学I
幾何学的フラストレーション
イジング3角格子
J < 0
(AF)
カゴメ格子
パイロクロア格子
?
幾何学的な理由で再近接相互作用では基底スピン配置を
一通りに決定できない
ネール秩序は著しく抑制される
絶対零度まで巨視的量の縮退(エントロピー)が残る可能性
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現代実験物理学I
非古典的な局在磁性
Localized spins
Disorder
Spin singlet
Spectra for Geometrical
fronzen spins: Spin ice Dy2Ti2O7
frustration
47Ti-NQR/zero-field
Spin glass
NMR
Spin
ice
Intensity (arb. units)
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= H+
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8
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(-,+
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-856$,
43 4,
2
The
coupled-dimer
antiferromagnet.
This is described
by the(-,+2
hamiltonian
.et al., H , with J'-0&)1
= J on the
red lines
J =
J/g on the
red lines. a, The
+*(, ("$3 '#$ +*(,+ *-(,' $('#$) 1()$"'75
-) solid
&0&5
.)-%and'#$
"$,')$+
-.dashed
'#$ '$')&#$1)&3
ground state,
Nature 399, 333large-g
(1999).
p with each ellipse representing a singlet valence bond
&,1 $&"# '$')&#$1)-, (+ "-,+')&(,$1 '- #&:$
'0- +*(,+ *-(,'(,6 (, &,1 '0- *-(,'(,6 -/'2
0
(|"#i
ij
ij
|#"i)/ 2. b, The S = 1 spin triplon excitation. The pair of parallel spins
Figure 3 Valence-bond solid states. a, Columnar VBS state of H0 + H1 with J ij = J
on all bonds. This state is the same as in Fig. 2a, but the square-lattice symmetry
has been broken spontaneously. Rotations by multiples of ⇡/2 about a lattice site
yield the four degenerate states. b, Four-fold degenerate plaquette VBS state with
現代実験物理学I
Quantum Spin Liquids by Theories
QSL must satisfy:
Unbroken symmetry
Not frozen spins
We frame our discussion in the context of a specific
perturbation, H , studied recently by Sandvik and others :
(insulator)
◆✓
◆
X✓
1
1
19
1
H1 = Q
hijkli
Si · Sj
4
Sk · Sl
Resonating Valence Bond
REVIEW ARTICLE FOCUS
P.W.Anderson 1987
20,21
4
,
where hijkli refers to sites on the plaquettes of the square lattice.
The ground state of the hamiltonian H0 + H1 will depend on
the ratio g = Q/J . For g = 0, we have the Néel ordered state of
Section IIA. Recent numerical studies19–21 have shown convincingly
that VBS order is present for large g (VBS order had also been found
earlier in a related ‘easy-plane’ model22 ). We characterize the VBS
state by introducing a complex field, , whose expectation value
measures the breaking of space-group symmetry:
QSL may have:
Broken ‘topological’
= ( 1) symmetry
S · S + i( 1) S · S .
(10)
This definition
satisfies the requirements of spin-rotation
Reduced gauge
invariance,
invariance, and measures the lattice symmetry broken by the
columnar VBS state because under a rotation about an even
U(1) to Z2 sublattice
or so
site by an angle = 0,⇡/2,⇡, 3⇡/2, we have
!e
.
(11)
Fractionalized excitations
Thus,
is a convenient measure of the Z rotation symmetry
of the square lattice, and h i = 0 in any state (such as the
Gapless excitations
Néel state) in which this Z symmetry is preserved. From these
jx
j
jy
j+x̂
j
j+ŷ
i
Sachdev, Nature Phys. 4, 173 (2008).
Spinon
Figure 4 Caricature of a spin-liquid state. The valence bonds are entangled
between different pairings of the spins, only one of which is shown. Also shown
are two unpaired S = 1/2 spinons, which can move independently in the
spin-liquid background.
excitation
4
4
definitions, we see that h i =
6 0 in the VBS states; states with
arg(h i) = 0, ⇡/2, ⇡, 3⇡/2 correspond to the four degenerate
VBS states with ‘columnar’ dimers in Fig. 3a, whereas states with
arg(h i) = ⇡/4, 3⇡/4, 5⇡/4, 7⇡/4 correspond to the plaquette
just the A µ photon that has only a single allowed polarization in
2 + 1 dimensions.) Nevertheless, these observations seem purely
formal because they only involve a restatement of the obvious
divergencelessness of the flux in equation (12), and the shift
symmetry is unobservable.
The key point made in ref. 23 was that the above shift symmetry
現代実験物理学I
Quantum Spin Liquids by Experiments
QSL must satisfy:
Unbroken symmetry
Not frozen spins
(insulator)
NMR spectrum,
Neutron Scattering, μSR
QSL may have:
Broken ‘topological’ symmetry
Reduced gauge invariance,
U(1) to Z2 or so
Fractionalized excitations
Gapless excitations
NMR T1, thermal conductivity
rihito Nakata1 , Yuichi Kasahara1,2 , Takahiko Sasaki2 , Naoki Yoneyama2 ,
hi Fujimoto1 , Takasada Shibauchi1 and Yuji Matsuda1
•
2.5
Present quantum spin liquid
candidates
A
2.0
LETTERS
1.5
κ (W K –1 m –1)
uids (QSLs), antiferromagnets
n disordered ground states, is
dimensional (1D) spin systems
ousins. The spin-1/2 organic
)-tetrathiafulvalene)2 Cu2 (CN)3
1) with a 2D triangular lattice
e first experimental realization
crucial importance is to unveil
lementary spin excitations2,3 ,
ence of a ‘spin gap’, which
on the universality class of
port on our thermal-transport
wn to 80 mK. We find, rather
vidence for the absence of a
y contradicts recent reports of
The low-energy physics of this
erpreted in light of the present
QSL phase.
