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Transcript of Strange metals and gauge-gravity 2012. 5. 17.¢  Strange metals and gauge-gravity duality...

  • Strange metals and

    gauge-gravity duality

    HARVARD

    Loughborough University, May 17, 2012

    Subir Sachdev

    sachdev.physics.harvard.edu

    Lecture at the 100th anniversary Solvay conference, Theory of the Quantum World, chair D.J. Gross.

    arXiv:1203.4565

    Thursday, May 17, 2012

  • Liza Huijse Max Metlitski Brian Swingle

    Thursday, May 17, 2012

  • Sommerfeld-Bloch theory of metals, insulators, and superconductors: many-electron quantum states are adiabatically

    connected to independent electron states

    Thursday, May 17, 2012

  • Sommerfeld-Bloch theory of metals, insulators, and superconductors: many-electron quantum states are adiabatically

    connected to independent electron states

    Band insulators

    E

    Metal Metal

    carrying a current

    Insulator Superconductor

    k An even number of electrons per unit cell

    Thursday, May 17, 2012

  • E

    Metal Metal

    carrying a current

    Insulator Superconductor

    k

    E

    Metal Metal

    carrying a current

    Insulator Superconductor

    k

    Sommerfeld-Bloch theory of metals, insulators, and superconductors: many-electron quantum states are adiabatically

    connected to independent electron states

    Metals

    An odd number of electrons per unit cell

    Thursday, May 17, 2012

  • E

    Metal Metal

    carrying a current

    Insulator Superconductor

    k

    Sommerfeld-Bloch theory of metals, insulators, and superconductors: many-electron quantum states are adiabatically

    connected to independent electron states

    Superconductors

    Thursday, May 17, 2012

  • Modern phases of quantum matter Not adiabatically connected

    to independent electron states:

    Thursday, May 17, 2012

  • Modern phases of quantum matter Not adiabatically connected

    to independent electron states: many-particle, long-range quantum entanglement

    Thursday, May 17, 2012

  • Mott insulator: Triangular lattice antiferromagnet

    H = J �

    �ij�

    �Si · �Sj

    Nearest-neighbor model has non-collinear Neel order Thursday, May 17, 2012

  • Spin liquid obtained in a generalized spin model with S=1/2 per unit cell

    =

    P. Fazekas and P. W. Anderson, Philos. Mag. 30, 23 (1974).

    Mott insulator: Triangular lattice antiferromagnet

    Thursday, May 17, 2012

  • P. Fazekas and P. W. Anderson, Philos. Mag. 30, 23 (1974).

    Spin liquid obtained in a generalized spin model with S=1/2 per unit cell

    =

    Mott insulator: Triangular lattice antiferromagnet

    Thursday, May 17, 2012

  • P. Fazekas and P. W. Anderson, Philos. Mag. 30, 23 (1974).

    Spin liquid obtained in a generalized spin model with S=1/2 per unit cell

    =

    Mott insulator: Triangular lattice antiferromagnet

    Thursday, May 17, 2012

  • P. Fazekas and P. W. Anderson, Philos. Mag. 30, 23 (1974).

    Spin liquid obtained in a generalized spin model with S=1/2 per unit cell

    =

    Mott insulator: Triangular lattice antiferromagnet

    Thursday, May 17, 2012

  • P. Fazekas and P. W. Anderson, Philos. Mag. 30, 23 (1974).

    Spin liquid obtained in a generalized spin model with S=1/2 per unit cell

    =

    Mott insulator: Triangular lattice antiferromagnet

    Thursday, May 17, 2012

  • P. Fazekas and P. W. Anderson, Philos. Mag. 30, 23 (1974).

    Spin liquid obtained in a generalized spin model with S=1/2 per unit cell

    =

    Mott insulator: Triangular lattice antiferromagnet

    Thursday, May 17, 2012

  • ssc

    non-collinear Néel state

    Mott insulator: Triangular lattice antiferromagnet

    Quantum “disordered” state with exponentially decaying spin correlations.

    Z2 spin liquid with long-range entanglement.

    Thursday, May 17, 2012

  • ssc

    non-collinear Néel state

    Mott insulator: Triangular lattice antiferromagnet

    N. Read and S. Sachdev, Phys. Rev. Lett. 66, 1773 (1991) X.-G. Wen, Phys. Rev. B 44, 2664 (1991)

    Quantum “disordered” state with exponentially decaying spin correlations.

    Z2 spin liquid with long-range entanglement.

