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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
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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


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