Quantum criticality and the phase diagram of the...

106
Quantum criticality and the phase diagram of the cuprates HARVARD Talk online: sachdev.physics.harvard.edu Monday, November 2, 2009

Transcript of Quantum criticality and the phase diagram of the...

  • Quantum criticalityand the phase diagram of

    the cuprates

    HARVARD

    Talk online: sachdev.physics.harvard.edu

    Monday, November 2, 2009

  • Victor Galitski, MarylandRibhu Kaul, Harvard Kentucky

    Max Metlitski, HarvardEun Gook Moon, Harvard

    Yang Qi, HarvardCenke Xu, Harvard Santa Barbara

    HARVARDMonday, November 2, 2009

  • The cuprate superconductors

    Monday, November 2, 2009

  • Γ Γ

    Central ingredients in cuprate phase diagram: antiferromagnetism, superconductivity, and

    change in Fermi surface

    Monday, November 2, 2009

  • Antiferro-magnetism

    Fermi surface

    d-wavesupercon-ductivity

    Monday, November 2, 2009

  • N. E. Hussey, J. Phys: Condens. Matter 20, 123201 (2008)

    Crossovers in transport properties of hole-doped cuprates

    0 0.05 0.1 0.15 0.2 0.25 0.3

    (K)

    Hole doping

    2

    + 2

    or

    FL?

    coh?

    ( )S-shaped

    *

    -wave SC

    (1 < < 2)AFM

    upturnsin ( )

    Monday, November 2, 2009

  • 0 0.05 0.1 0.15 0.2 0.25 0.3

    T(K)

    Hole doping x

    d-wave SC

    AFM

    Strange metal

    Crossovers in transport properties of hole-doped cuprates

    Pseudo-gap

    Monday, November 2, 2009

  • Antiferro-magnetism

    Fermi surface

    d-wavesupercon-ductivity

    Monday, November 2, 2009

  • Antiferro-magnetism

    Fermi surface

    d-wavesupercon-ductivity

    Monday, November 2, 2009

  • Ground state has long-range Néel order

    Square lattice antiferromagnet

    H =∑

    〈ij〉

    Jij !Si · !Sj

    Order parameter is a single vector field !ϕ = ηi !Siηi = ±1 on two sublattices

    〈!ϕ〉 #= 0 in Néel state.

    Monday, November 2, 2009

  • Square lattice antiferromagnet

    H =∑

    〈ij〉

    Jij !Si · !Sj

    J

    J/λ

    Weaken some bonds to induce spin entanglement in a new quantum phase

    Monday, November 2, 2009

  • Square lattice antiferromagnet

    H =∑

    〈ij〉

    Jij !Si · !Sj

    J

    J/λ

    Ground state is a “quantum paramagnet”with spins locked in valence bond singlets

    =1√2

    (∣∣∣↑↓〉−

    ∣∣∣ ↓↑〉)

    Monday, November 2, 2009

  • Pressure in TlCuCl3

    λλc

    Christian Ruegg, Bruce Normand, Masashige Matsumoto, Albert Furrer, Desmond McMorrow, Karl Kramer, Hans–Ulrich Gudel, Severian Gvasaliya,

    Hannu Mutka, and Martin Boehm, Phys. Rev. Lett. 100, 205701 (2008)Monday, November 2, 2009

  • Classicalspin

    waves

    Dilutetriplongas

    Quantumcritical

    Neel order Pressure in TlCuCl3Christian Ruegg et al. , Phys. Rev. Lett. 100, 205701 (2008)

    S. Sachdev and J. Ye, Phys. Rev. Lett. 69, 2411 (1992).

    Canonical quantum critical phase diagram of coupled-dimer antiferromagnet

    Monday, November 2, 2009

  • 0 0.05 0.1 0.15 0.2 0.25 0.3

    T(K)

    Hole doping x

    d-wave SC

    AFM

    Strange metal

    Crossovers in transport properties of hole-doped cuprates

    Pseudo-gap

    Monday, November 2, 2009

  • 0 0.05 0.1 0.15 0.2 0.25 0.3

    T(K)

    Hole doping x

    d-wave SC

    AFM

    Strange metal

    xm

    Crossovers in transport properties of hole-doped cuprates

    S. Sachdev and J. Ye, Phys. Rev. Lett. 69, 2411 (1992).

    A. J. Millis, Phys. Rev. B 48, 7183 (1993).

    C. M. Varma, Phys. Rev. Lett. 83, 3538 (1999).

    Pseudo-gap

    Strange metal: quantum criticality ofoptimal doping critical point at x = xm ?

    Monday, November 2, 2009

  • 0 0.05 0.1 0.15 0.2 0.25 0.3

    T(K)

    Hole doping x

    d-wave SC

    AFM

    xs

    Strange metal

    Only candidate quantum critical point observed at low T

    Spin density wave order presentbelow a quantum critical point at x = xswith xs ≈ 0.12 in the La series of cuprates

    Monday, November 2, 2009

  • Antiferro-magnetism

    Fermi surface

    d-wavesupercon-ductivity

    Monday, November 2, 2009

  • Antiferro-magnetism

    Fermi surface

    d-wavesupercon-ductivity

    Monday, November 2, 2009

  • “Large” Fermi surfaces in cuprates

    Γ

    Hole states

    occupied

    Electron states

    occupied

    Γ

    H0 = −∑

    i

  • Spin density wave theory

    The electron spin polarization obeys〈

    !S(r, τ)〉

    = !ϕ(r, τ)eiK·r

    where !ϕ is the spin density wave (SDW) order parameter,and K is the ordering wavevector. For simplicity, we con-sider K = (π, π).

    Monday, November 2, 2009

  • Spin density wave theory

    Spin density wave Hamiltonian

    Hsdw = !ϕ ·∑

    k,α,β

    c†k,α!σαβck+K,β

    Diagonalize H0 + Hsdw for !ϕ = (0, 0, ϕ)

    Ek± =εk + εk+K

    √(εk − εk+K

    2

    )+ ϕ2

    Monday, November 2, 2009

  • Increasing SDW order

    ΓΓΓ

    S. Sachdev, A. V. Chubukov, and A. Sokol, Phys. Rev. B 51, 14874 (1995). A. V. Chubukov and D. K. Morr, Physics Reports 288, 355 (1997).