pled spin systems, geometrical
fluctuations. Largely triggered
ating-valence-bond theory for
residing on a frustrated twottice5–7 and its possible applia doped 2D square lattice8,9 ,
D systems has been a longes of QSL states on S = 1/2
ported in organic compounds,
1, inset)1,10,11 , C2 H5 (CH3 )3 Sb
hiolate)2 ]2 (ref. 12) and 3 He thin
ular, the NMR spectrum of
ts no signs of magnetic ordering
four orders of magnitude below
refs 1,11). These findings aroused
lly believed that whereas a QSL
frustrated S = 1/2 2D kagome
as corner-sharing triangles, the
te is stable in the less frustrated
attice15,16 . Several ideas, such as
erate onsite repulsion17 , a ring
ionalization by• a slight distortion
attice19,20 , have been put forth
ng-range magnetic ordering in
rtheless, the origin for the QSL
現代実験物理学I
NATURE PHYSICS DOI: 10.1038/NPHYS1715
Perhaps holy grails
B
a
1.0
S
S
Organic
Triangular
t
Non¬magnetic
EtMe3Sb layer
0.5
0
0
C
S
t
C
C
S
S
Pd
S
S
C
S
C
S
C
S
Pd(dmit)2 molecule
κ-(BEDT-TTF)2Cu2(CN)3
t’
Pd(dmit)2 layer
R ES E A RC H | R E PO R TS
2
4
6
8
Gapless(C)1, Gapped(κ)2
No LRO down to J/
10000(NMR3/μSR)
10
frequency nQ(c). In addition, the central peak
using 17O NMR measurements of an isotopeenriched single crystal of ZnCu3(OH)6Cl2. We
frequency f for the Iz = –1/2 to +1/2 transition is
demonstrated that the spin excitation spectrum
shifted from the bare resonance frequency fo =
exhibits a finite gap D = 0.03 J to 0.07 J be(gn/2p)Bext by the effects of the hyperfine magFigure 1 | Temperature dependence of the in-plane thermal conductivity
netic fields from nearby Cu2+ sites, and the shift
tween a S = 0 spin-liquid ground state and the
excited
states, where J ~ 200 K represents the
of the peak (marked “main” in Fig. 2A) is probelow 10 K. (T) in zero fieldts for two
different single crystals of Jdeuterated
tr
Jr
s
portional to ckagome. Such an NMR frequency
3 He superexchange interaction (5, 21).
-(BEDT-TTF)2 Cu2 (CN)3 (sample A and sample B) measured in a Cu-Cu
shift may be expressed in terms of the Knight
A major advantage of using a single crystal for
tB
J
cryostat (black for sample A and green for sample B) and dilution NMR
B
shift, 17K (c) = f/fo – 1 = Ahf ckagome, where Ahf is
is that we can achieve high resolution by
between the 17O
refrigerator (blue for sample A and light green for sample B). As theapplying an external magnetic field Bext along the hyperfine coupling constant
Figure 1 | Crystal structure of EtMe3 Sb[Pd(dmit)2 ]2 . a, Side view of the crystal structure of EtMe
Pd(dmit)In
are
3 Sb[Pd(dmit)
2 ]2 . 2D magnetic
2 layers
specific
crystallographic
directions.
Fig. 2A,
nuclear spin and the Cu2+ electron spins. We can
temperature
lowered,layers
(T)of decreases
exhibits
broad
hump we
starting
17
separated byis
non-magnetic
the closed-shelland
monovalent
cationaEtMe
ofpresent
the crystalthe
structure
of the Pd(dmit)
The
3 Sb. b, Top view
O (nuclear
spin I2 layer.
= 5/2;
gyrofit the line shape in Fig. 2A with three sets of five
⇤ ' 6dimerized
Pd(dmit)2at
molecules
are T
strongly
(theThe
pairs crystal
are denoted
by dashed ovals).
with aratio
1/2-spin
is localized
each [Pd(dmit)
]2 dimer. peaks with three distinct values of 17K (c) and n (c).
to increase
around
K. Inset:
structure
of aOne electron
= 5.772onMHz/T)
NMR 2line
magnetic
gn/2p
Q
The arrows (tB , ts and tr ) indicate the transfer-integral network between the molecular orbitals of the [Pd(dmit)2 ]2 dimers. The three transfer integrals are
shape measured at 295 K in Bext = 9 T, applied
That is, the presence of the Cu2+ defects at the
2 . c, of
two-dimensional
layer
-(BEDT-TTF)
(CN)
viewed
2 Cu2to
non-equivalent butBEDT-TTF
close to each other
Localized
spin model applicable
EtMe33Sb[Pd(dmit)
2 ]2 . Three exchange interactions (JB , Js and Jr ) are close to
2+
17
along the c axis; the temperature dependence of
Zn sites results in three distinct O sites in
reflecting
of tB , ts and
tr . The exchange
interactions
are estimated
to be 220–250 K (ref. 2).
alongeach
theother,
long
axesthe
ofvalues
BEDT-TTF
molecules.
Pairs
of BEDT-TTF
molecules
the line shape is presented in Fig. 2C. Unlike preZnCu3(OH)6Cl2. Taking into account the differ35
17
formisdimers
arranged
in a triangular
lattice
in terms
t measured
powder-averaged Cl and O
ence in the transverse relaxation that affects the
100 viously
the stretching
exponent.
The decrease
in
means
thatofthetransfer integrals
S C S
S C S
13
0 between
NMR
(22,13C23),S we can resolve the five
apparent signal intensities (fig. S1), we estimated
function
becomes non-single-exponential,
indicating
the is nearly unity
S C 11line shapes
Pd
and trelaxation
the dimers.
The ratio of transfer
integrals
C
C
S
distribution of the relaxation. This distribution increases from
Iz = mS to mS + 1 Stransitions (Iz, z component of the
the population of the three sites as 13 ± 4%, 28 ±
1.
and the
spin-1/2
is realized
about
20 K andnearly
reaches isotropic
a maximumtriangular
around 1 Klattice
on cooling.
