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Quantum entanglement and the phases of matter

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Quantum entanglement

and the phases of matter

Colloquium Ehrenfestii

Leiden University

May 9, 2012

sachdev.physics.harvard.edu

metals, insulators, and superconductors:

many-electron quantum states are adiabatically connected to independent electron states

metals, insulators, and superconductors:

many-electron quantum states are adiabatically connected to independent electron states

Band insulators

An even number of electrons per unit cell

metals, insulators, and superconductors:

many-electron quantum states are adiabatically connected to independent electron states

Metals

An odd number of electrons per unit cell

metals, insulators, and superconductors:

many-electron quantum states are adiabatically connected to independent electron states

Superconductors

Modern phases of quantum matter

Not adiabatically connected

to independent electron states:

many-particle, long-range

quantum entanglement

superposition and entanglement

Quantum Entanglement: quantum superposition with more than one particle

Quantum Entanglement: quantum superposition with more than one particle

Hydrogen atom:

Quantum Entanglement: quantum superposition with more than one particle

Hydrogen atom:

Hydrogen molecule:

_

=

Superposition of two electron states leads to non-local correlations between spins

Quantum Entanglement: quantum superposition with more than one particle

_

Quantum Entanglement: quantum superposition with more than one particle

_

Quantum Entanglement: quantum superposition with more than one particle

_

Quantum Entanglement: quantum superposition with more than one particle

_

Einstein-Podolsky-Rosen “paradox”: Non-local correlations between observations arbitrarily far apart

Mott insulator: Triangular lattice antiferromagnet

Nearest-neighbor model has non-collinear Neel order

Mott insulator: Triangular lattice antiferromagnet

N. Read and S. Sachdev, Phys. Rev. Lett. 66, 1773 (1991)

X.-G. Wen, Phys. Rev. B44, 2664 (1991)

Mott insulator: Triangular lattice antiferromagnet

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

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

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

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

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

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).

Topological order in the Z2 spin liquid ground state

B

A

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. B84, 075128 (2011).

- Promising candidate: the kagome
- antiferromagnet

Numerical evidence for a gapped spin liquid:

Simeng Yan, D. A. Huse, and S. R. White, Science332, 1173 (2011).

Young Lee,

APS meeting, March 2012

superposition and entanglement

superposition and entanglement

Black holes

String theory

Quantum critical points of electrons

in crystals

superposition and entanglement

Black holes

String theory

Quantum critical points of electrons

in crystals

Spinning electrons localized on a square lattice

S=1/2

spins

J

J/

Examine ground state as a function of

Spinning electrons localized on a square lattice

J

J/

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

A. Oosawa, K. Kakurai, T. Osakabe, M. Nakamura, M. Takeda, and H. Tanaka, Journal of the Physical Society of Japan, 73, 1446 (2004).

An insulator whose spin susceptibility vanishes exponentially as the temperature T tends to zero.

Quantum paramagnet at ambient pressure

Neel order under pressure

A. Oosawa, K. Kakurai, T. Osakabe, M. Nakamura, M. Takeda, and H. Tanaka, Journal of the Physical Society of Japan, 73, 1446 (2004).

Excitations of TlCuCl3 with varying pressure

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

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

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

S. Sachdev,

arXiv:0901.4103

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)

Tensor network representation of entanglement wavefunction

at quantum critical point

M. Levin and C. P. Nave, Phys. Rev. Lett. 99, 120601 (2007)

G. Vidal, Phys. Rev. Lett. 99, 220405 (2007)

F. Verstraete, M. M. Wolf, D. Perez-Garcia, and J. I. Cirac, Phys. Rev. Lett. 96, 220601 (2006)

Entanglement entropy wavefunction

d

A

Entanglement entropy =

Number of links on optimal surface intersecting minimal number of links.

