88
References (review article) Yi Zhou, K. Kanoda and Ng Tai Kai Spin liquid states arXiv. 1607.03228 Spin liquid in organic materilas K. Kanoda, UTokyo Boulder School, July 28,29, 2016
References
(review article)
Yi Zhou, K. Kanoda and Ng Tai Kai
Spin liquid states
arXiv. 1607.03228
Spin liquid in organic materilas
K. Kanoda, UTokyo
Boulder School, July 28,29, 2016
Tohoku U.
(～1981)
Kyoto U.
Ph.D
(～1987)
U. Tokyo (～present)
Ins. Mol. Sci.
(～1997)
Gakushuin U. (～1991)
Physics of condensed matter
Understanding low-energy state of nucleus and electron assembly
+e -e+
++
Hydrogen LithiumHelium
atom (binding)
solid
molecule （pairing） liquid
superfluid
plasma
More is differently different.
+2e+3e-e-e
-e
-e-e
（different in 3He and 4He)
nucleus electron
Low
ener
gy
atom (binding)
plasma
liquid
metallic solid
superconductivity
atom (binding)
plasmaInteraction
Quantum
rk ie
e
k
kF-kF
e-
e-
e-
e
F
m
pk
2
2
e
)( kp
Fermi gas
Without interaction,
electrons are free waves with Fermi
surface.
Low dimensionality
With interaction, Fermi surface …..unstable
Fermi surface
Mott insulators
superconductors
attractive
On-site repulsive
Tomonaga-Luttinger liquid
Spin DW
Charge DW
p-wave d-waveS-wave
Quantum spin systems
Electronic glass
Electronic crystal
Electronic liq. Cryst.
＋el-ph / ele-ele
interaction
Electron pairing
Inter-site repulsive
Density waves Wigner crystal
Singlet, triplet
Contents
2. Electron correlation in organic materials
all-in-one systems for Mott physics
(Optional )
4. Massless Dirac Fermions in orhanic materials
Dirac cone reshaping and ferromagnetism
1. Fundamentals of organic materials
complex in real space, but simple in k-space
3. Spin liquid in quasi-triangular lattice
controlled frustration, correlation, disorder, doping
1. Molecular materials and electronic structures
Keywords;
a variety of lattice structures
concept of molecular orbital
simple band structure
highly compressible system
Inorganic solid
Simple metals,
oxides,….p electron systems
Organic (molecular) solid
Se Se
Se Se
Me
Me Me
Me
S S
S S
S
S S
S
Se Se
Se Se
S
S S
S
S S
S S
O
O O
O
Se
Se
Me
Me
S
S S
S
S S
S S S
S
TMTSF
BEDT-TTF (ET)
BEDO-TTF (BO)
DMET
BEDT-TSF (BETS)
MDT-TTF
giving (super)conductorsOrganic molecules
Molecular arrangement degrees of freedom
b type q type
e-
e-
e-
e-
e-
e-
Molecular arrangement degrees of freedom
superconductivity Electronic crystals
e-- e-
e-- e-
e- e- e-
Organic conductors
complex in real space, but simple band structure
atom molecule solid
eHOMO
LUMO
HOMO band
tight-binding
Molecular orbital Band formation
Electronic structure
Molecular orbital is a minimum entity for electrons
No need to look into atomic orbitals in a low energy scale
1) The simplest non-degenarate case; hydrogen molecule
bbaa cc R
e
r
e
r
e
mH
ba 0
2
0
2
0
22
2
4442 pepepe---
)()1(2
11 ba
S
)()1(2
12 ba
S -
-
dS ba *
eS Haa
S
HH abaa
11e
S
HH abaa
-
-
12e
dHH aaaa *
dHH abab *
It’s because of the hierarchy; atomic orbital molecular orbital electronic band
Molecular orbital Linear combination of atomic orbitals
bonding orbital
antibonding orbitalOverlapping integral
Transfer integral
Molecular material; structure is complicated in real space,
but electronic structure is surprisingly simple in k-space
Key concept
2) The degenarate case: carbon atom
),( lmnllmn YR
2pz
2p-
2p+
2s),(00 Y
),(10 Y
),(11 Y
),(11 -Y
For n=2, four orbitals (2s, 2p×3) are
degenarate.