In
nuclear spin angular momentum; magnetic quan5%, and 59 ± 8%, in agreement with earlier 2D NMR
10¬1
Fig. 2, we also show the relaxation rate determined from the initial
tum number m = –5/2, –3/2, –1/2, +1/2, +3/2),
observations of three corresponding sites in a
decay slope of the relaxation curve. The difference between the two
which are separated by a nuclear quadrupole
deuterated single crystal of ZnCu3(OD)6Cl2 (24).
relaxation
rates shows the
degree
of the distribution.
To
understand
the
nature
of novel QSL states, knowledge
0
In spite of the distribution, it is clear from Fig. 2 that there is
10¬2
10
on the
structure
the low-lying
an obvious
kink in the of
temperature
dependence ofexcitation
T1 1 at around spectrum in the
295K (9T||c)
Main
1.0 K. This stronglylimit,
suggestsparticularly
that a phase transition
occurs at
zero-temperature
the absence/presence
of a spin
NN
Main
10¬1
this temperature. As no discontinuous jump is observed in the
170K
¬3
f
10 the spin
gap,temperature
is indispensable,
bearing
implications
on
o
dependence of
T1 1 , thisimmediate
is not a first-order
but a
NNN
continuous transition.
importantstate,
point toas
notewell
here isas
thaton the quantum
correlations
of the The
ground
10¬2
continuous phase transitions always involve essential changes of
[1] S. Yamashita et al., Nat. Phys. 4, 459 (2008).
numbers
carried
by each
elementary
excitation.
For instance
in 1D,
states, that
is, symmetry
breaking
and/or topological
ordering.
10¬4
NN
(T1T)¬1 T
Therefore, our
resultHeisenberg
indicates that thechains
gapless spin
liquid changes
half-integer
spin
feature
a massless spectrum,
[2] M. Yamashita et al., Nat. Phys. 5, 44 (2009).
10¬3
to an essentially different spin state with symmetry breaking and/or
1
10 100300
0.01 0.10
2
fo
T1¬1 T
120K
which
enables
proliferation of low-energy spinon excitations,
topological
ordering.
Temperature (K)
[3] Y. Shimizu et al., PRL 91, 107001 (2003).
¬5
Thissuch
instability
is not considered
to be classicalin
magnetic
0.01 case,
0.1
1
10
100 300
whereas
excitations
are 1suppressed
the integer 10spin
ordering, because the anomaly in T1 21is not a critical divergence.
Temperature (K)
[4] M. Yamashita et al., Science 328, 1246 (2010).
51.5
52.0
52.5
53.0
53.5
which
hasanalysis
a massive
Spectral
directly spectrum
proves this point.. Figure 4 shows the NMR
Frequency
(MHz)
13 C nuclear
spectra down to 19.4 mK. The transition around 1.0 K does not Figure 2 | Temperature dependence of the
spin-lattice
[5] T. Itoh et al., Nat. Phys. 6, 673 (2010).
affect the spectral shape; all of the spectra are largely the same relaxation rate of EtMe3 Sb[Pd(dmit)2 ]2 . The main graph shows the
50K
over the whole temperature region and the spectral tails are at 13 C nuclear spin-lattice
2.0 relaxation rate T1 1 of EtMe3 Sb[Pd(dmit)2 ]2 , and1.0
[6] M.P.Shores et al., J. Am. Chem. Soc. 127, 13462 (2005).
Bext
= 9T
most within ±50 kHz. This width is much smaller than the scale the inset graph shows (T1 T) 1 , where T is temperature. The
circles
indicate
of the hyperfine coupling constant of the 13 C sites, which is about the values determined from the stretched-exponential analysis (see text),
[7] J.S. Helton et al., PRL 98, 107204 (2007).
1.5 the values determined from the initial decay slopes
ity, Kyoto 606-8502, Japan, 2 Institute for Materials
Research,
Tohoku University, Sendai 980-8577,and
Japan.
9⇥102 kHz/µ
the squares denote
B (ref. 2), where µB is the Bohr magneton. Therefore,
[8]
M. Fu et al., Science 350, 655 (2015).
classical
spin
ordering
and
freezing
are
clearly
absent
down
to
of
the
relaxation
curves.
The
dark
blue
circles
and
dark
red
squares
are
oto-u.ac.jp.
30K
19.4 mK, which proves that the present instability differs from obtained from the1.0
present measurements below 1.75 K in a dilution
0.5
2
classical ordering/freezing.
refrigerator. For clarity, we did the same analysis for previously reported
T1¬1 (s¬1)
Herbertsmithite
Kagome
EtMe3Sb[Pd(dmit)2]2
Gapless(κ)4, Nodal(T1)5
No LRO down to J/10000(NMR)5
ZnxCu4-x(OH)6Cl2
Gapless(INS)7, Gapped(NMR)8
No LRO down to J/6000(NS7)
Close to QSL?
Triangular: Cs2CuCl4, NiGa2S4, NaxCrO2, …
Hyperkagome: Na4Ir3O8
cho intensity (arb. units)
c
(T1T )¬1 (s¬1 K¬1)
T (K)
Spin echo intensity (arb. units)
Organic
Triangular
b
現代実験物理学I
Perfect Kagome : Herbertsmithite
Kagomé
Published on Web 09/09/2005
A Structurally Perfect S ) 1/2 Kagomé Antiferromagnet
Published on Web 09/09/2005
Matthew P. Shores, Emily A. Nytko, Bart M. Bartlett, and Daniel G. Nocera*
1
A Structurally Perfect S ) /2 Kagomé Antiferromagnet
Department of Chemistry, 6-335, Massachusetts Institute of
Technology,
Massachusetts
AVenue,
Matthew
P. Shores,77
Emily
A. Nytko, Bart M.
Bartlett, and Daniel G. Nocera*
Cambridge, Massachusetts
02139-4307
Department
of Chemistry, 6-335, Massachusetts Institute of Technology, 77 Massachusetts AVenue,
Cambridge, Massachusetts 02139-4307
Received June 13, 2005; E-mail: [email protected] June 13, 2005;
E-mail: [email protected]
Nearly two decades ago, Anderson proposed that the resonating
rly two decades ago, Anderson proposed that the resonating
valence bond (RVB) state may explain the scatterless hole transport
encountered in doped rare-earth cuprates.1 The quantum spin liquid
e bond (RVB) state may explain the scatterless hole transport
phase responsible for RVB is most likely to be found in lowntered in doped rare-earth cuprates.1 The quantum spin liquid
dimensional, low-spin, and geometrically frustrated systems.2
responsible for RVB is most likely to be found in Accordingly,
lowmost theoretical investigations of RVB have concen1/ antiferromagnets in kagomé (corner-sharing
trated
on
S
)
2
2
sional, low-spin, and geometrically frustrated systems.
