Competing orders in the cuprate 1.0 1.5 2.0 2.5 3.0 Regular QPSR Vortex Differential Conductance (nS)

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  • Eugene Demler (Harvard) Kwon Park

    Anatoli Polkovnikov Subir Sachdev

    Matthias Vojta (Augsburg) Ying Zhang

    Competing orders in the cuprate superconductors

    Talk online at http://pantheon.yale.edu/~subir

    (Search for “Sachdev” on )

  • ( )† † 0 cos cos 0

    Hole-doped cuprates are BCS superconductors w

    -wa

    ith

    ve pairing

    spin-single t

    x y dc c k k

    S

    ↑ ↓ ≡ ∆ = ∆ −

    = ��

    kk -k

    yk

    xk

    0 0

    2 2

    Low ener Superflow

    1/ 2 fermion

    gy excitat

    ic quasiparticles

    :

    :

    ons:

    i i

    S

    e

    E

    θ

    ε

    ∆ →

    ==

    + ∆k k k

    0

    0

    BCS theory also predicts that the Fermi surface, with gapless quasiparticles, will reveal itself when 0, either locally or globally at low temperatures. can be suppressed by a strong magnetic fiel

    ∆ → ∆

    d, and near vortices, impurities and interfaces.

  • Superconductivity in a doped Mott insulator

    Hypothesis: cuprate superconductors have low energy excitations associated with additional order parameters

    Theory and experiments indicate that the most likely candidates are spin density waves and associated

    “charge” order

    Competing orders are also revealed when superconductivity is suppressed locally, near impurities or around vortices.

    S. Sachdev, Phys. Rev. B 45, 389 (1992); N. Nagaosa and P.A. Lee, Phys. Rev. B 45, 966 (1992);

    D.P. Arovas, A. J. Berlinsky, C. Kallin, and S.-C. Zhang Phys. Rev. Lett. 79, 2871 (1997); K. Park and S. Sachdev Phys. Rev. B 64, 184510 (2001).

    Superconductivity can be suppressed globally by a strong magnetic field or large current flow.

  • Outline I. Experimental introduction

    II. Spin density waves (SDW) in LSCO Tuning order and transitions by a magnetic field.

    III. Connection with “charge” order – phenomenological theory STM experiments on Bi2Sr2CaCu2O8+δ

    IV. Connection with “charge” order – microscopic theory Theories of magnetic transitions predict bond-centered modulation of exchange and pairing energies with even periods---a bond order wave

    V. Conclusions

  • The doped cuprates

    2-D CuO2 plane Néel ordered ground state at zero doping

    2-D CuO2 plane with finite hole doping

    I. Experimental introduction

  • Phase diagram of the doped cuprates

    T

    δ

    d-wave SC

    3D AFM

    0

  • ky

    kx

    π/a

    π/a0

    Insulator• • • ••••

    Superconductor with Tc,min =20 K

    T = 0 phases of LSCO

    δ~0.12-0.140.055 SC+SDW SCNéel SDW

    0.020

    S. Wakimoto, G. Shirane et al., Phys. Rev. B 60, R769 (1999).

    G. Aeppli, T.E. Mason, S.M. Hayden, H.A. Mook, J. Kulda, Science 278, 1432 (1997).

    Y. S. Lee, R. J. Birgeneau, M. A. Kastner et al., Phys. Rev. B 60, 3643 (1999).

  • SDW order parameter for general ordering wavevector

    ( ) ( ) c.c.iS eα α ⋅= Φ +K rr r

    ( ) is a field and =(3 / 4, ) Spin density wave is (and not spiral):

    α π πΦ complex longitudi nal

    r K ie nθα αΦ =

    Bond-centered

    Site-centered

  • H

    δ~0.120.055 SC+SDW SCNéel SDW

    0.020

    ky

    kx

    π/a

    π/a0

    InsulatorSuperconductor with Tc,min =20 K

    • •• •

    II. Effect of a magnetic field on SDW order with co-existing superconductivity

  • H

    δ

    Spin singlet state

    SDW

    Effect of the Zeeman term: precession of SDW order about the magnetic field

    Characteristic field gµBH = ∆, the spin gap 1 Tesla = 0.116 meV

    Effect is negligible over experimental field scales

    δc

    ( )~ zcH νδ δ−

    2

    Elastic scattering intensity

    ( ) (0) HI H I a J

    � �= + � � � �

  • Dominant effect: uniformuniform softening of spin excitations by superflow kinetic energy

    2 2

    2

    Spatially averaged superflow kinetic energy 3 ln cs

    c

    HHv H H

    � �

    1 sv r �

    r

    ( ) 2 2

    The presence of the field replaces by 3 ln ceff

    c

    HHH C H H

    δ

    δ δ � �= − � � � �

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

    Competing order is enhanced in a

    “halo” around each vortex

  • • Theory should account for dynamic quantum spin fluctuations

    • All effects are ~ H2 except those associated with H induced superflow.