C
spx×2
py, pz
sp結合性軌道
sp反結合性軌道
)(2
1xx pssp
x s, px, py, pzpy, pz
配位子場効果（原子内混成）
H-C C-H
Acetylene
Atomic p-orbital molecular orbital
First, consider symmetry of coordination and reconstruct orbitals (intra-atomic hybridization)
Next, reconstruct orbitals between neighbors (inter-atomic hybridization like hydrogen molecule)
Finally, construct the overall molecular orbitals
i) Uniaxial 2-way coordination; sp hybridization
bonding orbital
antibonding orbital
intra-atomic
hybridization
(Chemical bond)
inter-atomic
hybridization
C
spx×2
py, pz
)(2
1xx pssp
xs, px, py, pz
py, pz
H-C C-H
Acetylene
Atomic p-orbital molecular orbital
Imagine alien atoms are approaching carbon intra-atomic hybridization
When the alien atoms get close to carbon inter-atomic hybridization like hydrogen molecule)
Finally, construct the overall molecular orbitals
i) Uniaxial 2-way coordination; sp hybridization
sp bonding orbital
sp antibonding orbital
intra-atomic
hybridization
(Chemical bond)
inter-atomic
hybridization
C
sp3×4
sp3結合性軌道
sp3反結合性軌道
s, px, py, pz
)(2
13
zyx pppssp
)(2
1zyx ppps --
)(2
1zyx ppps --
)(2
1zyx ppps --
メタン、 ダイヤモンドmethane, diamond
配位子場効果（原子内混成）
化学結合（原子間混成）
Tetrahedral 4-way coordination; sp3 hybridization
bonding orbital
antibonding orbital
intra-atomic inter-atomic hybridization
Chemical bond
sp3×4sp3 bonding orbital
sp3 antibonding orbitals, px, py, pz
intra-atomic inter-atomic
hybridization hybridization
Tetrahedral 4-way coordination
molecular orbital; the case of CH4 (sp3)
Molecular orbital
S S
S S
S
S S
S
BEDT-TTF (ET)
NN
DCNQI
Highest Occupied Molecular Orbital (HOMO)
Lowest Unoccupied Molecular Orbital (LUMO)
By Imamura
and Tanimura
e
Molecular orbital in molecular conductors
Y.-N. Xu et al., Phys. Rev. B 52, 12946
First-principles calculations
Band-structure calculations I; p electronic system
T. Mori et al., HOMO level
LUMO level
拡張Huckel法 ＋ 強束縛近似
unregistered
Antibonding
band
bonding
band
effectively 1/2-filled band
DOS
EF
DOS
EF
フェルミ面
HOMO＋tight-binding approx.
well described by tight-binding model of MO
k-(ET)2X
Notion of molecular orbital
S S
S S
S
S S
S
BEDT-TTF (ET)
HOMO
e
eHOMO
Cu
d orbital
Seemingly complicated structure in real space
but
Simple electronic structure in k space
( MO is a minimum electronic entity)
In many cases,
no orbital degeneracy
negligible spin-orbit interaction
Model systems to look into correlation effect
in simple electronic systems
Highly compressible
Molecular conductors
Contents
2. Electron correlation in organic materials
all-in-one systems for Mott physics
(Optional )
4. Massless Dirac Fermions in orhanic materials
Dirac cone reshaping and ferromagnetism
1. Fundamentals of organic materials
compex in real space, but simple in k-space
3. Spin liquid in quasi-triangular lattice
controlled frustration, correlation, disorder, doping
Correlation-induced insulating phases everywhere in organics
Quasi 1D 1/4-filled
(TMTSF)2X
Quasi 2D ¼-filled
a-(ET)2X
Quasi 2D 1/2-filled
k-(ET)2X
SDW/SC
Nesting
Mott/SC
On-site repulsion
CO/SCInter-site repulsion
Quasi 2D ¼-filled b-(meso, DMeET)2PF6
ka
kb
CO/DE
Inter-site repulsion
Pressure (kbar)
0 10 20
Tem
per
atu
re (
K)
10
100
1
Charge
order
Massless
Dirac
electrons
Sociology of electronic system
3. Mott transition
N. Mott (1949)
molecule
e-
e-
e-
e- e-
e- e-
e-
e-
Wbandwidth
Competition between kinetic energy and Coulomb
W ＞ U W ＜ U
U
量子力学的運動エネルギー
Coulomb
energy
Coulomb interaction
Wave-like
Kinetic
energy
metal insulator
particle-like
Mott transition
Mott transition
U >> W U << W
Mott insulator Metal
（localized electrons） （itinerant electrons）
(U : Coulomb repulsion)( W : bandwidth )
particlelike wavelike
3/18
Competition between kinetic energy and Coulomb energy
Strongly
interacting
Weakly
interacting
Temperature
U W
Hubbard model
),(,,,, .).(
ji iiiji nnUchctcH
k
k
Rk
,,
1ce
Nc ji
j
Hubbard Hamiltonian
e, ,,,
,,,,,,,
4321
43214321
k
kkk
kkkk
kkkkkkkkcccc
N
UccH
Wannier Bloch
H H’
H H’
t
U1particle/site
diagonal
diagonal off-diagonal
off-diagonal
e, ,,,
,,,,,,,
4321
43214321
k
kkk
kkkk
kkkkkkkkcccc
N
UccH
H0 H’ (perturbation)
Hubbard Hamiltonian
)1)(1()(12
)1)(1()(,',2
)(
1
'2
'12
'2
'121
'2
'12
'2
'121
'2
'12
'2
'12
'2
'121
,,,
2
2
,,
2
21
'
2
'
1
1
kkkkkkkkkk
kkkk
kkk
kkkkkkk
fffUN
fffkkHkkk
----
----
eeeep
eeeep
2
1)(
1T
k
TkT
k B
Fe
log
)(
1 2
1
Tkv
aU
kB
F
2
2
2
2
1 )(8
12
)(
1
p
p
TkB
× Non-Fermi liquid
Calculate life time of Bloch electron
Scattering term
Scattering rate
2D,
1D,
3D,
At low-Temperatures
Fermi liquid
In the weak correlation regime, W~2zt >> U
H0H’ (perturbation)
Hubbard Hamiltonian
Heisenberg Hamiltonian
In the strong correration regime, W~2zt << U
),(,,,, .).(
ji iiiji nnUchctcH
),( ji
jiJH SS
Antiferromagnetic insulator
J=4t2/U
Mott transition line
Imada et al. JPSJ (2003)
tt’
U
Mott transition occurs at W~U,
but depends on dimension and lattice geometry
1-D Hubbard models are always Mott insulators.