triangular)
lattices
due to the higher degree of geometric frustration.3
Herbertsmithite
dingly, most theoretical investigations of RVB have concenMaterials featuring such lattices are predicted to display no long1
range magnetic order due to competing antiferromagnetic interacon S ) /2 antiferromagnets in kagomé (corner-sharing
tions between
nearest-neighbor spin centers. Though long sought,
3
lar) lattices due to the higher degree of geometric frustration.
“no perfect S ) 1/2 Kagomé antiferromagnet has been up to now
Figure 1. Crystal structure of Zn-paratacamite (1), Zn0.33Cu3.67(OH)6Cl2.
als featuring such lattices are predicted to display no synthesized”,
long- 4 and accordingly, most theoretical predictions of such
Left: local coordination environment of intralayer Cu3(OH)3 triangles and
a lattice remain untested. Herein, we report the synthesis and
interlayer Zn2+/Cu2+ ion; the projection is parallel to the crystallographic
magnetic order due to competing antiferromagnetic interacpreliminary magnetic properties of a rare, phase-pure, copper
c axis. Right: the {Cu3(OH)6} kagomé lattice, projected perpendicular to
hydroxide chloride mineral featuring structurally perfect S ) 1/2
etween nearest-neighbor spin centers. Though long sought,
the c axis. The pure Zn2+-substituted compound 2 is isostructural to 1.
Selected
interatomic distances (Å) and angles (deg) for 2: Zn-O, 2.101kagomé
layers
separated
by
diamagnetic
Zn(II)
cations.
rfect S ) 1/2 Kagomé antiferromagnet has been up to now
(5); Cu-O, 1.982(2); Cu-Cl, 2.7698(17); Zn‚‚‚Cu, 3.05967(16); O-ZnWe have employed a redox-based hydrothermal protocol to
O, 76.21(18),
180.00(19);
O-Cu-O,
81.7(3), 98.3(3), 180.0;
Figure 1.jarosite-based
Crystal structure
of Zn-paratacamite
(1), 103.79(18),
Zn0.33Cu
3.67(OH)
6Cl2. 180.0; Cu-O-Cu, 119.1materials (AM
such pure, single-crystal
sized”,4 and accordingly, most theoretical predictions ofprepare
3(OH)6O-Cu-Cl, 82.31(11),
97.68(11);
Cl-Cu-Cl,
(SO4)2, A ) alkaliLeft:
metal ion,
M coordination
) V, Cr, Fe).5 These
compounds of intralayer
local
environment
Cu397.04(15).
(OH)3 triangles and
(2); Cu-O-Zn,
ce remain untested. Herein, we report the synthesisfeature
andkagomé lattices
2+ ion;
composed
of /Cu
M3(OH)
when M
6 triangles;
interlayer
Zn2+
the projection
is parallel to the crystallographic
inary magnetic properties of a rare, phase-pure, copper
) Fe(III), spins are
antiferromagnetically
coupled
and frustrated.6
c axis.
Right: the {Cu
projected
perpendicular
to
35(OH)6} kagomé lattice,
and refinement
are provided
in the Supporting
Information. Two
of the magnetic ion of Fe(III) (S ) /22+
) by Cu(II) (S )
1/
xide chloride mineral featuring structurally perfect S Substitution
)
geometrically
distinct
metal
sites
are
found.
On
the
first site, a Cuthe
c
axis.
The
pure
Zn
-substituted
compound
2
is
isostructural
to
1.
2
1
/2) was attempted, but charge imbalance on the kagomé layers
from the randomly oriented portion of the powder (i.e.,
80% of the sample) [10].現代実験物理学I
Notice that the whole 35 Cl NMR
line shape begins to tail-off toward lower frequencies
below #50 K. The resonance frequency of the sharp
c-axis central peak and its distribution depends on the
NMR Knight shift, 35 K, induced by !loc . Hence the observed line broadening implies that !loc varies depending
on the location within the sample below #50 K.
In Fig. 3, we summarize the 35 Cl NMR Knight shifts 35 K
35
K1=2 deduced from the line shapes, together with
EtMe 3Sb[Pd(dmit) 2] 2 and
35 K 6
ZnCu
Cl 2
3 (OH)
by SQUID.
corresponds
to the cen!bulk as observed
tral peak above #45 K as determined by FFT techniques.
13C-NMR
RAPID
COMMUNICATIONS
35
Below #45
K, where
the
central peak is smeared out by
Cl-NMR
T. Itoh et al., PRB 84, 094405 (2011).
line broadening,
we determined 35 K as the higher freT. Imai et al., PRL 100, 077203 (2008).
quencyBedge
the central !2006"
peak from point-by-point meaPHYSICAL REVIEW
73, of
140407!R"
NMR spectra for present QSL
candidates
κ-(BEDT-TTF) 2Cu 2(CN) 3
13C-NMR
Y. Shimizu et al., PRB 73, 140407(R) (2006).
∼0.1μB
T. ITOU et al.
PHYSICAL REVIEW B 84, 0
∼0.1μB
makes relaxation curves inhomogeneous. Therefor
likely that the power-law decrease is intrinsic and
state has a nodal gap structure in the magnetic
178 mK
We point out here that (T1 T )−1 is expected to foll
895 mK
temperature dependence in a typical nodally gappe
as a d-wave resonating-valence-bond state.22 Th
1080 mK
behavior (T1 T )−1 ∝ T is an open problem and m
1220 mK
factor in understanding the present spin liquid.
