Strongly Correlated Electrons on Frustrated Latticesykis07.ws/presen_files/26/Tsunetsugu.pdfStrongly Correlated Electrons on Frustrated Lattices (ISSP, Univ of Tokyo) Hirokazu Tsunetsugu

Strongly Correlated Electrons on Frustrated Lattices

(ISSP, Univ of Tokyo) Hirokazu Tsunetsugu

collab./w (Osaka Univ.) Takuma Ohashi(Kyoto Univ.) Norio Kawakami(RIKEN) Tsutomu Momoi

Yukawa International Seminar 2007 (YKIS2007) “Interaction and Nanostructural Effects in Low-Dimensional Systems”, Nov. 5-30, 2007, Kyoto University, Kyoto

sponsored by the MEXT of Japan:•a Grant-in-Aid for Scientific Research (Nos. 17071011 and 19052003) •the Next Generation Super Computing Project, Nanoscience Program

Nov. 26, 2007

OUTLINE

• Correlated electron systems with geometrical frustration

• [A] Kagomé Lattice Hubbard model – exotic spin correlation near metal-insulator transition

• [B] Anisotropic Triangular Lattice Hubbard Model – entropy and frustration effects – heavy quasiparticle formation and metal-insulator transition

• [C] Trimer phase of bilinear-biquadratic zigzag chain– antiferro spin nematic correlation

Pyrochlore

Triangular Kagomé

Many interesting systems:•Superconductivity NaxCoO2・yH2O, AOs2O6•Heavy Fermion LiV2O4, etc•Quantum spin liquid κ-(ET)2Cu2(CN)3

Correlated electron systems with geometrical frustration

Classical: Many states are degenerate in low-energy sector.Quantum effects hybridize these states -> new phase/correlations?

PART A

Mott Transition in Kagomé Lattice Hubbard Model

[ Ohashi, Kawakami, and Tsunetsugu, Phys. Rev. Lett. 97 (’06) 066401]

Kagomé Lattice Hubbard Model

• Typical frustrated lattice in 2D(thermodynamically degenerate ground states of AF Ising spins)

• 2D analog of pyrochlore lattice• Effective model of NaxCoO2・yH2O

Koshibae & Maekawa PRL 91, 257003 (2003)Bulut, Koshibae, & Maekawa PRL 95, 37001 (2005)

• Spin systems on Kagomé lattice⇒ unusual properties (gapped triplet, gapless singlet excitations)

H = −t ciσ+ c jσ

i, j ,σ∑ + U ni↑ni↓

i∑

Relation btw charge fluctuations and spin correlationsEffects on quasiparticle coherence

[ fix density at half filling n=1]

Cellular dynamical mean field theory(CDMFT)

Kotliar, et al. PRL 87, (2001)Lichtenstein & Katsnelson, PRB 62, (2000)...

Method

• strong correlation • geometrical frustration• short-range quantum

fluctuations ⇒ DMFT

Metal-insulator transition in Kagomé lattice at half filling

1 2

3

1 2

3

1 2

3

11 2

3

3 3

1 2

3

2

1 2

3

• Spatially extended correlation• Geometrical frustration

Σij ω( )Self-energy: 3x3 matrix Effective cluster model

DMFTVollhardtMuller-HartmannKotliarGeorgesJarrell...

Double occupancyMott transition

ni↑ni↓ • High temperatureT/W>1/80crossover U*～1.35

• Low temperatureT/W=1/801st order transition

with hysteresis: Uc～1.37

0.05

0.1

T/W=1/80T/W=1/50T/W=1/30T/W=1/20

1 1.2 1.4U/W

Band width: W=6t

Larger critical value Uc

square lattice: Uc～0.5-1.0

: measure of charge fluctuations

metallic

insulating

Crossover

1.0 1.2 1.40

0.2

0.4

T/W = 1/20T/W = 1/30T/W = 1/50T/W = 1/60

dDoc

c./d

U

U/W

Define metal-insulator crossover points U*(T) by largest change in double occupancy