    • Can treat SC order in a static Ginzburg-Landau theory

    ( ) 1/ 2 22 2 2 22 2 21 2

    0 2 2

    T

    b r g gd r d c sα τ α α α ατ

    ��= ∇ Φ + ∂ Φ + Φ + Φ + Φ �� �� � S

    2 22

    2c d rd ατ ψ

    � �= � �� �� S v

    ( ) 4

    222

    2GL r F d r iA

    ψ ψ ψ

    � � = − + + ∇ −� �

    � �� � �

    ( ) ( ) ( )

    ( )

    ,

    ln 0

    GL b cFZ r D r e

    Z r r

    ψ τ

    δ ψ δψ

    − − −= Φ� �� � � �� � =

    � S S

    (extreme Type II superconductivity)Effect of magnetic field on SDW+SC to SC transition

    Infinite diamagnetic susceptibility of non-critical superconductivity leads to a strong effect.

  • Main results

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

    ( )( ) ( )~

    ln 1/ c

    c

    H δ δ δ δ −

    2

    2

    Elastic scattering intensity 3( ) (0) ln c

    c

    HHI H I a H H

    � �= + � � � �

    ( ) ( ) 2 2

    1 exciton energy 30 ln c

    c

    S HHH b

    H H ε ε

    =

    � �= − � � � �

    T=0

    δδc

  • 2 4

    B. Khaykovich, Y. S. Lee, S. Wakimoto, K. J. Thomas, M. A. Kastner,

    Elas

    and

    tic neutron scatterin

    R.J. Birgeneau, cond-

    g off La C

    mat/01

    uO

    12505.

    y+

    ( ) ( )

    2

    2

    2

    Solid line --- fit to :

    is the only fitting parameter Best fit value - = 2.4 with

    3.01 l

    = 6

    n

    0 T

    0

    c

    c

    c

    I H HH H

    a

    a I H

    a H

    � �= + � � � �

    Neutron scattering measurements of static spin correlations of the superconductor+spin-density-wave (SC+SDW) in a magnetic field

  • 2- 4Neutron scattering of La Sr CuO at =0.1x x x

    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, T. E. Mason, Nature, 415, 299 (2002).

    2

    2

    Solid line - fit ( ) nto : l c c

    HHI H a H H

    � �= � � � �

  • Spin density wave order parameter for general ordering wavevector ( ) ( ) c.c.iS eα α ⋅= Φ +K rr r

    ( ) is a field and =(3 / 4, ) Spin density wave is (and not spiral):

    α π πΦ complex longitudi nal

    r K

    III. Connections with “charge” order – phenomenological theory

    ie nθα αΦ =

    Bond-centered

    Site-centered

  • ( ) ( ) ( )2 2 2 c.c.iS eα α α

    δρ ⋅∝ = Φ +� K rr r r

    J. Zaanen and O. Gunnarsson, Phys. Rev. B 40, 7391 (1989). H. Schulz, J. de Physique 50, 2833 (1989). K. Machida, Physica 158C, 192 (1989). O. Zachar, S. A. Kivelson, and V. J. Emery, Phys. Rev. B 57, 1422 (1998).

    A longitudinal spin density wave necessarily has an accompanying modulation in the site charge densities, exchange and pairing

    energy per link etc. at half the wavelength of the SDW

    “Charge” order: periodic modulation in local observables invariant under spin rotations and time-reversal.

    Order parmeter ( )2~ α α

    Φ� r

    Prediction: Charge order should be pinned in halo around vortex core

    K. Park and S. Sachdev Phys. Rev. B 64, 184510 (2001). E. Demler, S. Sachdev, and Ying Zhang, Phys. Rev. Lett. 87, 067202 (2001).

  • STM around vortices induced by a magnetic field in the superconducting state J. E. Hoffman, E. W. Hudson, K. M. Lang, V. Madhavan, S. H. Pan,

    H. Eisaki, S. Uchida, and J. C. Davis, Science 295, 466 (2002).

    -120 -80 -40 0 40 80 120 0.0

    0.5

    1.0

    1.5

    2.0

    2.5

    3.0

    Regular QPSR Vortex

    Di ffe

    re nti

    al Co

    nd uc

    tan ce

    (n S)

    Sample Bias (mV)

    Local density of states

    1Å spatial resolution image of integrated

    LDOS of Bi2Sr2CaCu2O8+δ ( 1meV to 12 meV)

    at B=5 Tesla.

    S.H. Pan et al. Phys. Rev. Lett. 85, 1536 (2000).

  • 100Å

    b 7 pA

    0 pA

    Vortex-induced LDOS of Bi2Sr2CaCu2O8+δ integ