∼0.1μ
Our results
refute the presence of fully gaples
B
4.04 K
∼J/60
magnetic excitations. This seemingly contrasts wi
10.15 K
for the thermal quantities,16,17 which implies g
tations like metals. A possible explanation for t
19.9 K
contradiction may be the fact that NMR relaxation
29.0 K
only magnetic excitations, while thermal quan
both magnetic and nonmagnetic excitations. Our
42.8 K
do not provide any information on nonmagnetic
105.9 K
which do not contribute to the dynamic spin su
The thing that ∼J/40
can be definitely determined from
299.7 K
data is that the dynamic susceptibility disappears
-200 -150 -100 -50
0
50 100 150 200
If we accept the thermal-quantities result as true
Shift from TMS (kHz)
scenario for the present ground state is a spin
with (maybe
FIG. 3. NMR
spectra
of
the
inner
carbons
of
35 nodally) gapped magnetic excitations
FIG. 2 (color
online).
Cl NMR line shapes of the Iz $ % 12 to
nonmagnetic
which is
sometimes
EtMe3 Sb[Pd(dmit)2 ]2 at several temperatures.
in 8.4 excitations,
Tesla in a partially
(#20%)
uniax-re
& 1 central transition
Intensity (arb. units)
18.7 mK
∼J/60
2
6,7,9
NMR relaxation for present QSL
candidates
EtMe 3Sb[Pd(dmit) 2] 2
κ-(BEDT-TTF) 2Cu 2(CN) 3
13C-NMR
RAPID COMMUNICATIONS
PHYSICAL
B 73, 140407!R"
!2006"
Y. Shimizu et
al., PRB REVIEW
73, 140407(R)
(2006).
∼J/60
ZnCu (OH) Cl
13C-NMR
T. Itoh et al., PRB 84, 094405 (2011).
NUCLEAR MAGNETIC RESONANCE OF THE . . .
∼J/60
102
(a)
101
10
T1-1inner
T1-1outer
0
inhomogeneity of EtMe3 Sb[Pd(dmit)2 ]2 also grows on cooling
in the region from about 10 to 1 K, where the growth bears
a notable resemblance to that of κ-(BEDT-TTF)2 Cu2 (CN)3 . 10 5
However, in contrast8.3
to the
case of κ-(BEDT-TTF)2 Cu2 (CN)3 ,
T (34.6MHz)
βinner and βouter in 4.4T
EtMe(18.4MHz)
stop de3 Sb[Pd(dmit)2 ]2 suddenly
OH freezes?
creasing at 1 K and recover
rapidly
to
the
homogeneous
values
2.4T (10MHz)
on further cooling, as
shown
in Fig. 2(b).
1.5T
(6.3MHz)
The most important
of the present results is that the
1.0Tpoint
(4.2MHz)
temperature dependence of T1−1 inner shows an obvious kink at
around 110
K and then shows a steep decrease on cooling, which
is essentially the same behavior as for the outer carbons. In the
low-temperature limit, T1−1 inner is proportional to the square 10 4
of the temperature, as is also observed in T1−1 outer . If these
behaviors were caused by the cation molecular motion, the two
relaxations should show different temperature dependencies.
This is because the inner-carbon relaxation is hardly affected
by the molecular motion, while the outer-carbon relaxation
is more readily affected. Therefore, the accordance between
the two relaxations
is very strong evidence that the observed
1
behaviors of the relaxation rates are not affected by the molecular motion of the cation but reflect the electron spin dynamics. 1000
(8T,
93MHz)
Thus, the kinks at around 1 K and 63-Cu
the rapid
decreases
below
(0.9T,
38MHz)
this temperature in T1−1 inner and T1−11-H
are
reliable
indicators
outer
of the spin state. The kinks strongly suggest that the spin
1
1
0
100 which
state undergoes an abrupt change at this temperature,
involves a change of the excitation spectrum.
T [K ] The change of the
spin state is likely to be a continuous phase transition, because
T1−1 inner and T1−1 outer appear to show singular behaviors with 35
FIG.
online). at Temperature
dependence
of Cl NMR
respect4 to(color
the temperature
the kinks (that is,
discontinuities
35
−1various magnetic fields
"
at
spin-lattice
relaxation
of the first-order
derivativesrate
of T1−1!1=T
and
T
).
It
is
im1
1 outer
inner
portant to
note that continuous
phaserepresents
transitions always
(filled
symbols).
Solid line
a fit involve
to a power law,
35
symmetry
breaking
and/or
!1=T1 " $
T # with
# $topological
0:47 (8.3ordering.
T), 0.44Therefore,
(4.4 T), 0.2 (2.4 T
the ground state
of the present system is likely to have a1
rate in low field (0.9 T), !1=T1 ", and
and
1.0 T). 1 H relaxation
broken-symmetry and/or topological structure.
63
63 !1=T ", are also
Cu
relaxation
rate
in
high
T),carbons
Figure 3 shows the NMR spectrafield
of the(8
inner
at1
35
" measured
in comparable
field.
superposed
on !1=T
several temperatures.
