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

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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

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,

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