2D ½-filled Hubbard model on anisotropic triangular lattice
Square lattice Triangular lattice
Tremblay et al. PRL (2006)
PIRG Cluster-DMFT
Layered molecular conductors, (BEDT-TTF)2X
XA variety of in-plane structures
X- Gorund state U/t t’/t
Cu2(CN)3 Mott insulator 8.2 1.06
Cu[N(CN)2]Cl Mott insulator 7.5 0.75
Cu[N(CN)2]Br Metal (SC) 7.2 0.68
Cu(NCS)2 Metal (SC) 6.8 0.84
Cu(CN)[N(CN)2] Metal (SC) 6.8 0.68
Ag(CN)2 H2O Metal (SC) 6.6 0.60
I3 Metal (SC) 6.5 0.58
e-
e- e-
e-
e-e-e-
e-
e- e-
e-
e-
U : on-site Coulomb
t : inter-dimer transfer integral
k-(ET)2X family are on the verge of Mott transition
in-plane structure
t’
t t
t’
t t
t’
t t
t’
t t
Triangular lattice
Half-filled band
dimer model
Kino & Fukuyama
Resistivity of k-(BEDT-TTF)2X
1 10 10010-6
10-4
10-2
102
104
1
Re
sist
ivit
y(W
cm
)
温度 （K）
k typeX= Cu[N(CN)2]Br
X= Cu[N(CN)2]Cl
X= Cu(NCS)2
b type X=I3
X
SIT by fine pressure tuning or isotope substitution
d [0,0]
d [1,1]
d [2,2]
d [4,4]
d [3,3]
Fig.1. Hiromi Taniguchi et al. (submitted to Nature)
A
C
B
resi
stiv
ity
HH
H
H
HH
H
Hdeuterated
k-(deuterated ET)2Cu[N(CN)2]Br
1x10-4
1x10-3
1x10-2
1x10-1
1x100
1x101
1x102
1x103
1x104
1x105
4 5 6 7 8 9 10
P(MPa)
T = 6 K
圧力 (MPa)
電気抵抗
(オーム
)
絶対温度
P
Res
ista
nce
(W
)
Mott
Ins.SC
e- e- e-e-- e-
e-- e-
U/t
b -
I 3
k -
Cu
(NC
S) 2
k–
Cu
[N(C
N) 2
]Br
k –
Cu
[N(C
N) 2
]Cl
k –
Cu
[N(C
N) 2
]Br
b’
-IC
l 2
Packing type of
BEDT-TTF
X
T（
K）
Paramagnetic insulator
Metal
supreconductor AF insulator
100
10
1
Commensurate AF
deu
tera
ted
d- SC
e- e- e-e-- e-
e-- e-
e-- e-
k-(ET)2X family are on the verge of Mott transition
e-
e-
e-
)()(22111221
- nnnnUcccctH
2
16 22 tUU -
2
16 22 tUU
U
),,(4
16,00,
22
--
t
tUU
- ,00,
,,
),00,(16
4,,
22
-
tUU
t
,, ,,
2 electrons/dimer with on-site Coulomb energy Hubbard model of a hydrogen molecule
2-particle ground state1-particle ground state
0
t
-t
ttUU
tttUU
22
162)(2
2
16 2222
-
---
Ueff
e
From band-structure calculations, Ueff ~ 2t ~ 0.5 eV and bandwidth W ~ 0.4 eV
comparable
bonding
antibonding
tU U
1 2
e-
tU U
1 2
e- e-
Mott physics in 2D organics
Mott transition
Charge
Magnetism
Spin
Superconductivity
Charge/Spin
U/W (Mottness)
Tem
per
atu
re
AF/SLSC
Mott insulator Metal
Spin liquid/order
on lattices
Quantum criticality
Preformed pairs
Interacting spins Order or not ?
1900 1950 2000
Heisenberg
J Si・Sj
Antiparrallel
interaction
Neel
proposed
proved
by neutron Anderson
proposed
1936 1949 1973
?
