Strongly Correlated Electrons on Frustrated Latticesykis07.ws/presen_files/26/Tsunetsugu.pdfStrongly...

57
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

Transcript of Strongly Correlated Electrons on Frustrated Latticesykis07.ws/presen_files/26/Tsunetsugu.pdfStrongly...

  • 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