Long-range entanglement in a CFT3 wavefunction

B

A

M. A. Metlitski, C. A. Fuertes, and S. Sachdev, Physical Review B 80, 115122 (2009).

H. Casini, M. Huerta, and R. Myers, JHEP 1105:036, (2011)

I. Klebanov, S. Pufu, and B. Safdi, arXiv:1105.4598

Characteristics of wavefunction

quantum critical point

Characteristics of wavefunction

quantum critical point

Characteristics of wavefunction

quantum critical point

Characteristics of wavefunction

quantum critical point

Quantum wavefunction

superposition and entanglement

Black holes

String theory

Quantum critical points of electrons

in crystals

Quantum wavefunction

superposition and entanglement

Black holes

String theory

Quantum critical points of electrons

in crystals

Quantum wavefunction

superposition and entanglement

Black holes

String theory

Quantum critical points of electrons

in crystals

String theory wavefunction

Tensor network representation of entanglement wavefunction

at quantum critical point

d

Emergent direction

of AdS4

Brian Swingle, arXiv:0905.1317

Entanglement entropy wavefunction

d

A

Entanglement entropy =

Number of links on optimal surface intersecting minimal number of links.

Emergent direction

of AdS4

Quantum matter with wavefunction

long-range entanglement

r

Emergent holographic direction

J. McGreevy, arXiv0909.0518

Quantum matter with wavefunction

long-range entanglement

A

Area measures

entanglement

entropy

r

Emergent holographic direction

S. Ryu and T. Takayanagi, Phys. Rev. Lett. 96, 18160 (2006).

Quantum wavefunction

superposition and entanglement

Black holes

String theory

Quantum critical points of electrons

in crystals

Quantum wavefunction

superposition and entanglement

Black holes

String theory

Quantum critical points of electrons

in crystals

CFT3

Quantum wavefunction

superposition and entanglement

Black holes

String theory

Quantum critical points of electrons

in crystals

Thermally excited wavefunction

spin waves

Thermally excited

triplon particles

Short-range entanglement

Short-range entanglement

Excitations of a ground state with long-range entanglement wavefunction

Thermally excited

spin waves

Thermally excited

triplon particles

Excitations of a ground state with long-range entanglement wavefunction

Needed: Accurate theory of quantum critical spin dynamics

Thermally excited

spin waves

Thermally excited

triplon particles

String theory at non-zero temperatures wavefunction

A 2+1 dimensional system at its quantum critical point

String theory at non-zero temperatures wavefunction

A 2+1 dimensional system at its quantum critical point

A “horizon”, similar to the surface of a black hole !

Black Holes wavefunction

Objects so massive that light is gravitationally bound to them.

Black Holes wavefunction

Objects so massive that light is gravitationally bound to them.

In Einstein’s theory, the region inside the black hole horizon is disconnected from the rest of the universe.

Black Holes + Quantum theory wavefunction

Around 1974, Bekenstein and Hawking showed that the application of the quantum theory across a black hole horizon led to many astonishing conclusions

Quantum Entanglement across a black hole horizon wavefunction

_

Quantum Entanglement across a black hole horizon wavefunction

_

Quantum Entanglement across a black hole horizon wavefunction

There is a non-local quantum entanglement between the inside and outside of a black hole

Black hole

horizon

Quantum Entanglement across a black hole horizon wavefunction

There is a non-local quantum entanglement between the inside and outside of a black hole

Black hole

horizon

Quantum Entanglement across a black hole horizon wavefunction

There is a non-local quantum entanglement between the inside and outside of a black hole

This entanglement leads to a

black hole temperature

(the Hawking temperature)

and a black hole entropy

(the Bekenstein entropy)

String theory at non-zero temperatures wavefunction

A 2+1 dimensional system at its quantum critical point

A “horizon”,

whose temperature and entropy equal those of the quantum critical point

String theory at non-zero temperatures wavefunction

A 2+1 dimensional system at its quantum critical point

A “horizon”,

whose temperature and entropy equal those of the quantum critical point

Friction of quantum criticality = waves falling into black brane

String theory at non-zero temperatures wavefunction

A 2+1 dimensional system at its quantum critical point

An (extended) Einstein-Maxwell provides successful description of dynamics of quantum critical points at non-zero temperatures (where no other methods apply)