?
Landau Anderson
Triangle-based lattices
Triangular lattice
Hyper Kagome lattice
Pyrochlore lattice
Kagome lattice
Anderson’ idea of spin liquids:
Resonating Valence Bond (RVB) state
In analogy with benzene
= +
+ …… ………..
Heisenberg model
j
ji
iJH SS ),(
2 on
120°Neel order
?
Solution
However, ………. ～1990
………the end ?No spin liquid material in 20th century
Q2D organics k-(ET)2X; spin-1/2 on triangular lattice
dimer model
Kino & Fukuyama
t’
t t
t’
t t
t’
t t
t’
t t
t’/t = 0.4 ~ 1.1
0.68SCCu[N(CN)2]Br
0.75Mott insulatorCu[N(CN)2]Cl
0.84SCCu(NCS)2
1.06Mott insulatorCu2(CN)3
t’/tGround StateX-
0.68SCCu[N(CN)2]Br
0.75Mott insulatorCu[N(CN)2]Cl
0.84SCCu(NCS)2
1.06Mott insulatorCu2(CN)3
t’/tGround StateX-
0.80
0.44
Ab initio; Kandpal et al. PRL (2009)
Nakamura et al. JPSJ (2009)
Huckel & tight binding?Quantum spin liquid ?
Anderson (1973)
Spin frustration
Magnetism
H=0
w0 = gn (H0 +DH)
= w0 +Dw
I = 1/2
w0
Local fields
w0+Dw
Knight shift, K=Dw/w0
w0=10-1000 MHzelectron
Hyperfine coupling
nuclear SpinH0
Nuclear Magnetic Resonance
electron density of HOMO
Y. Imamura, et al., JCP111(1999)5986
w
Spin ordering or not ?
Triangular latticeHeisenberg model J = 250 K
(pade[7,7])
k-Cu2(CN)3
k-Cu[N(CN)2]Cl
Elstner et al.
PRL 71(1993)1629.
PRL 91 (2003) 107001
PRL 75 (1995) 1174
Canted AF transition
0.0000
0.0001
0.0002
0.0003
0.0004
0.0005
0.0006
0.0007
0 50 100 150 200 250 300
T (K)
t’/t = 0.75
t’/t = 1.06
Triangular latticeHeisenberg model J = 250 K
(pade[7,7])
k-Cu2(CN)3
k-Cu[N(CN)2]Cl
Elstner et al.
PRL 71(1993)1629.
PRL 91 (2003) 107001
PRL 75 (1995) 1174
Canted AF transition
0.0000
0.0001
0.0002
0.0003
0.0004
0.0005
0.0006
0.0007
0 50 100 150 200 250 300
T (K)
t’/t = 0.75
t’/t = 1.06
1H NMR spectrum
94.32 94.4 94.5 94.6 94.7
Frequency (MHz)
901 mK
1.4 K
2.8 K
9.7 K
32 mK
56 mK
164 mK
36.1 K
508 mK
156.6 156.7 156.8 156.9 157.0
Frequency (MHz)
4.9 K
10.3 K
14.1 K
18.1 K
25.1 K
22.1 K
27.2 K
30.2 K
164 K
k-(ET)2Cu2(CN)3
t’/t =1.06
k-(ET)2Cu[N(CN)2]Cl
t’/t =0.75
Magnetic susceptibility
No ordering AF ordered (0.45mB)
t’
t t
t’
t t
t’
t t
t’
t t
Also see Zheng et al. PRB 71 (2005) 134422
Mott insulators k-(ET)2X X t’/t
Cu2(CN)3
Cu[N(CN)2]Cl
1.06 (0.8)
0.75 (0.44)
Spin anomaly around 5-6K in k-(ET)2Cu2(CN)3
1/T1
0.01
0.1
1
10
100
0.01 0.1 1 10 100300Temperature (K)
~T1/2
~T3/2
0
0.5
1
0.01 0.1 1 10T (K)
inner
outer
Inhomogeneous
relaxation
a in
stretched exp
Shimizu et al., PRB 70 (2006) 060510
NMR relaxation rate
8 T layer
13C13C
13C NMR
Gapless
Why a spin liquid realized instead of
near Mott transition
“Hubbard spin liquid”
expected in Heisenberg model ?
Spin liq. emerges Hubbard model ?
Tocchio et al. PRB (2013)PIRG; Morita, Mizusaki, Imada (2002), (2006)
Cellular DMFT; Kyung, Tremblay (2006) VCA + LDFA Laubach et al. PRB (2015)
Why not
near the Mott transition
expected in Heisenberg model ?
Metal NMI stripe
Metal NMI
120°NeelMetal
120°Neel
U/t
U/t
U/t
Triangular lattice Hubbard model
Morita et al. (2002)
Mizusaki et al. (2006)
Sahebsara et. al. (2008)
Yoshioka et al. (2009)
T. Watanabe et. al. (2008)
Inaba et al. (2008)
Mott transition
0
0
0
Possible spin liquid
Interaction
(exchange)
Excitation
(magnon)Order
(AF/F)
Absence of
ordering Exotic excitation
?