Phase diagram

1.0 1.1 1.2 1.3 1.4 1.50

0.02

0.04

0.06Crossover1st order transition

metal

insulator

U/W

T/W

Mott transition in Kagome Hubbard model

Charge susceptibility

0 0.5 10

0.5

1U/W = 1.1U/W = 1.3U/W = 1.5

U/W = 0.5

T/W

χcloc

Charge response grows once again in low-T metallic region

-1 0 10

0.5

1

1.5

Density of States: U/W=1.1

T/W=1/80T/W=1/50T/W=1/20

ω/W

Evolution of heavy quasiparticles at low temperatures

measurable by PE/IPEexperiments

electron spectral function (k-summed)

-1 0 10

0.5

1

1.5

Density of States: U/W=1.3

T/W=1/80T/W=1/50T/W=1/20

ω/W

Evolution of heavy quasiparticles at low temperatures

Precursor of insulating behavior

+

-1 0 10

0.5

1

1.5

Density of States: U/W=1.5

T/W=1/80T/W=1/50T/W=1/20

ω/D

Clear formationof Mott gap

DOS (T/W=1/80)

U/W=0W

Dispersion (U=0)

Insulator: Uc/W ～ 1.37

-1 0 1ω /W

U/W=1.1

U/W=1.36

U/W=1.4

U/W=1.3

Density of states

Strongly correlated

metal

• Whole bands are renormalized• Heavy quasiparticles

Local spin susceptibility

0 0.5 10

0.1

0.2

Tχs loc

T/W

U/W = 1.1U/W = 1.3U/W = 1.5

U/W = 0.5

1-site DMFT:Free spins in insulating phase

cluster DMFT:Spins are screened/correlated

SizSi+1

z

Metallic phase:Nonmonotnic behavior

Insulating phase: AF correlation ⇒monotonic enhancement

Recover of itinerancy

AF spin correlation

Spin correlation functionNearest-neighbor spin correlation

T/W=1/80

1 1.1 1.2 1.3 1.4 1.5-0.08

-0.07

-0.06

-0.05

-0.04

U/W

T/W=1/20T/W=1/30T/W=1/50T/W=1/80

Temperature Dependence of Spin Correlations

0 0.05 0.1 0.15-0.08

-0.06

-0.04

-0.02

0

T/W

nearest-neighbor corr.

U/W=0.5U/W=1.0U/W=1.3U/W=1.5

Antiferromag.

correlated metal:Nonmonotonic T-depSuppressed AF correlationat low-T

insulator:Monotonic growth of AF correlation

recovery of coherence

relax frustrationCharacteristic for frustrated systems near MIT

metal ⇒ insulatorDouble peak

Dynamical Susceptibility near Mott Transition

ω /W

U/W=1.30U/W=1.36U/W=1.40

0 0.1 0.2 0.3 0.40

2

4

6

8

InsulatorUc/W ～ 1.37

T/W=1/80

χ loc ω( )= −i Siz t( ),Siz 0( )[ ] e− itω dt∫Imaginary part of local susceptibility

Metallic phase:Renormalized single peak

Suppressed Magnetic Instability

Mott transition : Uc/W ～ 1.35

U /0.6 0.8 1 1.2 1.4

0

0.2

0.4

0.6

W

T/W=1/30

1/χmax

Wavevector dependence of dominant mode

Metallic phase

Leading magnetic mode: 6 points ⇒ nesting

Insulating phaseLeading magnetic mode: Three lines ⇒1-dimensional order

temperature: T/W=1/30

U/W=0.0

U/W=1.1

U/W=1.3

χmax(q)

Configuration in the unit cell

Spin correlations in the real space

1-dim. spin correlations

no phase coherencefrom chain to chain

Self Energy of Single-Particle Green’s Fn.

Im part of self energy T/W=1/50

Im part of self energy T/W=1/50

1 2

3

1 2

3

D=6t=W

Quasiparticle Renormalization Factor

1 2

3 Renormalization factor Zloc

Mass enhancement〜 10-20near Mott transition

Σ11(iωn)self energy

U/W

• Metal-insulator transition– 1st order transition :Uc/W～1.37

• Strongly correlated metal– Whole bands are renormalized – large mass enhancement– nonmonotonic temperature dependence of spin correlation

functions

• Magnetic instability – one-dimensional spin correlations

Summary (1)Kagome lattice Hubbard model

Cellular dynamical mean field theory

PART B

Mott Transition in Anisotropic Triangular-Lattice Hubbard

Model

•Phase Boundary Topology•Heavy Quasiparticles

[ Ohashi, Momoi, Tsunetsugu, and Kawakami, cond-mat.st-el/0709.1700]

Mott transition in κ-type organic materials

κ-(ET)2-Cu[N(CN)2]Clκ−(ET)2-Cu2(CN)3F. Kagawa et al., PRB 69, 064511 (2004) Y. Kurosaki et al., PRL 95, 177001 (2005)

metalPI

Reentrant !!

