Strongly Correlated Electrons on Frustrated Latticesykis07.ws/presen_files/26/Tsunetsugu.pdfStrongly...
Transcript of Strongly Correlated Electrons on Frustrated Latticesykis07.ws/presen_files/26/Tsunetsugu.pdfStrongly...
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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
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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
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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?
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PART A
Mott Transition in Kagomé Lattice Hubbard Model
[ Ohashi, Kawakami, and Tsunetsugu, Phys. Rev. Lett. 97 (’06) 066401]
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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]
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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...
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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
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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
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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
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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
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-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)
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-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
+
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-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
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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
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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
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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
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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
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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
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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
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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)
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Configuration in the unit cell
Spin correlations in the real space
1-dim. spin correlations
no phase coherencefrom chain to chain
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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
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Quasiparticle Renormalization Factor
1 2
3 Renormalization factor Zloc
Mass enhancement〜 10-20near Mott transition
Σ11(iωn)self energy
U/W
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• 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
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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]
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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
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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?
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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
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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
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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?
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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
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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
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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
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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
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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
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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)
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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
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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
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-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
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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
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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
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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
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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
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• 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
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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.)
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3-sublattice Antiferro Nematic Order
[ Tsunetsugu and Arikawa, JPSJ 75, 083701 (2006) ]
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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
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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)
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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)
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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
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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
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Antiferro Nematic Order
3-sublattice ordermagnetic quadrupoles
K>00
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1D Analog of 3-sublattice Nematic Order?
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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
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Phase Diagram (S=1 BLBQ zigzag spin chain)
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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
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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