A “horizon”,

whose temperature and entropy equal those of the quantum critical point

Quantum wavefunction

superposition and entanglement

Black holes

String theory

Quantum critical points of electrons

in crystals

Quantum wavefunction

superposition and entanglement

Black holes

String theory

Quantum critical points of electrons

in crystals

Metals, “strange metals”, and high temperature superconductors

Insights from gravitational “duals”

High temperature superconductors superconductors

Iron pnictides superconductors:

a new class of high temperature superconductors

Ishida, Nakai, and Hosono

arXiv:0906.2045v1

AF superconductors

S. Kasahara, T. Shibauchi, K. Hashimoto, K. Ikada, S. Tonegawa, R. Okazaki, H. Shishido,

H. Ikeda, H. Takeya, K. Hirata, T. Terashima, and Y. Matsuda,

Physical Review B81, 184519 (2010)

Short-range entanglement superconductors

in state with Neel (AF) order

AF

S. Kasahara, T. Shibauchi, K. Hashimoto, K. Ikada, S. Tonegawa, R. Okazaki, H. Shishido,

H. Ikeda, H. Takeya, K. Hirata, T. Terashima, and Y. Matsuda,

Physical Review B81, 184519 (2010)

AF superconductors

Superconductor

Bose condensate of pairs of electrons

Short-range entanglement

S. Kasahara, T. Shibauchi, K. Hashimoto, K. Ikada, S. Tonegawa, R. Okazaki, H. Shishido,

H. Ikeda, H. Takeya, K. Hirata, T. Terashima, and Y. Matsuda,

Physical Review B81, 184519 (2010)

AF superconductors

Ordinary metal

(Fermi liquid)

S. Kasahara, T. Shibauchi, K. Hashimoto, K. Ikada, S. Tonegawa, R. Okazaki, H. Shishido,

H. Ikeda, H. Takeya, K. Hirata, T. Terashima, and Y. Matsuda,

Physical Review B81, 184519 (2010)

Sommerfeld-Bloch theory of ordinary metals superconductors

Momenta with

electron states

occupied

Momenta with

electron states

empty

Sommerfeld-Bloch theory of ordinary metals superconductors

Key feature of the theory:

the Fermi surface

115

AF superconductors

Ordinary metal

(Fermi liquid)

S. Kasahara, T. Shibauchi, K. Hashimoto, K. Ikada, S. Tonegawa, R. Okazaki, H. Shishido,

H. Ikeda, H. Takeya, K. Hirata, T. Terashima, and Y. Matsuda,

Physical Review B81, 184519 (2010)

Strange superconductors

Metal

AF

S. Kasahara, T. Shibauchi, K. Hashimoto, K. Ikada, S. Tonegawa, R. Okazaki, H. Shishido,

H. Ikeda, H. Takeya, K. Hirata, T. Terashima, and Y. Matsuda,

Physical Review B81, 184519 (2010)

Ordinary superconductors

Metal

Strange superconductors

Metal

AF

S. Kasahara, T. Shibauchi, K. Hashimoto, K. Ikada, S. Tonegawa, R. Okazaki, H. Shishido,

H. Ikeda, H. Takeya, K. Hirata, T. Terashima, and Y. Matsuda,

Physical Review B81, 184519 (2010)

Excitations of a ground state with long-range entanglement superconductors

Strange

Metal

AF

S. Kasahara, T. Shibauchi, K. Hashimoto, K. Ikada, S. Tonegawa, R. Okazaki, H. Shishido,

H. Ikeda, H. Takeya, K. Hirata, T. Terashima, and Y. Matsuda,

Physical Review B81, 184519 (2010)

Key (difficult) problem: superconductors

Describe quantum critical points and phases of systems with Fermi surfaces leading to metals with novel types of long-range entanglement

+

Challenge to string theory: superconductors

Describe quantum critical points

and phases of metals

Can we obtain gravitational theories

of superconductors and

ordinary Sommerfeld-Bloch metals ?