The1features
of the spectra
are almost magnetic
the
101
10-5
0.01
(T1T )-1∝T
(1/T1) [sec-1]
(T1T)-1 (s-1 K-1)
100
10-1
10-2
0.1
0.1
1
10
Temperature (K)
1
10
Temperature (K)
100
35
10-3
0.01
100 300
1
Stretching Exponent β
(b)
0.8
0.6
β inner
0.4
β outer
0.2
0.01
0.1
1
10
Temperature (K)
1
-3 T1
-1∝ T 2
100 300
FIG. 2. (Color) Temperature dependencies of the 13 C nuclear spin-lattice relaxations of the inner and outer carbons of
EtMe3 Sb[Pd(dmit)2 ]2 . The red circles show data for the inner
12
35* (1/T1) [sec-1]
10-2
10-4
FIG. 2. !a" Temperature dependence of the NMR linewidth $1/2
and 1 / T2 measured at 8 T. !b" 1 / T1 for the inner !triangle", outer
!circle" and total 13C cites !diamond". The inset shows the exponent
(1/T1),
T1-1 (s-1)
63
10-1
10
T
T
y
n
e
g
m
K
perature dependence is different from the smaller !local as
represented by 35 K. !bulk simply represents a bulk average
現代実験物理学I
of !local .
In passing, we recall that earlier "SR Knight shift K"SR
measurements by Ofer et al. [10] showed identical behavior between K"SR and !bulk . They concluded that the upturn of !bulk below 50 K is not caused by impurity spins but
is a bulk phenomenon. Our new results in Fig. 3 do not
contradict these "SR data. K"SR was deduced by assuming
a Gaussian distribution of !local ; hence by default K"SR
represents the central value of the presumed Gaussian
distribution. That explains why K"SR shows behavior simi6
2
lar to !bulk and 35 K1=2 . 3
Next, we turn1our35
attention to the dynamics of lattice and
H, Cl,63Cu-NMR
spin degrees of freedom. Figure 4 shows the temperature
T. Imaiofetthe
al.,35 Cl
PRL
100, spin-lattice
077203 (2008).
dependence
nuclear
relaxation rate,
PHYSICAL REVIEW B 84, 094405 (2011)
35
!1=T1 ", measured at the central peak frequency in various
∼J/6
ca
ag
in
35
th
tio
w
bi
co
in
35
at
ha
tu
ne
tio
pr
N
(1
th
35
th
no
an
th
ou
sa
ab
th
m
su
la
st
R
m
[1
en
!
re
C
35
tio
th
takes one of three orientations, labeled using the variable
4. Representing
spins
by Majorana
operators
state
to other
flux sectors
are OðJÞ
現代実験物理学I
!jk ¼ x, y, or z as shown in Fig. 1(a). A spin one-half
Jx=1,
!.
one can project onto aJ =given
flux
variable at site j is represented by the Pauli matrices
"
4.1. Aj general spin-fermion transformation
y Jz=0
batively in h. For the undiluted la
Exchange of strength J!jk acts between spin components
Fig. 5. Phase diagram of the model. The
the the
Zeeman
energy
between
sta
Let
us
remind
reader
formalism
J + J +some
J = 1. general
The diagrams
for the ot
!jk and the Hamiltonian is
sector
areisall
zerodescribed
and thebylead
with n fermionic
modes
usually
the
X
y
!jk !jk
is can
second
in hc
ak (k(1)
= 1, . . .,perturbation
n). Instead,
one
use their
linear
H ¼ " J!jk "j "k :
is impossible,
even
iforder
we introduc
Majorana operators:
a vacancy,
thephase
individua
eighthowever,
copies of each
(corres
Another route to QSL: Kitaev model
x
hjki
Annals of Physics 321 (2006) 2–111
c2k%1
www.elsevier.com/locate/aop
y
z
y
a
%
a
k
k
thethree
same
; properties
¼ rounding
ak þ
c2k ¼translational
plaquettes
lose p
i
phase are algebraically
different.
ayk ;
It is exactly soluble, for example, with a local transformaonly theirWe
Z2now
sum
enters
the defi
consider
the
zeros
of t
!
!
which
are called
Majorana
operators.
The
operato
tion "Anyons
¼ ibin canwhich
represents
each
spin
using
the
four
Because
of
themodulo
Zeeman
e
exactly
solved
model and beyond
tum
q isthis,
defined
the recip
the lattice contains
one
are divided
into
three
types,
depending
20
A.
Kitaev
/
Annals
of
Physics
321
(2006)
2–111
x vertex
y
z of each kind. Links
obey
the
following
relations:
QSL phases
A,B: elements
Majorana fermions b , b , b , and c [10]. Then
momentum space
by a
thesector,
parallel
matrix
within
on their direction (see
Alexei Fig.
Kitaev3B); we call them ‘‘x-links,’’ ‘‘y-links,’’ and ‘‘z-links.’’ The
symmetric
Jy = Jzl)
x = now
J¼
Jlxc=jJy=0if jcase
c2j ¼ 1; projected
cj cl the
%c
6¼ l.(Jare
Hamiltonian
z=1,
Hamiltoniani X
is as follows:
!jk !jk
J!jk u^ jk cj ck ; X u^ jk ¼ ibX
(2)gapless arise from the specific field compo
H¼
j bk :
X
2
on equal
operators cj can be
H ¼ "Jj;k
rxj rxk " J y
ryj ryk " J z
rzj rzk ; Note that allin
ð4Þtreatedthe
x
Fig.
1(b):
employing
site
la
A zrepresentation of
We
now
describe
a
a
spin
b
gapped
x-links
y -links
z-links
Abstract
A. Kitaev / Annals of Physics 321 (2006) 2–111
15
projected
Zeeman
energy
is
H
with each other and with
H. operators. Let us denote these operato
The operators
u^ jk commute
Majorana
A spin-1/2 system on a honeycomb
lattice is studied. The interactions between nearest neighbors
z
are of XX, YY or ZZ type, depending on the direction of the link; different types of interactions may
"
Written operators
using Majorana
fer
h
c
,
c
,
and
c
).
The
act on the
z
2
3
4
,
J
,
and
J
are
model
parameters.