    Γ

    Hole pockets

    Electron pockets

    Large Fermi surface breaks up intoelectron and hole pockets

    Hole-doped cuprates

    Monday, November 2, 2009

  • Increasing SDW order

    ΓΓΓ Γ

    Hole pockets

    Electron pockets

    Large Fermi surface breaks up intoelectron and hole pockets

    Hole-doped cuprates

    S. Sachdev, A. V. Chubukov, and A. Sokol, Phys. Rev. B 51, 14874 (1995). A. V. Chubukov and D. K. Morr, Physics Reports 288, 355 (1997).

    Monday, November 2, 2009

  • Spin density wave theory in hole-doped cuprates

    A. J. Millis and M. R. Norman, Physical Review B 76, 220503 (2007). N. Harrison, Physical Review Letters 102, 206405 (2009).

    Incommensurate order in YBa2Cu3O6+x

    ΓΓ

    Monday, November 2, 2009

  • Increasing SDW orderIncreasing SDW order

    ΓΓΓΓ

    Electron pockets

    Hole pockets

    Electron-doped cuprates

    Large Fermi surface breaks up intoelectron and hole pockets

    D. Senechal and A.-M. S. Tremblay, Physical Review Letters 92, 126401 (2004)J. Lin, and A. J. Millis, Physical Review B 72, 214506 (2005).

    Monday, November 2, 2009

  • Nd2−xCexCuO4

    T. Helm, M. V. Kartsovnik, M. Bartkowiak, N. Bittner,

    M. Lambacher, A. Erb, J. Wosnitza, and R. Gross,

    Phys. Rev. Lett. 103, 157002 (2009).

    Increasing SDW orderIncreasing SDW order

    Quantum oscillations

    Monday, November 2, 2009

  • Nature 450, 533 (2007)

    Quantum oscillations

    Monday, November 2, 2009

  • Nature 450, 533 (2007)

    Quantum oscillations

    Monday, November 2, 2009

  • FluctuatingFermi

    pocketsLargeFermi

    surface

    StrangeMetal

    Spin density wave (SDW)

    Theory of quantum criticality in the cuprates

    Underlying SDW ordering quantum critical pointin metal at x = xm

    Increasing SDW orderIncreasing SDW order

    Monday, November 2, 2009

  • Broken rotational symmetry in the pseudogap phase of a high-Tc superconductorR. Daou, J. Chang, David LeBoeuf, Olivier Cyr-Choiniere, Francis Laliberte, Nicolas Doiron-Leyraud, B. J. Ramshaw, Ruixing Liang, D. A. Bonn, W. N. Hardy, and Louis TailleferarXiv: 0909.4430

    S. A. Kivelson, E. Fradkin, and V. J. Emery, Nature 393, 550 (1998).

    Monday, November 2, 2009

    http://arxiv.org/find/cond-mat/1/au:+Daou_R/0/1/0/all/0/1http://arxiv.org/find/cond-mat/1/au:+Daou_R/0/1/0/all/0/1http://arxiv.org/find/cond-mat/1/au:+Chang_J/0/1/0/all/0/1http://arxiv.org/find/cond-mat/1/au:+Chang_J/0/1/0/all/0/1http://arxiv.org/find/cond-mat/1/au:+LeBoeuf_D/0/1/0/all/0/1http://arxiv.org/find/cond-mat/1/au:+LeBoeuf_D/0/1/0/all/0/1http://arxiv.org/find/cond-mat/1/au:+Cyr_Choiniere_O/0/1/0/all/0/1http://arxiv.org/find/cond-mat/1/au:+Cyr_Choiniere_O/0/1/0/all/0/1http://arxiv.org/find/cond-mat/1/au:+Cyr_Choiniere_O/0/1/0/all/0/1http://arxiv.org/find/cond-mat/1/au:+Cyr_Choiniere_O/0/1/0/all/0/1http://arxiv.org/find/cond-mat/1/au:+Laliberte_F/0/1/0/all/0/1http://arxiv.org/find/cond-mat/1/au:+Laliberte_F/0/1/0/all/0/1http://arxiv.org/find/cond-mat/1/au:+Doiron_Leyraud_N/0/1/0/all/0/1http://arxiv.org/find/cond-mat/1/au:+Doiron_Leyraud_N/0/1/0/all/0/1http://arxiv.org/find/cond-mat/1/au:+Doiron_Leyraud_N/0/1/0/all/0/1http://arxiv.org/find/cond-mat/1/au:+Doiron_Leyraud_N/0/1/0/all/0/1http://arxiv.org/find/cond-mat/1/au:+Ramshaw_B/0/1/0/all/0/1http://arxiv.org/find/cond-mat/1/au:+Ramshaw_B/0/1/0/all/0/1http://arxiv.org/find/cond-mat/1/au:+Liang_R/0/1/0/all/0/1http://arxiv.org/find/cond-mat/1/au:+Liang_R/0/1/0/all/0/1http://arxiv.org/find/cond-mat/1/au:+Bonn_D/0/1/0/all/0/1http://arxiv.org/find/cond-mat/1/au:+Bonn_D/0/1/0/all/0/1http://arxiv.org/find/cond-mat/1/au:+Hardy_W/0/1/0/all/0/1http://arxiv.org/find/cond-mat/1/au:+Hardy_W/0/1/0/all/0/1http://arxiv.org/find/cond-mat/1/au:+Taillefer_L/0/1/0/all/0/1http://arxiv.org/find/cond-mat/1/au:+Taillefer_L/0/1/0/all/0/1

  • Nematic order in YBCOV. Hinkov, D. Haug, B. Fauqué, P. Bourges, Y. Sidis, A. Ivanov, C. Bernhard, C. T. Lin, and B. Keimer , Science 319, 597 (2008)

    Monday, November 2, 2009

  • Nematic order in YBCO

    V. Hinkov, D. Haug, B. Fauqué, P. Bourges, Y. Sidis, A. Ivanov, C. Bernhard, C. T. Lin, and B. Keimer , Science 319, 597 (2008)

    Monday, November 2, 2009

  • Evidence for connection between linear resistivity andstripe-ordering in a cuprate with a low Tc