AFISC

STRONG frustrationnearly perfect regular triangle

INTERMED. frustrationdistorted towards square

Georges et al., RMP, 68, 13 (1996)

DMFT

Mott transition line in Phase Diagram

U/W

T/W

d=∞ Hubbard model organic conductorκ-(ET)2-Cu[N(CN)2]Cl

T/W

U/W∝1/(pressure)

metal insulator

Cellular-DMFT

Reentrant !

Anisotropic Hubbard model

effects of 1-site approx. or frustration?

Metal

Ins.Crossover

Metal Ins.1st order

Mott transition at finite temperature

Georges et al., RMP, 68, 13 (1996) Moukouri & Jarrell PRL 87, 167010 (2001)

DMFTDCA on square lattice

CPM for small t’

Onoda & Imada, PRB 67, 161102 (2003) Parcollet et al., PRL 92, 226402 (2004)

CDMFT on triangular lattice

Double occupnacy

Anisotropic Triangular Lattice Model

t-t’-U Hubbard model

effective cluster model

4-sitecluster

t’t

• t’/t=0: regular square• t’/t=1: regular triangular

anisotropic triangular lattice

κ−(BEDT-TTF)2-Cu2(CN)3

κ-(BEDT-TTF)2-Cu[N(CN)2]Cl

t’/t ~ 1

t’/t ~ 0.8

t’/t controls frustration

Temperature-Dependence of Double Occupancy

0 0.2 0.4 0.6 0.8 1

0.05

0.06

0.07

T/t

t’/t=0.8

t’/t=0.7

t’/t=0.6

t’/t=0.5

Repulsion U/t = 8

small t’-weak GF:almost monotonicinsulating

large t’-strong GF:nonmonotonic T-dep.

insulatingmetallic

insu

latin

ginsula

ting

Insulator-Metal-InsulatorTransition?

Electron Spectral Function

-5 0 5ω/t

high T

low T

T/t=0.7

U/t = 8, t’/t=0.5 U/t = 8, t’/t=0.8

T/t=0.4

T/t=0.2

ω/t-5 0 5

T/t=0.7

T/t=0.25

T/t=0.1

gap emerges :insulating

insulating

metallicfrustration

insulating

Reentrant: I→M → I

Local Spectral Function – single site DMFT

[ Zhang, Rosenberg, and Kotliar, PRL, 1993 ]

Mott transition is driven by transfer of spectral weight between high-energy Mott band and low-energy quasiparticle band

Mott Transition

8 9

0.05

0.06

0.07

0.08

0.09

U/t

T/t=0.1

T/t=0.2

T/t=0.27

1st order Mott tansition

U-dep of double occupancy

higher T ⇒ largerUc

d=∞: DMFT

7 8 9 1000.20.40.60.81

T/t

U/t

metallic

insulating

t’/t=0.8

T-dep reversed

Crossover at a higher temperature

U-dep of double occupancy

T-dep changescrossover

8 9

0.05

0.06

0.07

7 8 9 1000.20.40.60.81

T/t

U/t ∝ 1/(pressure)

metallic

insulating

t’/t=0.8T/t=0.6T/t=0.5T/t=0.4

higher Tlower T

U/t

Reentrant !!

insulating metallic

Electron Spectral Function Ak(ω): high-T insulating phase

high-T insulating phaseU/t=8.0, T/t=0.7

7 8 9 1000.20.40.60.81

T/t

U/t

M

I

t’/t=0.8

no quasiparticlesHubbard gap

Ak(ω)

Local moment

Electron Spectral Function Ak(ω): intermediate-T metallic phase

U/t=8.0, T/t=0.25

7 8 9 1000.20.40.60.81

T/t

U/t

M

I

t’/t=0.8

sharp quasiparticle peaks

GF-induced metal

Ak(ω)

Electron Spectral Function Ak(ω): low-T insulating phase

U/t=8.0, T/t=0.1

7 8 9 1000.20.40.60.81

T/t

U/t

M

I

t’/t=0.8

quasiparticle peak splitsdifferent from high-T insulating

phase

Ak(ω)

Magnetic exchange

4 5 6 7 8 9 100

0.1

0.2

0.3t’/t = 0.5 (diverge)t’/t = 0.5

t’/t = 0.8 t’/t = 1.0 t’/t = 1.0 (diverge)

t’/t = 0.8 (diverge)

U/t

1/χmax

Magnetic susceptibility for different t’at T/t=0.2

Susceptibilities diverge.