Conventional
Frustration
Thermodynamics
Specific heat by Yamashita and Nakazawa (Osaka Univ.)
At low temperatures At higher temperatures;
Hidden order or some crossover ?
Field-insensitive anomaly
0
25
50
75
100
125
150
0 1 2 3 4 5 6
■ 0T
▼ 1T
● 4T
◆ 8T
■ 0T
▼ 1T
● 4T
◆ 8T
CPT
-1 /
mJ
K-2
mo
l-1
T2 (K2)
Finite g
k-(ET)2Cu2(CN)3
AF Mott insulators
CPT
-1/
mJK
-2m
ol-1
0
250
500
750
1000
1250
1500
0 20 40 60 80 100
0 T
2 T
8 T
T 2 ( K2 )
T / K
DC
PT
-1/
mJK
-2m
ol-1
0
50
100
150
0 2 4 6 8 10
0 T 2 T 8 T
k-(ET)2Cu2(CN)3
Low-lying spin excitations
● k-(d8:BEDT-TTF)2Cu[N(CN)2]Br
× k-(BEDT-TTF)2Cu[N(CN)2]Cl
○ b’-(BEDT-TTF)2ICl2
● k-(d8:BEDT-TTF)2Cu[N(CN)2]Br
× k-(BEDT-TTF)2Cu[N(CN)2]Cl
○ b’-(BEDT-TTF)2ICl2 Anomaly at 5-6 K
C=gT+bT3
g=(2p2kB2/3)D(eF)
Wilson ratio ~ 1.6
Degenerate Fermionic objects in Mott insulator
Triangular latticeHeisenberg model J = 250 K
(pade[7,7])
k-Cu2(CN)3
Elstner et al.
PRL 71(1993)1629.
PRL 91 (2003) 107001
0.0000
0.0001
0.0002
0.0003
0.0004
0.0005
0.0006
0.0007
0 50 100 150 200 250 300
T (K)
t’/t = 1.06
Triangular latticeHeisenberg model J = 250 K
(pade[7,7])
k-Cu2(CN)3
Elstner et al.
PRL 71(1993)1629.
PRL 91 (2003) 107001
0.0000
0.0001
0.0002
0.0003
0.0004
0.0005
0.0006
0.0007
0 50 100 150 200 250 300
T (K)
t’/t = 1.06
Triangular latticeHeisenberg model J = 250 K
(pade[7,7])
k-Cu2(CN)3
Elstner et al.
PRL 71(1993)1629.
PRL 91 (2003) 107001
0.0000
0.0001
0.0002
0.0003
0.0004
0.0005
0.0006
0.0007
0 50 100 150 200 250 300
T (K)
t’/t = 1.06
Triangular latticeHeisenberg model J = 250 K
(pade[7,7])
k-Cu2(CN)3
Elstner et al.
PRL 71(1993)1629.
PRL 91 (2003) 107001
0.0000
0.0001
0.0002
0.0003
0.0004
0.0005
0.0006
0.0007
0 50 100 150 200 250 300
T (K)
t’/t = 1.06
0
25
50
75
100
125
150
0 1 2 3 4 5 6
■ 0T
▼ 1T
● 4T
◆ 8T
■ 0T
▼ 1T
● 4T
◆ 8T
CPT
-1 /
mJ
K-2
mo
l-1
T2 (K2)
k-(ET)2Cu2(CN)3
cspin = 3 ×10-4 emu/mol g = 13 mJ/K2mol
g=(2p2kB2/3)D(eF)c=2mB
2D(eF)
)}3/2/({
)}2/({22
2
B
BW
kR
pg
mc
Spin liquid in k-(ET)2Cu2(CN)3; Gapless or gapped
0
25
50
75
100
125
150
0 1 2 3 4 5 6
■ 0T
▼ 1T
● 4T
◆ 8T
■ 0T
▼ 1T
● 4T
◆ 8T
CPT
-1 /
mJ
K-2
mo
l-1
T2 (K2)
S. Yamashita et al., , Nature Phys. 4 (2008) 459
Specific heat gapless (g = 13-14 mJ/K2mol)
0
50
T2 (K2
)
Thermal conductivity gapped; 0.46 K
M. Yamashita et a., Nature Phys. 5 (2009) 44
k-(ET)2Cu2(CN)3
g = 13-14 mJ/K2mol
Wilson ratio ~ 1.1
Degenerate chargeless Fermionic objects
A[Pd(dmit)2]2; quasi-triangular lattice systems
Spin liquid in EtMe3Sb[Pd(dmit)2]2
Specific heat Thermal conductivity
M. Yamashita et al, Science 328, 1246 (2010)S. Yamashita et al., Nat. Commun. 2, 275 (2011)
Wilson ratio
cspin =4.5 emu/mol g =20 mJ/mol K2
R. Kato, Bull. Chem. Soc. Jpn. 87, 355 (2014) S. Yamashita et al., Nat. Commun. 2, 275 (2011)
RW=1.6
Charge excitation in antiferromagnet and spin liquid
AFIAFIAFI
Optical conductivity
Kezsmarki et al.