    Linear temperature dependence of resistivity and change in the Fermi surface at the pseudogap critical point of a high-Tc superconductorR. Daou, Nicolas Doiron-Leyraud, David LeBoeuf, S. Y. Li, Francis Laliberté, Olivier Cyr-Choinière, Y. J. Jo, L. Balicas, J.-Q. Yan, J.-S. Zhou, J. B. Goodenough & Louis Taillefer, Nature Physics 5, 31 - 34 (2009)

    Magnetic field of upto 35 T

    used to suppress superconductivity

    Monday, November 2, 2009

    http://www.nature.com.ezp-prod1.hul.harvard.edu/nphys/journal/v5/n1/full/nphys1109.htmlhttp://www.nature.com.ezp-prod1.hul.harvard.edu/nphys/journal/v5/n1/full/nphys1109.htmlhttp://www.nature.com.ezp-prod1.hul.harvard.edu/nphys/journal/v5/n1/full/nphys1109.htmlhttp://www.nature.com.ezp-prod1.hul.harvard.edu/nphys/journal/v5/n1/full/nphys1109.html

  • FluctuatingFermi

    pocketsLargeFermi

    surface

    StrangeMetal

    Spin density wave (SDW)

    Theory of quantum criticality in the cuprates

    Underlying SDW ordering quantum critical pointin metal at x = xm

    Increasing SDW orderIncreasing SDW order

    Monday, November 2, 2009

  • Antiferro-magnetism

    Fermi surface

    d-wavesupercon-ductivity

    Monday, November 2, 2009

  • Antiferro-magnetism

    Fermi surface

    d-wavesupercon-ductivity

    Monday, November 2, 2009

  • FluctuatingFermi

    pocketsLargeFermi

    surface

    StrangeMetal

    Spin density wave (SDW)

    Theory of quantum criticality in the cuprates

    Underlying SDW ordering quantum critical pointin metal at x = xm

    Increasing SDW orderIncreasing SDW order

    Monday, November 2, 2009

  • LargeFermi

    surface

    StrangeMetal

    Spin density wave (SDW)

    d-wavesuperconductor

    Small Fermipockets with

    pairing fluctuations

    Theory of quantum criticality in the cuprates

    Onset of d-wave superconductivityhides the critical point x = xm

    Fluctuating, paired Fermi

    pockets

    Monday, November 2, 2009

  • LargeFermi

    surface

    StrangeMetal

    Spin density wave (SDW)

    d-wavesuperconductor

    Small Fermipockets with

    pairing fluctuationsE. Demler, S. Sachdevand Y. Zhang, Phys.Rev. Lett. 87,067202 (2001).

    E. G. Moon andS. Sachdev, Phy.Rev. B 80, 035117(2009)

    Theory of quantum criticality in the cuprates

    Competition between SDW order and superconductivitymoves the actual quantum critical point to x = xs < xm.

    Fluctuating, paired Fermi

    pockets

    Monday, November 2, 2009

  • LargeFermi

    surface

    StrangeMetal

    Spin density wave (SDW)

    d-wavesuperconductor

    Small Fermipockets with

    pairing fluctuationsE. Demler, S. Sachdevand Y. Zhang, Phys.Rev. Lett. 87,067202 (2001).

    E. G. Moon andS. Sachdev, Phy.Rev. B 80, 035117(2009)

    Theory of quantum criticality in the cuprates

    Competition between SDW order and superconductivitymoves the actual quantum critical point to x = xs < xm.

    Fluctuating, paired Fermi

    pockets

    Monday, November 2, 2009

  • Small Fermipockets with

    pairing fluctuationsLargeFermi

    surface

    StrangeMetal

    Magneticquantumcriticality

    Spin density wave (SDW)

    Spin gap

    Thermallyfluctuating

    SDW

    d-wavesuperconductor

    Theory of quantum criticality in the cuprates

    Competition between SDW order and superconductivitymoves the actual quantum critical point to x = xs < xm.

    Fluctuating, paired Fermi

    pockets

    E. Demler, S. Sachdevand Y. Zhang, Phys.Rev. Lett. 87,067202 (2001).

    E. G. Moon andS. Sachdev, Phy.Rev. B 80, 035117(2009)

    Monday, November 2, 2009

  • E. Demler, S. Sachdevand Y. Zhang, Phys.Rev. Lett. 87,067202 (2001).

    E. G. Moon andS. Sachdev, Phy.Rev. B 80, 035117(2009)

    Small Fermipockets with

    pairing fluctuationsLargeFermi

    surface

    StrangeMetal

    Magneticquantumcriticality

    Spin density wave (SDW)

    Spin gap

    Thermallyfluctuating

    SDW

    d-wavesuperconductor

    Theory of quantum criticality in the cuprates

    Competition between SDW order and superconductivitymoves the actual quantum critical point to x = xs < xm.

    Criticality of the coupled dimer antiferromagnet at x=xs

    Classicalspin

    waves

    Dilutetriplon

    gas

    Quantumcritical

    Neel order

    Fluctuating, paired Fermi

    pockets

    Monday, November 2, 2009

  • Small Fermipockets with

    pairing fluctuationsLargeFermi

    surface

    StrangeMetal

    Magneticquantumcriticality

    Spin density wave (SDW)

    Spin gap

    Thermallyfluctuating

    SDW

    d-wavesuperconductor

    Theory of quantum criticality in the cuprates

    Competition between SDW order and superconductivitymoves the actual quantum critical point to x = xs < xm.