    Thursday, May 17, 2012

  • |Ψ� ⇒ Ground state of entire system, ρ = |Ψ��Ψ|

    ρA = TrBρ = density matrix of region A

    Entanglement entropy SEE = −Tr (ρA ln ρA)

    B

    A

    Topological order in the Z2 spin liquid ground state

    Thursday, May 17, 2012

  • Entanglement entropy of a band insulator: SEE = aP − exp(−bP )

    where P is the surface area (perimeter) of the boundary between A and B.

    A

    B

    Topological order in the Z2 spin liquid ground state

    Thursday, May 17, 2012

  • Entanglement entropy of a Z2 spin liquid: SEE = aP − ln(2)

    where P is the surface area (perimeter) of the boundary between A and B.

    The ln(2) is a universal characteristic of the Z2 spin liquid, and implies long-range quantum entanglement.

    A

    B

    M. Levin and X.-G. Wen, Phys. Rev. Lett. 96, 110405 (2006); A. Kitaev and J. Preskill, Phys. Rev. Lett. 96, 110404 (2006); Y. Zhang, T. Grover, and A. Vishwanath, Phys. Rev. B 84, 075128 (2011).

    Topological order in the Z2 spin liquid ground state

    Thursday, May 17, 2012

  • Young Lee, APS meeting, March 2012

    Promising candidate: the kagome antiferromagnet

    Numerical evidence for a gapped spin liquid: Simeng Yan, D. A. Huse, and S. R. White, Science 332, 1173 (2011).

    Thursday, May 17, 2012

  • Quantum superposition and

    entanglement

    Thursday, May 17, 2012

  • Quantum superposition and

    entanglement

    Quantum critical points of electrons

    in crystals String theory

    Thursday, May 17, 2012

  • Quantum superposition and

    entanglement

    Quantum critical points of electrons

    in crystals String theory

    Thursday, May 17, 2012

  • Spinning electrons localized on a square lattice

    H = �

    �ij�

    Jij �Si · �Sj

    J

    J/λ

    Examine ground state as a function of λ

    S=1/2 spins

    Thursday, May 17, 2012

  • H = �

    �ij�

    Jij �Si · �Sj

    J

    J/λ

    At large ground state is a “quantum paramagnet” with spins locked in valence bond singlets

    = 1√ 2

    ����↑↓ � −

    ��� ↓↑ ��

    λ

    Spinning electrons localized on a square lattice

    Thursday, May 17, 2012

  • H = �

    �ij�

    Jij �Si · �Sj

    J

    J/λ

    = 1√ 2

    ����↑↓ � −

    ��� ↓↑ ��

    Nearest-neighor spins are “entangled” with each other. Can be separated into an Einstein-Podolsky-Rosen (EPR) pair.

    Spinning electrons localized on a square lattice

    Thursday, May 17, 2012

  • H = �

    �ij�

    Jij �Si · �Sj

    J

    J/λ

    For λ ≈ 1, the ground state has antiferromagnetic (“Néel”) order, and the spins align in a checkerboard pattern

    Spinning electrons localized on a square lattice

    Thursday, May 17, 2012

  • H = �

    �ij�

    Jij �Si · �Sj

    J

    J/λ

    For λ ≈ 1, the ground state has antiferromagnetic (“Néel”) order, and the spins align in a checkerboard pattern

    No EPR pairs

    Spinning electrons localized on a square lattice

    Thursday, May 17, 2012

  • λλc

    = 1√ 2

    ����↑↓ � −

    ��� ↓↑ ��

    Thursday, May 17, 2012

  • 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)

    Excitations of TlCuCl3 with varying pressure

    0 0.5 1 1.5 2 2.5 30

    0.2

    0.4

    0.6

    0.8

    1

    1.2

    Pressure [kbar]

    En er

    gy [m

    eV ]

    Thursday, May 17, 2012

  • 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)

    Excitations of TlCuCl3 with varying pressure

    0 0.5 1 1.5 2 2.5 30

    0.2

    0.4

    0.6

    0.8

    1

    1.2

    Pressure [kbar]

    En er

    gy [m

    eV ]

    S. Sachdev, arXiv:0901.4103

    Higgs boson

    First observation of the Higgs boson

    at the theoretically predicted energy!

    Thursday, May 17, 2012

  • λλc

    A. W. Sandvik and D. J. Scalapino, Phys. Rev. Lett. 72, 2777 (1994).

    Quantum critical point with non-local entanglement in spin wavefunction

    = 1√ 2

    ����↑↓ � −

    ��� ↓↑ ��

    Thursday,