J
3 Þ.Majorana
^
i
¼
u
¼
#1,
Onewhere
can
therefore
fix
the
values
of
h
u
Majorana
x The ymodel is solved zexactly by a reduction
differ in strength.
to free operators
fermions in a staticjk
Z gauge
jk
B|Jx| and |Jy| decrease! while |Jz|
If
field. A phase diagram in the parameter space is obtained. One of the phases has an energy gap
the Hilbert
space
a spinthe
is identified
a twoform
h!with
"i ¼
ihun
to Hof
Z have
!
and carries
excitations
that are Abelianof
anyons.
The other
phaseHamiltonian,
is gapless, but acquires a gap in
move
to
a
subspace
the
full
and
obtain
a
(within
the
parallelogram)
y
y
x
x
A
the presence of magnetic field. In the latter case excitations are non-Abelian anyons whose braiding
A
y
by thisJ condition:
x
sented
in
model,
|Jx|the
+ |Jtight
|. The
q* and
rules coincide with those of conformal blocks for the Ising model. We also consider a general theory
Jy=1,points
y| = |Jzbinding
x=1,
’s.
Numbering
the
sites
around
a
bilinear
form
in
the
c
u
z fermions with a gapped
z spectrum, which
j zis characterized by a spectral Chern number m. The
of free
Abelian and non-Abelian phases of the original model correspondspins
to m = 0 and m = ±1, respectively.
Jy= Jz=0
of new allelogram.
sites, with(Note
theJx=that
coupling
sh
Jz=0 the equati
y of 1excitation
y on m mod
flux
through
a
plaquette
from
to 6depend
[see
Fig.
1(a)],
theZ2edge
Zflux
The anyonic properties
16, whereas
m itself governs
thermal
2
At the points ±q* the spectrum
transport. The paper also provides mathematical background on anyons as well as an elementary
The
task
now
ispositive
to calculate
Fig.
5.
Phase
diagram
of
the
model.
The
triangle
is
the
section
of the
octant (J , J , Jthe
P
x
x
theory
of
Chern
number
for
quasidiagonal
matrices.
plaquette
is
defined
to
be
w
¼
u
u
u
u
u
u
.
p
21
23
43
45
65
61
! 2005 Elsevier Inc. All rights reserved.
J + J + J = 1. The diagrams for the other octants are similar.
ground-state energy of the Kitaev
Physical properties
of
the
system
depend
only
on
these
Fig. 4. Graphic representation of Hamiltonian (13).
Gapless
using
the tight binding model of Fi
is
impossible,
even
if
we
introduce
new terms in the Hamiltonian. On the oth
g
give
fluxes
[10],
but
note
that
many
choices
of
the
set
fu
1.
Comments
to
the
contents:
what
is
this
paper
about?
Note that each pair of connected sites is counted twice, and û = !û . The structure of jk
Majoranato
behavior
indifferent
a gapped
phase. A of
con
eight copies of each phase (corresponding
sign combinations
Jx
his Hamiltonian
is shown
Fig.
4. ofthe
Certainly,
the
main
result
thesector.
paper is anisexact
solution
ofIta is
particular
two-dimen(3) considered
inin
[24],
but
solution
less
trivial.
not clear
how
to realizeis
this
model
rise
to
the
same
flux
The
ground-state
sector
fluxRemarkably,
thequantum
operators
û However,
commute
Hamiltonian
with to
each
other. translational properties.
the
same
It is unknown
whether
the
eight copies
of
sional
model.
I waswith
sittingthe
on that
result for too and
long, trying
perfect
structure
of
this
tight
binding
mod
Fermion
in solid
state,
but
an
optical
lattice
implementation
has
been
proposed
[42].
e
derive
some
properties
ofall
the
model,
and¼
put them
more
framework.
Thus
herefore
the it,Hilbert
space
L splits
into
eigenspaces
ofgeneral
û , which
indexed
phase
algebraically different.
þ1into afor
sites
j are
on
aareparticular
free,
e.g.,
with
ucommon
many ramifications have come along.jk
Some of them stem from the desireeto avoid e
the use
*
California Institute of Technology, Pasadena, CA 91125, USA
Received 21 October 2005; accepted 25 October 2005
2
cj
z
bj
jk
z
bk
ck
x
x
kj
jk
jk
jk
y
z
y
z
現代実験物理学I
Toward Kitaev QSL: 2D honeycomb
iridates
α-A2IrO3
5d5 in cubic symmetry
Edge-shared IrO6
IrO6
eg
Δ
A
z
x
λSO
Jeff
∼3 eV ∼0.4 eV
1/2
t2g
3/2
y
Spin
Isospin
Jeff=1/2 Kramers doublet:
the lattice contains one vertex of each kind. Links are divided into three types, depending
1
on their direction (see Fig. 3B); we call them ‘‘x-links,’’
The
J eff 1/ 2‘‘y-links,’’
=
xyand
,± 1‘‘z-links.’’
2 ± yz ,∓
1 2 + i zx,∓ 1 2
Hamiltonian is as follows:
3
PHYSICAL REVIEW
PRL 102, 017205 (2009)
X
X y y
X
x x
z z
H ¼ "J x
rj rk " J y
rj rk " J z
rj rk ;
ð4Þ
Heis
x-links
y -links
z-links
(
)
+
=
where Jx, Jy, and Jz are model parameters.
isospin up
spin up, lz=0
spin down, lz=1
G. Jackeli and G. Khaliullin, PRL 102, 017205 (2009)
FIG. 1 (color online).
Density profile of a hole in the isospin
wher
simp
ij bo
cubic symmetry (! ¼ 0, This peculiarity
on a given to
bond,
ofentirely
a 90due tobond
leads
A 180" bond:
For this geometry,
thethe
nearestenough, its (A)
anisotropy
is
Hund’s
couthe xy-plane. Ho
現代実験物理学I
he
effects
of
a
tetragonal
neighbor
t
hopping
matrix
is
diagonal
in
the
orbital
space
2g
such a system
nondiagonal
elem
pling. This
is
opposite
to atwoconventional
situation:
typiHamiltonian
drastically
different
from
that
o
and,
on
a
given
bond,
only
orbitals
are
active,
e.g.,
jxyi
a
charge
transfer
Toward
Kitaev
QSL:
2D
h
oneycomb
mon cases of TM-O-TM
cally, the anisotropy
obtained
in powers
of
and jxzi orbitalscorrections
along a bond in are
x-direction
[Fig. 2(a)].