    Criticality of the topological change in Fermi surface at x=xm

    E. Demler, S. Sachdevand Y. Zhang, Phys.Rev. Lett. 87,067202 (2001).

    E. G. Moon andS. Sachdev, Phy.Rev. B 80, 035117(2009)

    Fluctuating, paired Fermi

    pockets

    Increasing SDW orderIncreasing SDW order

    Monday, November 2, 2009

  • Small Fermipockets with

    pairing fluctuationsLargeFermi

    surface

    StrangeMetal

    Magneticquantumcriticality

    Spin gapThermallyfluctuating

    SDW

    d-waveSC

    T

    Fluctuating, paired Fermi

    pockets

    Monday, November 2, 2009

  • Small Fermipockets with

    pairing fluctuationsLargeFermi

    surface

    StrangeMetal

    Magneticquantumcriticality

    Spin gapThermallyfluctuating

    SDW

    d-waveSC

    T

    Fluctuating, paired Fermi

    pockets

    E. Demler, S. Sachdev and Y. Zhang,Phys. Rev. Lett. 87, 067202 (2001).

    Monday, November 2, 2009

  • Small Fermipockets with

    pairing fluctuationsLargeFermi

    surface

    StrangeMetal

    Magneticquantumcriticality

    Spin gapThermallyfluctuating

    SDW

    d-waveSC

    T

    Fluctuating, paired Fermi

    pockets

    E. Demler, S. Sachdev and Y. Zhang,Phys. Rev. Lett. 87, 067202 (2001).

    H

    SC

    M"Normal"

    (Large Fermisurface)

    SDW(Small Fermi

    pockets)

    SC+SDWHc2

    Monday, November 2, 2009

  • Small Fermipockets with

    pairing fluctuationsLargeFermi

    surface

    StrangeMetal

    Magneticquantumcriticality

    Spin gapThermallyfluctuating

    SDW

    d-waveSC

    T

    Fluctuating, paired Fermi

    pockets

    E. Demler, S. Sachdev and Y. Zhang,Phys. Rev. Lett. 87, 067202 (2001).

    H

    SC

    M"Normal"

    (Large Fermisurface)

    SDW(Small Fermi

    pockets)

    SC+SDWHc2

    Quantum oscillations

    Monday, November 2, 2009

  • Small Fermipockets with

    pairing fluctuationsLargeFermi

    surface

    StrangeMetal

    Magneticquantumcriticality

    Spin gapThermallyfluctuating

    SDW

    d-waveSC

    T

    Fluctuating, paired Fermi

    pockets

    E. Demler, S. Sachdev and Y. Zhang,Phys. Rev. Lett. 87, 067202 (2001).

    H

    SC

    M"Normal"

    (Large Fermisurface)

    SDW(Small Fermi

    pockets)

    SC+SDW

    Hsdw

    Monday, November 2, 2009

  • Small Fermipockets with

    pairing fluctuationsLargeFermi

    surface

    StrangeMetal

    Magneticquantumcriticality

    Spin gapThermallyfluctuating

    SDW

    d-waveSC

    T

    Fluctuating, paired Fermi

    pockets

    E. Demler, S. Sachdev and Y. Zhang,Phys. Rev. Lett. 87, 067202 (2001).

    H

    SC

    M"Normal"

    (Large Fermisurface)

    SDW(Small Fermi

    pockets)

    SC+SDW

    Hsdw

    Monday, November 2, 2009

  • Small Fermipockets with

    pairing fluctuationsLargeFermi

    surface

    StrangeMetal

    Magneticquantumcriticality

    Spin gapThermallyfluctuating

    SDW

    d-waveSC

    T

    Fluctuating, paired Fermi

    pockets

    E. Demler, S. Sachdev and Y. Zhang,Phys. Rev. Lett. 87, 067202 (2001).

    H

    SC

    M"Normal"

    (Large Fermisurface)

    SDW(Small Fermi

    pockets)

    SC+SDW

    Hsdw

    Neutron scattering & muon resonance

    Monday, November 2, 2009

  • B. Lake, G. Aeppli, K. N. Clausen, D. F. McMorrow, K. Lefmann, N. E. Hussey, N. Mangkorntong, M. Nohara, H. Takagi, T. E. Mason, and A. SchröderScience 291, 1759 (2001).

    B. Lake, H. M. Rønnow, N. B. Christensen, G. Aeppli, K. Lefmann, D. F. McMorrow, P. Vorderwisch, P. Smeibidl, N. Mangkorntong, T. Sasagawa, M. Nohara, H. Takagi, and T. E. Mason, Nature 415, 299 (2002)

    Monday, November 2, 2009

  • Monday, November 2, 2009

  • J. Chang, N. B. Christensen, Ch. Niedermayer, K. Lefmann,

    H. M. Roennow, D. F. McMorrow, A. Schneidewind, P. Link, A. Hiess,

    M. Boehm, R. Mottl, S. Pailhes, N. Momono, M. Oda, M. Ido, and

    J. Mesot, Phys. Rev. Lett. 102, 177006

    (2009).

    J. Chang, Ch. Niedermayer, R. Gilardi, N.B. Christensen, H.M. Ronnow,

    D.F. McMorrow, M. Ay, J. Stahn, O. Sobolev, A. Hiess, S. Pailhes, C. Baines, N. Momono,

    M. Oda, M. Ido, and J. Mesot, Physical Review B 78, 104525 (2008).

    Monday, November 2, 2009

  • 0.3 0.4 0.5 0.6 0.7

    0

    500

    1000

    1500

    0.3 0.4 0.5 0.6 0.7

    0

    500

    1000

    1500

    0.4 0.5 0.6

    1500

    1600

    1700

    0.3 0.4 0.5 0.6 0.7

    -500

    0

    500

    0.4 0.5 0.6

    0.4

    0.5

    0.6

    0.7

    QH (r.l.u.)

    I(14.9T,2K) ! I(0T,2K)

    (d)

    (c)

    H = 14.9T

    (b)

    (a)

    ba

    I (14.9T, 2K) ! I (0T, 2K)

    H = 0T

    Inte

    nsity (

    counts

    / ~

    10 m

    in)

    QH (r.l.u.)

    H = 14.9T

    2K

    50K

    QH (r.l.u.)

    QK (r.l.u.)

    Inte

    nsity (

    counts

    / ~

    1 m

    in)

    Inte

    nsity (

    counts

    / ~

    10 m

    in)

    Inte

    nsity (

    counts

    / ~

    10 m

    in)

    QK

    QH

    0.3 0.4 0.5 0.6 0.7

    20

    40

    60

    80

    100

    (b)

    (b)(a)

    Inte

    nsity (

    a.u

    .)

    H = 0T

    H = 13.5T

    Inte

    nsity (

    counts

    / ~

    12 m

    in)

    QH (r.l.u.)