PRB 74(2006)201101
Kornelsen et al., SSC 81 (1992)343
t’
t t
t’
t t
t’
t t
t’
t t
t’/t = 1.06t’/t = 0.75
Charge gap is clearly opened on AF ordering, but remains undeveloped in spin liquid.
Thermodynamic anomaly at 6K in k-(ET)2Cu2(CN)3
Specific heatS. Yamashita et al.,
Nature Phys. 4 (2008)
459
Thermal expansion coefficientManna et al., PRL 104 (2010) 016403
Thermal conductivityM. Yamashita et al.,
Nature Phys. 5 (2009) 44
NMR Relaxation rateShimizu et al.,
PRB 70 (2006) 060510
0
1
2
3
4
5
6
7
8
0 1 2 3 4 5 6 7 8 9 10Temperature (K)
(a)
13C NMRrelaxation rate
Inhomogeneous relaxation0
1
2
3
4
5
6
7
8
0 1 2 3 4 5 6 7 8 9 10Temperature (K)
(a)
13C NMRrelaxation rate
Inhomogeneous relaxation0
1
2
3
4
5
6
7
8
0 1 2 3 4 5 6 7 8 9 10Temperature (K)
(a)
13C NMRrelaxation rate
Inhomogeneous relaxation
Ultrasound velocityPoirier et al.,
Can you distinguish SL and SC ?
-400 -200 0 200 400 600 800
SHIFT from TMS (ppm)
20 K
18 K
16 K
14 K
12 K
10 K
8 K
6 K
4 K
2 K
-400 -200 0 200 400 600 800
SHIFT from TMS (ppm)
20.2 K
18.2 K
16.2 K
14.2 K
12.4 K
9.9 K
8.1 K
6.0 K
5.7K
k-ET2Cu2(CN)3 k-(ET,d44)2Cu[N(CN)2Br
TC
NMR spectra
Thermal conductivity
B a axis
k-ET2Cu2(CN)3
M. Yamashita et al., Nature Phys. 5 (2009) 44
k-ET2Cu(NCS)2
TC
Y. Matsuda et al.,
J. Phys: Condens. Matter 5 (2006) R705
SL
SC
SL
SC
Pressurize AFI and spin liquid
½-filled Hubbard model (Cluster DMFT)
Kyung, Tremblay PRL (2006)
Frustrated k-(ET)2Cu2(CN)3
Less frustratedk-(ET)2Cu[N(CN)2]Cl
t’
t t
t’
t t
t’
t t
t’
t t
Kandpal et al.
PRL 103 (2009) 067004
Material parameters
Thermodynamics of Mott transition
Clausius Clapeyron dT/dP = (VA-VB)/(SA-SB)
T
P
A phaseB phase
>0
SA > SB
T
P
A phase
B phase
SA < SB
Entropy balance known from phase diagram
k-(ET)2Cu2(CN)3 t’/t ~ 0.80-1.06
AFIAFIAFI
k-(ET)2Cu[N(CN)2]Cl t’/t ~ 0.44-0.75Kagawa et al., Nature 2005 , PRL 2004; PRB 2004,
Kurosaki et a., PRL 2005, Furukawa et al.unpublished
Thermodynamics of Mott transition
Clausius Clapeyron
dT/dP = DV/DS=(VA-VB)/(SA-SB)
>0
T
P
A phase B phase
SA > SB
SA < SB
T
P
A phase B phase
DS=SSL-Smetal= (dP/dT) DV
SSLgmetal=27.5 mJ/mol K2
Smetal =gmetalT
Entropy of spin liquidparameter
SSL
Phase diagram
NMR
1/T1
resistivity
Mott transition of SL; drastic change in charge transport but not in spin
Under He-gas pressure
Experimental test of scaling
Pressure (MPa)
Tem
pe
ratu
re (
K)
R(o
hm
)R
/Rc(
T)
P-Pc(T)(MPa)
R/Rc vs T/(c|P- Pc|0.6)
10
0.1
1
0.1 1 10 100
T/T0
R/R
c(T)
R/Rc vs (P- Pc)*T-1/zv
{P-Pc(T)}T-1/zv
zv=0.6
QSL FLSC
Critical endpoint
P (MPa)
T(K
)
Perfect scaling
for T >1.5Tc
Furukawa et al.,
Nat. Phys 11 (2015) 221
Nat. Phys 11 (2015) 221
QC scaling --- nearly material -independent
SL/SC SL/MetalAFI/SC
k-(ET)2Cu2(CN)3 k-(ET)2Cu[N(CN)2]Cl EtMe3Sb[Pd(dmit)2]2