This peculiarity
Ref.
[12].
After
Two
transfer
amplitudes
via upper
an
ond formed by corner- ometry.
The spin-orbital
exchange Hamiltonian
for such a system
Hamiltonian dras
!
while
the
Hund’s
coupling
is
not
essential.
has
already been reported: see Eq. (3.11) in Ref. [12]. After
" -bond
iridates
ometry. Two tran
et,
we(B)find
), and
a 90an
"
gen(B)interfere
in
a
destructive
manner
and
the
the are
ground
state doublet,
we find
an
A 90projecting
bond:it onto
There
again
only two
orbitals
active
gen
interfere
in a
exchange Hamiltonian for isospins in a form of
2(b).
of the Hamiltonia
on
a
given
bond,
e.g.,
jxzi
and
jyzi
orbitals
along
a
bond
in
a
form
of
Edge-shared
IrO
6
α-A
2IrOof
3
the
Hamiltonian
exactly
vanishes.
The
finit
interaction
appea
geometry, the nearestthe xy-plane.(a)However, the hopping
matrix has now
tureonly
of the excite
(b)
gonal in the orbital
space interaction
Edge-shared
IrO6
180
of theJ
exchange
i
appears,
to paths
the
-m
H
nondiagonal elements,
and however,
there are two90due
possible
for
tion of a given bo
bitals are IrO
active, e.g., jxyi ture
perpendicut
a charge
transfer
[via upper
or lowerMost
oxygen,importantly,
see Fig.plane
2(b)].
of
the
excited
levels.
6
A
With this in mind
n x-direction [Fig. 2(a)].
This peculiarity
of a 90" bond leadspz to an exchange
ofxHamiltonian
the exchange
interaction
depends
on" gethe sp
z
onian
for such a system
drastically
different
from
that
of
a
180
o
yz
xz
90
(3.11) in Ref. [12]. After
4
ometry.
transfer
amplitudes
upper a
andbond
lower
of Two
a given
bond.
Wevialabel
ij
y tion
withoxyJ ¼ layi
py
xy
xy
3 #2 . Re
quantum
analog o
ate doublet, we find an
gen
interfere
in
a
destructive
manner
and
the
isotropic
part
perpendicular
to
the
'ð¼
x;
y;
zÞ
axis
b
introduced
origin
ospins in a form of plane
of the Hamiltonian exactly vanishes.xz The finite,
anisotropic
Jahn-Teller syste
yz
studies
as a prot
With
this
in
mind,
the
Hamiltonian
can
be
w
xz
xz
pinto
interaction
appears,
however,
due
to
the
J
-multiplet
strucz
the
lattice
contains
one
vertex
of
each
kind.
Links
are
divided
three
types,
depending
H
pz
degeneracy of to
pThe
z
on
their
direction
(see
Fig.
3B);
we
call
them
‘‘x-links,’’
‘‘y-links,’’
and
‘‘z-links.’’
ture of the excited levels. Most importantly, the very
form to our
)
However,
Hamiltonian is as follows:
the compass mod
FIG. 2 (color
online). Two possible
geometries
of
a TM-O-TM
ð'Þ
'
'
of
the
exchange
interaction
depends
on
the
spatial
orientaactive along
these
bonds. The S ;Implementing
90o X
H
¼
'JS
X y y
Xbond with corresponding orbitals
ij
i in the
jKitaev%&
x x
z
z
model is e
of a given
Weforlabel
a bond
laying
large
dots stand
the transition
metalij
(oxygen)
ions.
H ¼ xz
"J x
rj rk " J y
rtion
rj rk(small)
; bond.
ð4Þ
j rk " J z
"
a honeycomb latt
formed by corner-shared octahedra, and (b) a
x-links
y -links plane perpendicular
z-links(a) A 180 -bond to
the
'ð¼
x;
y;
zÞ
axis
by
a
(')-bond.
⊥
plane
"
properties such
90 -bond formed by edge-shared octahedra.
4
With J
this¼in mind,
the
Hamiltonian canthis
be written
as
with
#
.
Remarkably,
Hamiltonian
017205-2
, zand Jz are model parameters.
where Jx, Jyp
And J > 0 (FM)
3 2
Kitaev FM QSL is expected
' '
quantum analogHofð'Þ
the
so-called
compass(3)mod
¼
'JS
G. Jackeli and
ij
i SG.
j ;Khaliullin, PRL 102, 017205 (2009)
o
yz
xz
o
現代実験物理学I
Real Kitaev QSL?: NMR spectra for
H3LiIr2O6
1 Li site only
@K~0.1%
Li-NMR 5T
Original
α-Li2IrO3
H3LiIr2O6
IrO6
110 K
2Li
Intensity (arb. units)
70 K
50 K
30 K
15 K
1
H-NMR 2T
6H
110 K
70 K
50 K
30 K
20 K
10 K
10 K
5K
5K
1.2 K
1.0 K
-0.2
0.0
Shift (%)
0.2
0.4
H3LiIr2O6
A
Intensity (arb. units)
7
1 H site only
@K~0.02%
-0.2
0.0
Shift (%)
0.2
現代実験物理学I
Real Kitaev QSL?: Knight shift as
Intrinsic susceptibility
Knight shifts
Bulk M/H
0.10
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H3LiIr2O6
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0.08
7
Li-NMR 5T, oriented
0.06
K (%)
7
Li-NMR 2T, oriented
0.04
�����
Raw data
������
After
�����
subtraction
������
�
0.02
1
H-NMR 1T, unoriented
0.00
1
2
3
4 5 6 78
10
T (K)
2
3
4 5 6 78
for impurities
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Supports gapless QSL
& first time detecting finite K(T=0) in QSLs
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