    H = 0T

    H = 14.9T

    0 1 2 3 4

    100

    1000

    T = 2K

    T = 2K

    Energy E (meV)

    D. Haug, V. Hinkov, A. Suchaneck, D. S. Inosov, N. B. Christensen, Ch. Niedermayer, P. Bourges, Y. Sidis, J. T. Park, A. Ivanov, C. T. Lin, J. Mesot, and B. Keimer, Phys. Rev. Lett. 103, 017001 (2009)

    Monday, November 2, 2009

  • Small Fermipockets with

    pairing fluctuationsLargeFermi

    surface

    StrangeMetal

    Magneticquantumcriticality

    Spin gapThermallyfluctuating

    SDW

    d-waveSC

    T

    H

    SC

    M"Normal"

    (Large Fermisurface)

    SDW(Small Fermi

    pockets)

    SC+SDW

    Fluctuating, paired Fermi

    pockets

    Monday, November 2, 2009

  • Small Fermipockets with

    pairing fluctuationsLargeFermi

    surface

    StrangeMetal

    Magneticquantumcriticality

    Spin gapThermallyfluctuating

    SDW

    d-waveSC

    T

    H

    SC

    M"Normal"

    (Large Fermisurface)

    SDW(Small Fermi

    pockets)

    SC+SDW

    Theory of theonset of d-wave

    superconductivityfrom a

    large Fermi surface

    Fluctuating, paired Fermi

    pockets

    Monday, November 2, 2009

  • Increasing SDW order

    ΓΓΓ Γ

    Spin-fluctuation exchange theory of d-wave superconductivity in the cuprates

    D. J. Scalapino, E. Loh, and J. E. Hirsch, Phys. Rev. B 34, 8190 (1986) K. Miyake, S. Schmitt-Rink, and C. M. Varma, Phys. Rev. B 34, 6554 (1986)

    Fermions at the large Fermi surface exchangefluctuations of the SDW order parameter !ϕ.

    Monday, November 2, 2009

  • Increasing SDW order

    ++_

    _

    Γ

    d-wave pairing of the large Fermi surface

    〈ck↑c−k↓〉 ∝ ∆k = ∆0(cos(kx)− cos(ky))

    K

    D. J. Scalapino, E. Loh, and J. E. Hirsch, Phys. Rev. B 34, 8190 (1986) K. Miyake, S. Schmitt-Rink, and C. M. Varma, Phys. Rev. B 34, 6554 (1986)

    Monday, November 2, 2009

  • :

    • Tc increases upon approaching the SDW transition.

    • Pairing from SDW fluctuations: SDW and SC ordersdo not compete, but attract each other.

    Ar. Abanov, A. V. Chubukov and J. Schmalian, Advances in Physics 52, 119 (2003).

    Approaching the onset of antiferromagnetism in the spin-fluctuation theory

    Monday, November 2, 2009

  • :

    • Tc increases upon approaching the SDW transition.

    • Pairing from SDW fluctuations: SDW and SC ordersdo not compete, but attract each other.

    Ar. Abanov, A. V. Chubukov and J. Schmalian, Advances in Physics 52, 119 (2003).

    Approaching the onset of antiferromagnetism in the spin-fluctuation theory

    Monday, November 2, 2009

  • H

    SC

    M"Normal"

    (Large Fermisurface)

    SDW(Small Fermi

    pockets)

    SC+SDW

    Small Fermipockets with

    pairing fluctuationsLargeFermi

    surface

    StrangeMetal

    Magneticquantumcriticality

    Spin gapThermallyfluctuating

    SDW

    d-waveSC

    T

    Fluctuating, paired Fermi

    pockets

    Monday, November 2, 2009

  • H

    SC

    M"Normal"

    (Large Fermisurface)

    SDW(Small Fermi

    pockets)

    SC+SDW

    Small Fermipockets with

    pairing fluctuationsLargeFermi

    surface

    StrangeMetal

    Magneticquantumcriticality

    Spin gapThermallyfluctuating

    SDW

    d-waveSC

    T

    Theory of theonset of d-wave

    superconductivityfrom small

    Fermi pockets

    Fluctuating, paired Fermi

    pockets

    Monday, November 2, 2009

  • H

    SC

    M"Normal"

    (Large Fermisurface)

    SDW(Small Fermi

    pockets)

    SC+SDW

    Small Fermipockets with

    pairing fluctuationsLargeFermi

    surface

    StrangeMetal

    Magneticquantumcriticality

    Spin gapThermallyfluctuating

    SDW

    d-waveSC

    T

    Fluctuating, paired Fermi

    pockets

    Physics ofcompetition:

    d-wave SC andSDW “eat up’same pieces ofthe large Fermi

    surface.

    B. Kyung, J.-S. Landry, andA.-M. S. Tremblay,Phys. Rev. B 68, 174502 (2003).

    Monday, November 2, 2009

  • Begin with SDW ordered state, and rotate to a framepolarized along the local orientation of the SDW order !̂ϕ

    (c↑c↓

    )= R

    (ψ+ψ−

    ); R† !̂ϕ · !σR = σz ; R†R = 1

    Theory of underdoped cuprates

    Increasing SDW order

    ΓΓΓ Γ

    H. J. Schulz, Physical Review Letters 65, 2462 (1990)B. I. Shraiman and E. D. Siggia, Physical Review Letters 61, 467 (1988).

    J. R. Schrieffer, Journal of Superconductivity 17, 539 (2004)Monday, November 2, 2009

  • Theory of underdoped cuprates

    With R =(

    z↑ −z∗↓z↓ z∗↑

    )or !̂ϕ = z∗α!σαβzβ

    the theory is invariant under

    zα → eiθzα ; ψ+ → e−iθψ+ ; ψ− → eiθψ.−

    We obtain a U(1) gauge theory of

    • bosonic neutral spinons zα;

    • spinless, charged fermions ψ±;

    • an emergent U(1) gauge field Aµ.

    X.-G. Wen, Phys. Rev.B 39, 7223 (1989).P. A. Lee, Phys. Rev. Lett.63, 680 (1989).R. Shankar, Phys. Rev.Lett. 63, 203 (1989).L. B. Ioffe and P. B. Wieg-mann, Phys. Rev. Lett.65, 653 (1990).R. K. Kaul, A. Kolezhuk,M. Levin, S. Sachdev, andT. Senthil, Phys. Rev. B75 , 235122 (2007).