zn=0.62±0.02 zn=0.49±0.01 zn=0.68±0.04
Furukawa et al., Nat. Phys 11 (2015) 221
0 21
experiement
theory
Terletska, Dbrosavljevic, et al
(2011)
DMFT Continuous Mott
Krempa, Kim, senthil, et al
(2012)
Marginal Quantum Mott
Imada, et al
(2007)
k-(ET)2Cu2(CN)3
k-(ET)2Cu[N(CN)2]Cl
EtMe3Sb[Pd(dmit)2]2
Si-MOSFET
(Kravchenko)
Q2D Mott
Critical exponents, zn, in metal-insulator transitions
0.5-0.7
Mott phase diagrams of quasi-triangular lattices
k-(ET)2Cu2(CN)3t’/t=0.80-1.0
t’
t t
t’
t t
t’
t t
t’
t t
frustrated less frustrated
k-(ET)2Cu[N(CN)2]Cl
t’/t=0.44-0.75
0.33
>10
1
R/Rc
Weak Mott, Strong Mott,
QSL FLSC
Critical endpoint
P (MPa)
T(K
)
QSL FLSC
Critical endpoint
P (MPa)
T(K
)
QSL
P (MPa)
T (K)
AFI
Critical endpoint
FL
SC
P (MPa)
T (K)
AFI
Critical endpoint
FL
SCAF
Low Tc High Tc, Pseudo-gap
Similar QC behavior at high T
Dissimilar at low T
Single-site DMFT of Hubbard modelH.Terletska , V.Dobrosavljevic et al.,Phys. Rev. Lett 107, 026401(2011)
U-T Phase diagram
insulator metal
TC
δU-T Phase diagram
Fermi liquidMott insulator
Tc
T0
Quantum critical region
Instability line
Uc1(T)Uc2(T)
U/Uc1 0.61.51.7
U=UC(T)
UC(T)
T/T0
R/R
C
Possible quantum critical behavior in an intermediate energy range
FermiLiquid
MottInsulator
U
WT
P
Organic Mott system
TK
J
Heavy fermion system
TRKKY
T
Quantum criticality ( Tc < T << t,U)
AF FL
TNTF
~20 K
~5000 K
Why eager for spin liquid ?
electron
spin
e
e
e
e
e
ee
e
e
e
e
e
e
ee
Classical Quantum And more
superconductivity
electronics
……
…..
Wigner Xtal Fermi liquid
Magnet Quatum spin liquid
!?
Electron correlation in massless Dirac fermions
M. Hirata et al., Nat. Commun. (2016) in pressD. Liu et al., PRL (2016)K. Miyagawa et al., JPSJ (2016)K. Ishikawa et al., PRB (2016)
Dirac cone reshaping Ferrimagnetism
AB
C
b
a
A’ (= A)B
Michihiro Hirata Kyohei Ishikawa Kazuya Miyagawa
Claude Berthier Denis Basko Akito Kobayashi Genki Matsuno
Masafumi Tamura
NMR Sample preparation
NMRRG calculation
(continuum model)Mean-field calculation
(lattice model)
NMR NMR
Organic Conductor a-(BEDT-TTF)2I3
Phae diagram
0 4 8 12 16 200
40
80
120
160
T (
K)
P (kbar)
P at room T
charge order
anomalous metal
Dirac
D. Liu et al., PRL, in pressH. Schwenk et al., Mol. Cryst. Liq. Cryst. 119 (1985)
13C
Charge order is suppressed by pressure and a Dirac semimetal emerges !
Vertical cone in Graphene Atomic Orbs. (A & B sublat.)
Unit cell
EF
E
qx
qy
kD-kD
S. Katayama et al., Eur. Phys. J. B 67 (2009)M.O. Goerbig et al., PRB 78 (2008)K. Kajita et al., JPSJ 83 (2014)
General cones exist in various systems: d-wave SC, 3He, Topological Ins., Organic Solids K. Asano et al., PRB (2011), T. O. Wehling et al., Adv. Phys. (2014)
Titled cone in a-(BEDT-TTF)2I3
Molecular Orbs. (A, A’, B, C sublat.)