At T/t=0.2UN～7.0 for t’/t=0.5UN～9.3 for t’/t=0.8UN～9.7 for t’/t=1.0

-5 0 50

0.1

0.2

0.3

-5 0 50

0.1

0.2

0.3

Density of States for different t’ at T/t=0.2U/t = 8.0

t’/t = 0.5t’/t = 0.8t’/t = 1.0

ω/t ω/t

t’/t = 0.8 (Ins.)t’/t = 1.0

U/t = 9.0

Geometrical frustration tends to stabilize the metallic phase

DOS on the triangular lattice (t’/t=1.0)

ω/t-5 0 50

0.05

0.1

0.15

-5 0 50

0.05

0.1

0.15

T/t = 0.25 T/t = 0.20

ω/t

U/t = 8.0U/t = 9.0U/t = 10.0

Insulating gap

Magnetic OrderCellular-DMFT magnetic LRO at T>0

AFI

TN

0

0.2

0.4

0.6

0.8

1

T/t

U/t7 8 9 10

MPI

crossover1st order Mott tr.

anisotropy: t’/t=0.8

Georges et al. RMP 68, 13 (1996) Zitzler et al. PRL 93 016406 (2004)

PIT

U

DMFT unfrustrated

Mott transition is masked.

Frustrated system: Mott transition is NOT maskedparamagnetic insulator phase

AFI

weak 3-dim couplings stabilize LRO

Comparison with ∞–dim frustrated stsytems

DMFT: frustrated Bethe latticeZitzler et al. PRL 93 016406 (2004)

0

0.2

0.4

0.6

0.8

1

T/t

U/t7 8 9 10

MI

AFI

crossover1st order Mott tr.TN

anisotropy: t’/t=0.8

T

U

PM PIAFI

intermediately frustrated

effects of short-range fluctuationsUc-curve changes its directionnonmagnetic insulating phaseCellular-DMFT

Comparison with Organic Materials

0

0.2

0.4

0.6

0.8

1

T/t

U/t7 8 9 10

MIns.

AFI

crossovercTN

anisotropy: t’/t=0.8 Maier et al., PRL 85, 1524 (2000)

κ-(BEDT-TTF)2-Cu[N(CN)2]Cl

MPI

AFI

Cellular-DMFT magnetic order at T>0stabilized by weak 3-dimensionality

consistent with experiments ~t/U

• Metal-insulator transition– different slope of transition line from unfrustrated systems

– entropy effects• Intermediate Correlation Regime:

– “reentrant” insulator → metal → insulator transition– heavy quasiparticle formation in the intermediate metallic

phase– gap formation inside heavy qp band

• Magnetic instability – transition to paramagnetic insulator phase– magnetic phase appears at lower temperature

Summary (2)Anisotropic Triangular Lattice Hubbard model (mainly t’/t=0.8)Cellular dynamical mean field theory

PART C

Trimer Phase of bilinear-biquadratic zigzag chain

[ Corboz, Lauchli, Totsuka and Tsunetsugu, cond-mat.st-el/0707.1195in press in Phys. Rev. B ]

collab. with Philippe Corboz (ETH Zurich)Andreas Läuchli (EPF Lausanne)Keisuke Totsuka (YITP, Kyoto U.)