Challenge to string theory: superconductors

Describe quantum critical points

and phases of metals

Can we obtain gravitational theories

of superconductors and

ordinary Sommerfeld-Bloch metals ?

Yes

T. Nishioka, S. Ryu, and T. Takayanagi, JHEP 1003, 131 (2010)

G. T. Horowitz and B. Way, JHEP 1011, 011 (2010)

S. Sachdev, Physical Review D 84, 066009 (2011)

Challenge to string theory: superconductors

Describe quantum critical points

and phases of metals

Do the “holographic” gravitational theories

also yield metals distinct from

ordinary Sommerfeld-Bloch metals ?

Challenge to string theory: superconductors

Describe quantum critical points

and phases of metals

Do the “holographic” gravitational theories

also yield metals distinct from

ordinary Sommerfeld-Bloch metals ?

Yes, lots of them, with

many “strange” properties !

S.-S. Lee, Phys. Rev. D79, 086006 (2009);

M. Cubrovic, J. Zaanen, and K. Schalm, Science325, 439 (2009);

T. Faulkner, H. Liu, J. McGreevy, and D. Vegh, arXiv:0907.2694;

F. Denef, S. A. Hartnoll, and S. Sachdev, Phys. Rev. D80, 126016 (2009)

Challenge to string theory: superconductors

Describe quantum critical points

and phases of metals

How do we discard artifacts, and choose the

holographic theories applicable to condensed matter physics ?

Challenge to string theory: superconductors

Describe quantum critical points

and phases of metals

How do we discard artifacts, and choose the

holographic theories applicable to condensed matter physics ?

Choose the theories with the

proper entropy density

Checks: these theories also have the proper entanglement entropy and Fermi surface size !

N. Ogawa, T. Takayanagi, and T. Ugajin, arXiv:1111.1023

L. Huijse, S. Sachdev, B. Swingle, Physical Review B 85, 035121 (2012)

Entanglement entropy of Fermi surfaces superconductors

B

A

Y. Zhang, T. Grover, and A. Vishwanath, Physical Review Letters107, 067202 (2011)

Quantum matter with superconductors

long-range entanglement

r

Emergent holographic direction

J. McGreevy, arXiv0909.0518

Quantum matter with superconductors

long-range entanglement

r

Emergent holographic direction

Abandon conformal invariance, and only require scale invariance at long lengths and times.....

L. Huijse, S. Sachdev, B. Swingle, Physical Review B superconductors85, 035121 (2012)

L. Huijse, S. Sachdev, B. Swingle, Physical Review B superconductors85, 035121 (2012)

L. Huijse, S. Sachdev, B. Swingle, Physical Review B superconductors85, 035121 (2012)

N. Ogawa, T. Takayanagi, and T. Ugajin, arXiv:1111.1023 superconductors

Conclusions superconductors

Phases of matter with long-range quantum entanglement are prominent in numerous modern materials.

Conclusions superconductors

Simplest examples of long-range entanglement are in

insulating antiferromagnets:

Z2 spin liquids and

quantum critical points

Conclusions superconductors

More complex examples in metallic states are experimentally ubiquitous, but pose difficult strong-coupling problems to conventional methods of field theory

Conclusions superconductors

String theory and gravity in emergent dimensions

offer a remarkable new approach to describing states with long-range quantum entanglement.

Conclusions superconductors

String theory and gravity in emergent dimensions

offer a remarkable new approach to describing states with long-range quantum entanglement.

Much recent progress offers hope of a holographic description of “strange metals”

anti-de Sitter space superconductors

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