1. Dirac Cones Everywhere
BEDT-TTF
T. O. Wehling et al., Adv. Phys. 63, 1-76 (2014)
Fermi surface Fermi point
kx
ky
Gapped
2. Short-ranged or Long-ranged?
Short-ranged Long-rangedCoulomb interaction is
Unscreened
EF
k
E
Long-range part preserved
Logarithmic divergence of vF
A. A. Abrikosov et al., JETP 32 (1971)
• Reshaping of vertical axes cones in graphene
D. C. Elias et al., Nat. Phys. 7 (2011)C. Faugeras et al., PRL 114 (2015)V. N. Kotov et al., Rev. Mod. Phys. 84 (2012)
Unscreened Long-range Coulomb Interaction
A. A. Abrikosov
b
a
B
C
Steep slope (Large vF)Gentle slope (Small vF)S. Katayama et al., Eur. Phys. J. B 67 (2009)
Molecular site to k-space correspondence
E (m
eV)
E (m
eV)
0 50 100 150 200 250 300
0
2
4
6
8
T (K)
j = A (A') ( = 60o)
j = B ( = 120o)
j = C ( = 120o)
c j s
(10
-5 m
B/k
Oe
)
2.3 GPa, 6 T || ab
13C NMR
a-(BEDT-TTF)2I3
-0.05 0.00 0.050
2
4
6
8
10Tight-binding
Density o
f sta
tes p
er
spin
(eV
-2)
E (eV)
A (= A')
B
C
Kino et al., J. Phys. Soc. Jpn. 75 (2006); Katayama et al., Eur. Phys. J. B 67 (2009)Kobayashi et al., J. Phys. Soc. Jpn. 82 (2013)
J-sublattice Electron Spin Susceptibility
C > A (A’) > B
Average
Large vF
Small vF
C
B
A (=A’)
0
2
4
6
c
j s
(10
-5 m
B/k
Oe
)
j = C
2.3 GPa, 13
C NMR
a-(BEDT-TTF)2I3
6 T || ab
0 50 100 150
0
2
4
6
T (K)
j = B
c j s
(10
-5 m
B/k
Oe
)Linear spectrum RG calculation
Gentle slope:Renormalized at lower E
Steep slope:Renormalized from higher E
A Non-uniform vF Renormalization
2-times different
B
C
Renormalization Group (RG) Calculation
Anisotropic tilted cone
RPA self energy
RG equations (leading order in 1/N; N >> 1)
N = 4: fermion flavor, q = (k – kD) = q(cos , sin ), l = ln(L/q)
Linear spectrum
RG calculation
100
101
102
103
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
vx
w0x
w0y
vx,y,
w0x,
0y
(10
5 m
/s)
L/q
vy
(L = 0.8 Å-1, e = 1) Initial w0, v: (band-structure value)/3.2
EF
cf. H. Isobe et al., J. Phys. Soc. Jpn. 81 (2012): using HF exchange term
S. Katayama et al., Eur. Phys. J. B 67 (2009)
A site
AB
C
b
a
A’ (= A)B
0.0 0.2 0.4 0.6
-0.1
0.0
0.1
0.2
0.3
0.4
13C NMR
2.3 GPa, 6 T || ab
a-(BEDT-TTF)2I3
j = B
j = A, C
c j s(T
)/c
j
s(T
j
Fle
x)
T/T j
Flex
0
0.0 0.5 1.0 1.5
0.0
0.5
1.0
1.5
2.0j = A (A') ( = 60
o)
j = B ( = 120o)
j = C ( = 120o)
T/T j
Flex
0=B
< 0
0 20 40 60 80
-0.2
0.0
0.2
0.4
T (K)
c j s
(10
-5 m
B/k
Oe
)
RG calculation
Ferrimagnetic spin Polarization exist on top of the vF renormalization
Negative Susceptibility on the B SublatticeNegative susceptibility on B Anomaly observed uniquely on B
Hubbard model; Mean-field calculationHubbard model (nearest neighbor)
RPA spin susceptibility (Q = 0)
Intra-band
E
qx
kD
Inter-band
E
qx
kD
0.12
0.14
0.16
USpin
su
scep
tib
ility
(ar
b. u
nit
s)
Intra-band + Inter-band
U = 0j = B
0 50 100 150 200 250 300
0
1
2
3
4Hubbard model
+ RPA (U = 0.14 eV)
RP
A s
ub
lattic
e s
pin
su
sce
ptib
ility
(a
rb. u
nits)
T (K)
Intra-band + Inter-band
j = C
j = A(A’)
j = B
Intra-band
Spin
su
scep
tib
ility
(ar
b. u
nit
s)
U = 0.14 eV
A. Kobayashi et al., J. Phys. Soc. Jpn. 82 (2013)
Hubbard model; Mean-field calculationHubbard model (nearest neighbor)
RPA spin susceptibility (Q = 0)
Intra-band
E
qx
kD
Inter-band
E
qx
kD
0.12
0.14
0.16
USpin
su
scep
tib
ility
(ar
b. u
nit
s)
Intra-band + Inter-band
U = 0j = B
0 50 100 150 200 250 300
0
1
2
3
4Hubbard model
+ RPA (U = 0.14 eV)
RP
A s
ub
lattic
e s
pin
su
sce
ptib
ility
(a
rb. u
nits)
T (K)
Intra-band + Inter-band
j = C
j = A(A’)
j = B
Intra-band
Spin
su
scep
tib
ility
(ar
b. u
nit
s)
Ferrimagnetic Polarization due to SR Coulomb Int.
U = 0.14 eV
A. Kobayashi et al., J. Phys. Soc. Jpn. 82 (2013)
Conclusions
EF
E
qx
qy
kD-kD
Long-range Coulomb, 1/r
Cone reshaping
Short-range Coulomb, U, V
Ferrimagnetism
Collapse of band structureinto charge order state