3-sublattice Antiferro Nematic Order

[ Tsunetsugu and Arikawa, JPSJ 75, 083701 (2006) ]

NiGa2S4 - structureS=1 spin system (Ni2+)Quasi-2D triangular structure

a= 3.6 [Å]c=12.0 [Å]a= 3.6 [Å]c=12.0 [Å]

Ref: Nakatsuji et al., Science 309, 1697(‘05)Ref: Nakatsuji et al., Science 309, 1697 (‘05)

NO orbital degrees of freedomNO orbital degrees of freedom

NiGa2S4: spin liquid behavior• No phase transition down to 0.35[K]• C(T)∝T2

-> presence of gapless excitations • Finite χ≈8x10-3[emu/mole] at T≈0 • Finite ξ≈25[Å] at T≈0• Spatial modulation in spin correlations

Q≈(1/6,1/6,0)

Nakatsuji et al., Science 309, 1697 (‘05)Nakatsuji et al., Science 309, 1697 (‘05)

Difficulty of Ordinary Scenarios

consistent:no singularity in C(T) and χ(T)

NOT consistent:non-divergent ξ(T) neutron scattering

consistent:no singularity in C(T) and χ(T)

NOT consistent:non-divergent ξ(T) neutron scattering

[A] magnetic LRO with Tc < 0.3K[A] magnetic LRO with Tc < 0.3K

consistent:non-divergent ξ(T)

NOT consistent:(a) : C(T) ∝ T 2 (b) : χ(T →0) = finite

consistent:non-divergent ξ(T)

NOT consistent:(a) : C(T) ∝ T 2 (b) : χ(T →0) = finite

(a)gap(S=1 excitations)＞０gap(S=0 excitations) ＞０(eg. Haldane chain, Shastry-Sutherland system SrCu2(BO3)2)

(b)gap(S=1 excitations) ＞０gap(S=0 excitations) ＝０(eg. S=1/2 Kagome)

[B] spin gap state[B] spin gap state

T = 0: magnetic LRO (eg, 120-degree structure)

T > 0: paramagnetic(Mermin-Wagner)

Possibility of Unconventional OrderHidden non-”magnetic” order？Antiferro order of spin quadrupoles

spontaneous breaking of spin rotation symmetryspin inversion sym. is NOT broken

Hidden non-”magnetic” order？Antiferro order of spin quadrupoles

spontaneous breaking of spin rotation symmetryspin inversion sym. is NOT broken

S≧１S≧１

Blume, Chen&Levy...Blume, Chen&Levy...

anisotropy of spin fluctuationsanisotropy of spin fluctuations

directordirector

Phenomenological ModelBilinear-Biquadratic modelBilinear-Biquadratic model

low-energy effective modellow-energy effective model

Biquadratic termBiquadratic termBiquadratic term(cf. 4th order of hopping process)(cf. 4th order of hopping process)

quadrupole couplingsquadrupole couplings

Chen & Levy (‘71)Matveev (‘74)Andreev & Grishchuk (‘84)Fath & Solyom (‘95)Schollwock, Jolicoeur & Garel (‘96)Harada & Kawashima (‘02)Lauchli, Schmidt & Trebst (‘03)

Chen & Levy (‘71)Matveev (‘74)Andreev & Grishchuk (‘84)Fath & Solyom (‘95)Schollwock, Jolicoeur & Garel (‘96)Harada & Kawashima (‘02)Lauchli, Schmidt & Trebst (‘03)

1-dim system[Lauchli, Schmid, Trebst, ‘03]1-dim system[Lauchli, Schmid, Trebst, ‘03]

JJ

KK Mean FieldMean Field

Antiferro Nematic Order

3-sublattice ordermagnetic quadrupoles

K>00

1D Analog of 3-sublattice Nematic Order?

Model

1

2

inter-chainintra-chainij

JJ

J⎧

= ⎨⎩

S=1 bilinear-biquadratic (BLBQ) zigzag chain:

2 BLBQ decoupled chains

1 BLBQ simple chain

SU(3) symmetricHeisenberg zigzag chain

J2/J1

θ

∞

0π/4 π/2

1 Symmetriccoupling

T=0

Phase Diagram (S=1 BLBQ zigzag spin chain)

RG Flow

∼J2−(J2)c

∼tanθ−1

Trimerized

Haldane

[ Itoi and Kato, PRB, 1997, Totsuka and Lecheminant 2007]

N=3

liquid ↔ trimerized:liquid ↔ Haldane:

Kosterlitz-Thouless tr

trimerized ↔ Haldane: 1st order

SummaryS=1 BLBQ zigzag spin chain

•Existence of trimerized phase

•Order of transitions

•Reentrant transition to liquid phase in the large-J2 regionat SU(3) sym. line