    Monday, November 2, 2009

  • Increasing SDW order

    Γ

    Pairing across Tc

    Focus on pairing near (π, 0), (0, π), where ψ± ≡ g±,and the electron operators are

    (c1↑c1↓

    )= Rz

    (g+g−

    );

    (c2↑c2↓

    )= Rz

    (g+−g−

    )

    Rz ≡(

    z↑ −z∗↓z↓ z∗↑

    ).

    Electron c1α,spinless fermion g±

    Electron c2α,spinless fermion g±

    R. K. Kaul, M. Metlitksi, S. Sachdev, and Cenke Xu, Phys. Rev. B 78, 045110 (2008).

    Monday, November 2, 2009

  • Increasing SDW order

    Γ

    Pairing across Tc

    Focus on pairing near (π, 0), (0, π), where ψ± ≡ g±,and the electron operators are

    (c1↑c1↓

    )= Rz

    (g+g−

    );

    (c2↑c2↓

    )= Rz

    (g+−g−

    )

    Rz ≡(

    z↑ −z∗↓z↓ z∗↑

    ).

    Electron c1α,spinless fermion g±

    Electron c2α,spinless fermion g±

    R. K. Kaul, M. Metlitksi, S. Sachdev, and Cenke Xu, Phys. Rev. B 78, 045110 (2008).

    Monday, November 2, 2009

  • Fluctuating pocket theory forelectrons near (0, π) and (π, 0)

    Attractive gauge forces lead to simple s-wave pairing of the g±

    〈g+g−〉 = ∆

    For the physical electron operators, this pairing implies

    〈c1↑c1↓〉 = ∆〈|zα|2

    〈c2↑c2↓〉 = −∆〈|zα|2

    i.e. d-wave pairing !

    R. K. Kaul, M. Metlitksi, S. Sachdev, and Cenke Xu, Phys. Rev. B 78, 045110 (2008).

    Monday, November 2, 2009

  • Increasing SDW order

    ++_

    _

    Γ

    Strong pairing of the g± electron pockets

    〈g+g−〉 = ∆

    Monday, November 2, 2009

  • 〈f+1(k)f−1(−k)〉 ∼ (kx − ky)J〈g+g−〉;〈f+2(k)f−2(−k)〉 ∼ (kx + ky)J〈g+g−〉;〈f+1(k)f−2(−k)〉 = 0,

    Increasing SDW order

    f±v ++_

    _

    Γ

    Weak pairing of the f± hole pockets

    Monday, November 2, 2009

  • V. B. Geshkenbein, L. B. Ioffe, and A. I. Larkin, Phys. Rev. B 55, 3173 (1997).

    Γ

    -2e bosons at antinodes,+e fermion “arcs” at nodes,and proximity “Josephson”

    coupling

    Similar features in our theoryV. Galitski and S. Sachdev, Physical Review B 79, 134512 (2009).

    Monday, November 2, 2009

  • • d-wave superconductivity.

    • Nodal-anti-nodal dichotomy: strong pairing near (π, 0),(0, π), and weak pairing near zone diagonals.

    • Tc decreases as spin correlation increases (competingorder effect).

    • Shift in quantum critical point of SDW ordering:gauge fluctuations are stronger in the superconduc-tor.

    • After onset of superconductivity, monopoles condenseand lead to confinement and nematic and/orvalence bond solid (VBS) order.

    Features of superconductivity

    V. Galitski and S. Sachdev, Physical Review B 79, 134512 (2009).

    Monday, November 2, 2009

  • • d-wave superconductivity.

    • Nodal-anti-nodal dichotomy: strong pairing near (π, 0),(0, π), and weak pairing near zone diagonals.

    • Tc decreases as spin correlation increases (competingorder effect).

    • Shift in quantum critical point of SDW ordering:gauge fluctuations are stronger in the superconduc-tor.

    • After onset of superconductivity, monopoles condenseand lead to confinement and nematic and/orvalence bond solid (VBS) order.

    Features of superconductivity

    Eun Gook Moon and S. Sachdev, Physical Review B 80, 035117 (2009).

    Monday, November 2, 2009

  • • d-wave superconductivity.

    • Nodal-anti-nodal dichotomy: strong pairing near (π, 0),(0, π), and weak pairing near zone diagonals.

    • Tc decreases as spin correlation increases (competingorder effect).

    • Shift in quantum critical point of SDW ordering:gauge fluctuations are stronger in the superconduc-tor.

    • After onset of superconductivity, monopoles condenseand lead to confinement and nematic and/orvalence bond solid (VBS) order.

    Features of superconductivity

    R. K. Kaul, M. Metlitksi, S. Sachdev, and Cenke Xu, Phys. Rev. B 78, 045110 (2008).Monday, November 2, 2009

  • H

    SC

    M"Normal"

    (Large Fermisurface)

    SDW(Small Fermi

    pockets)

    SC+SDW

    Small Fermipockets with

    pairing fluctuationsLargeFermi

    surface

    StrangeMetal

    Magneticquantumcriticality

    Spin gapThermallyfluctuating

    SDW

    d-waveSC

    T

    Fluctuating, paired Fermi

    pockets

    Monday, November 2, 2009

  • H

    SC

    M"Normal"

    (Large Fermisurface)

    SDW(Small Fermi

    pockets)

    SC+SDW

    Small Fermipockets with

    pairing fluctuationsLargeFermi

    surface

    StrangeMetal

    d-waveSC

    T

    Tsdw

    R. K. Kaul, M. Metlitksi, S. Sachdev, and Cenke Xu, Physical Review B 78, 045110 (2008).

    Fluctuating, paired Fermi

    pockets

    VBSand/ornematic

    Monday, November 2, 2009

  • H

    SC

    M"Normal"

    (Large Fermisurface)

    SDW(Small Fermi

    pockets)

    SC+SDW

    Small Fermipockets with

    pairing fluctuationsLargeFermi

    surface

    StrangeMetal

    d-waveSC

    T

    Tsdw

    R. K. Kaul, M. Metlitksi, S. Sachdev, and Cenke Xu, Physical Review B 78, 045110 (2008).

    Fluctuating, paired Fermi

    pockets

    VBSand/ornematic

    Monday, November 2, 2009

  • H

    SC

    M"Normal"

