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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
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