    (Large Fermisurface)

    SDW(Small Fermi

    pockets)

    SC+SDW

    Small Fermipockets with

    pairing fluctuationsLargeFermi

    surface

    StrangeMetal

    d-waveSC

    T

    Tsdw

    R. K. Kaul, M. Metlitksi, S. Sachdev, and Cenke Xu, Physical Review B 78, 045110 (2008).

    Fluctuating, paired Fermi

    pockets

    VBSand/ornematic

    Monday, November 2, 2009

  • Quantum phase transitions in metal

    〈z〉 #= 0 ; 〈N〉 #= 0SDW order

    small Fermi pockets 〈z〉 #= 0 ; 〈N〉 = 0Fermi liquidlarge Fermi surface

    〈z〉 = 0 ; 〈N〉 #= 0small critical Fermi pockets

    gapless U(1) photon

    〈z〉 = 0 ; 〈N〉 = 0large critical Fermi surface

    gapless SU(2) photons

    S. Sachdev, M. A. Metlitski, Y. Qi, and C. Xu, Physical Review B 80, 155129 (2009)

    Increasing SDW orderIncreasing SDW order

    Monday, November 2, 2009

  • Quantum phase transitions in metal

    〈z〉 #= 0 ; 〈N〉 #= 0SDW order

    small Fermi pockets 〈z〉 #= 0 ; 〈N〉 = 0Fermi liquidlarge Fermi surface

    〈z〉 = 0 ; 〈N〉 #= 0small critical Fermi pockets

    gapless U(1) photon

    〈z〉 = 0 ; 〈N〉 = 0large critical Fermi surface

    gapless SU(2) photons

    S. Sachdev, M. A. Metlitski, Y. Qi, and C. Xu, Physical Review B 80, 155129 (2009)

    Fermi liquid phases

    Increasing SDW orderIncreasing SDW order

    Monday, November 2, 2009

  • Quantum phase transitions in metal

    〈z〉 #= 0 ; 〈N〉 #= 0SDW order

    small Fermi pockets 〈z〉 #= 0 ; 〈N〉 = 0Fermi liquidlarge Fermi surface

    〈z〉 = 0 ; 〈N〉 #= 0small critical Fermi pockets

    gapless U(1) photon

    〈z〉 = 0 ; 〈N〉 = 0large critical Fermi surface

    gapless SU(2) photons

    S. Sachdev, M. A. Metlitski, Y. Qi, and C. Xu, Physical Review B 80, 155129 (2009)

    non Fermi liquid phases

    Increasing SDW orderIncreasing SDW order

    Monday, November 2, 2009

  • Quantum phase transitions in metal

    〈z〉 #= 0 ; 〈N〉 #= 0SDW order

    small Fermi pockets 〈z〉 #= 0 ; 〈N〉 = 0Fermi liquidlarge Fermi surface

    〈z〉 = 0 ; 〈N〉 #= 0small critical Fermi pockets

    gapless U(1) photon

    〈z〉 = 0 ; 〈N〉 = 0large critical Fermi surface

    gapless SU(2) photons

    S. Sachdev, M. A. Metlitski, Y. Qi, and C. Xu, Physical Review B 80, 155129 (2009)

    Hertz-Millis-Moriya theory

    Increasing SDW orderIncreasing SDW order

    Monday, November 2, 2009

  • Max Metlitski

    M. Metlitski and S. Sachdev, to appear

    Ar. Abanov, A.V. Chubukov, and J. Schmalian, Advances in Physics 52, 119 (2003)

    Sung-Sik Lee, arXiv:0905.4532.

    Monday, November 2, 2009

  • Hertz-Millis-Moriya (HMM) theory: mean field theory

    + Gaussian fluctuations of overdamped paramagnons.

    Monday, November 2, 2009

  • Theory for the onset of spin density wave order in metals is strongly coupled in two dimensions

    Monday, November 2, 2009

  • Start from the “spin-fermion” model

    Z =∫DcαD!ϕ exp (−S)

    S =∫

    dτ∑

    k

    c†kα

    (∂

    ∂τ− εk

    )ckα

    −λ∫

    dτ∑

    i

    c†iα!ϕi · !σαβciβeiK·ri

    +∫

    dτd2r

    [(∂r !ϕ)

    2 +1c2

    (∂τ !ϕ)2]

    Monday, November 2, 2009

  • ! = 1

    ! = 2

    ! = 4

    ! = 3

    Low energy fermionsψ!1α, ψ!2α

    " = 1, . . . , 4

    Lf = ψ!†1α(ζ∂τ − iv!1 · ∇r

    )ψ!1α + ψ

    !†2α

    (ζ∂τ − iv!2 · ∇r

    )ψ!2α

    v!=11 = (vx, vy), v!=12 = (−vx, vy)Monday, November 2, 2009

  • Lf = ψ!†1α(ζ∂τ − iv!1 · ∇r

    )ψ!1α + ψ

    !†2α

    (ζ∂τ − iv!2 · ∇r

    )ψ!2α

    v1 v2

    ψ2 fermionsoccupied

    ψ1 fermionsoccupied

    Monday, November 2, 2009

  • Order parameter: Lϕ =12

    (∇r !ϕ)2 +s

    2!ϕ2 +

    u

    4!ϕ4

    Lf = ψ!†1α(ζ∂τ − iv!1 · ∇r

    )ψ!1α + ψ

    !†2α

    (ζ∂τ − iv!2 · ∇r

    )ψ!2α

    Monday, November 2, 2009

  • Order parameter: Lϕ =12

    (∇r !ϕ)2 +s

    2!ϕ2 +

    u

    4!ϕ4

    “Yukawa” coupling: Lc = −!ϕ ·(ψ!†1α!σαβψ

    !2β + ψ

    !†2α!σαβψ

    !1β

    )

    Lf = ψ!†1α(ζ∂τ − iv!1 · ∇r

    )ψ!1α + ψ

    !†2α

    (ζ∂τ − iv!2 · ∇r

    )ψ!2α

    Monday, November 2, 2009

  • Order parameter: Lϕ =12

    (∇r !ϕ)2 +s

    2!ϕ2 +

    u

    4!ϕ4

    “Yukawa” coupling: Lc = −!ϕ ·(ψ!†1α!σαβψ

    !2β + ψ

    !†2α!σαβψ

    !1β

    )

    HMM theoryIntegrate out fermions and obtain non-local corrections to Lϕ

    Lϕ =12

    !ϕ2[q2 + γ|ω|

    ]/2 ; γ =

    2πvxvy

    Exponent z = 2 and mean-field criticality (upto logarithms)

    Lf = ψ!†1α(ζ∂τ − iv!1 · ∇r

    )ψ!1α + ψ

    !†2α

    (ζ∂τ − iv!2 · ∇r

    )ψ!2α

    Monday, November 2, 2009

  • Order parameter: Lϕ =12

    (∇r !ϕ)2 +s

    2!ϕ2 +

    u

    4!ϕ4

    “Yukawa” coupling: Lc = −!ϕ ·(ψ!†1α!σαβψ

    !2β + ψ

    !†2α!σαβψ

    !1β

    )

    But, higher order terms contain an infinitenumber of marginal couplings . . . . . .

    Ar. Abanov and A.V. Chubukov, Phys. Rev. Lett. 93, 255702 (2004).

    Integrate out fermions and obtain non-local corrections to Lϕ

    Lϕ =12

    !ϕ2[q2 + γ|ω|

    ]/2 ; γ =

    2πvxvy

    Exponent z = 2 and mean-field criticality (upto logarithms)

    Lf = ψ!†1α(ζ∂τ − iv!1 · ∇r

    )ψ!1α + ψ

    !†2α

    (ζ∂τ − iv!2 · ∇r

    )ψ!2α

    HMM theory

    Monday, November 2, 2009

  • Order parameter: Lϕ =12

    (∇r !ϕ)2 +s

    2!ϕ2 +

    u

    4!ϕ4

    “Yukawa” coupling: Lc = −!ϕ ·(ψ!†1α!σαβψ

    !2β + ψ

    !†2α!σαβψ

    !1β

    )

    Perform RG on both fermions and !ϕ,using a local field theory.

    Lf = ψ!†1α(ζ∂τ − iv!1 · ∇r

    )ψ!1α + ψ

    !†2α

    (ζ∂τ − iv!2 · ∇r

    )ψ!2α

    Monday, November 2, 2009

  • !ϕ propagator

    fermion propagator

    1N

    1(q2 + γ|ω|)

    1

    v · q + iζω + i 1N√

    γv

    √ωF

    (v2q2

    ω

    )

    New infra-red singularities as ζ → 0 at higher loops(Breakdown of Migdal-Eliashberg)

    Monday, November 2, 2009

  • !ϕ propagator

    fermion propagator

    Dangerous

    1N

    1(q2 + γ|ω|)

    1

    v · q + iζω + i 1N√

    γv

    √ωF

    (v2q2

    ω

    )

    New infra-red singularities as ζ → 0 at higher loops(Breakdown of Migdal-Eliashberg)

    Monday, November 2, 2009

  • Ignoring fermion self energy: ∼ 1N2

    × 1ζ2× 1

    ω

    New infra-red singularities as ζ → 0 at higher loops(Breakdown of Migdal-Eliashberg)

    Monday, November 2, 2009

  • Actual order ∼ 1N0

    Ignoring fermion self energy: ∼ 1N2

    × 1ζ2× 1

    ω

    New infra-red singularities as ζ → 0 at higher loops(Breakdown of Migdal-Eliashberg)

    Monday, November 2, 2009

  • Double line representation• A way to compute the order of a diagram.

    • Extra powers of N come from the Fermi-surface

    • What are the conditions for all propagators to be on the Fermi surface?

    • Concentrate on diagrams involving a single pair of hot-spots

    • Any bosonic momentum may be (uniquely) written as

    R. Shankar, Rev. Mod. Phys. 66, 129 (1994).S. W. Tsai, A. H. Castro Neto, R. Shankar, and D. K. Campbell, Phys. Rev. B 72, 054531 (2005).

    Monday, November 2, 2009

  • =

    New infra-red singularities as ζ → 0 at higher loops(Breakdown of Migdal-Eliashberg)

    Actual order ∼ 1N0

    Singularities as ζ → 0 appear when fermions in closed blueand red line loops are exactly on the Fermi surface

    Monday, November 2, 2009

  • Graph is planar after turning fermion propagators also into double lines by drawing additional dotted single line loops for each fermion loop

    Actual order ∼ 1N0

    =

    New infra-red singularities as ζ → 0 at higher loops(Breakdown of Migdal-Eliashberg)

    Sung-Sik Lee, arXiv:0905.4532Monday, November 2, 2009

  • A consistent analysis requires resummation of all planar graphs

    Actual order ∼ 1N0

    =

    New infra-red singularities as ζ → 0 at higher loops(Breakdown of Migdal-Eliashberg)

    Monday, November 2, 2009

  • H

    SC

    M"Normal"

    (Large Fermisurface)

    SDW(Small Fermi

    pockets)

    SC+SDW

    Small Fermipockets with

    pairing fluctuationsLargeFermi

    surface

    StrangeMetal

    d-waveSC

    T

    Tsdw

    Fluctuating, paired Fermi

    pockets

    VBSand/ornematic

    Monday, November 2, 2009

  • 98H

    SC

    M"Normal"

    (Large Fermisurface)

    SDW(Small Fermi

    pockets)

    SC+SDW

    LargeFermi

    surface

    StrangeMetal

    d-waveSC

    T

    Tsdw

    Fluctuating, paired Fermi

    pockets

    Fluctuating, paired Fermi

    pockets

    VBS and/or Ising nematic order

    SDWInsulator

    T*

    Tc

    Hc2

    Monday, November 2, 2009

  • Identified quantum criticality in cuprate superconductors with a critical point at optimal

    doping associated with onset of spin density wave order in a metal

    Conclusions

    Elusive optimal doping quantum critical point has been “hiding in plain sight”.

    It is shifted to lower doping by the onset of superconductivity

    Monday, November 2, 2009

  • Conclusions

    Theory for the onset of spin density wave order in metals is strongly coupled in two dimensions

    Monday, November 2, 2009