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HARVARD. Quantum entanglement and the phases of matter. Colloquium Ehrenfestii Leiden University May 9, 2012. sachdev.physics.harvard.edu. Max Metlitski. Liza Huijse. Brian Swingle. Sommerfeld-Bloch theory of metals, insulators, and superconductors:

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slide1

HARVARD

Quantum entanglement

and the phases of matter

Colloquium Ehrenfestii

Leiden University

May 9, 2012

sachdev.physics.harvard.edu

slide2

Max Metlitski

Liza Huijse

Brian Swingle

slide3

Sommerfeld-Bloch theory of

metals, insulators, and superconductors:

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

slide4

Sommerfeld-Bloch theory of

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

slide5

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

slide6

Sommerfeld-Bloch theory of

metals, insulators, and superconductors:

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

Superconductors

slide7

Modern phases of quantum matter

Not adiabatically connected

to independent electron states:

slide8

Modern phases of quantum matter

Not adiabatically connected

to independent electron states:

many-particle, long-range

quantum entanglement

slide9

Quantum

superposition and entanglement

slide12

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

slide16

Quantum Entanglement: quantum superposition with more than one particle

_

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

slide17

Mott insulator: Triangular lattice antiferromagnet

Nearest-neighbor model has non-collinear Neel order

slide19

Mott insulator: Triangular lattice antiferromagnet

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

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

slide20

=

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).

slide21

=

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).

slide22

=

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).

slide23

=

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).

slide24

=

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).

slide25

=

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).

slide28

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).

slide29

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

slide30

Quantum

superposition and entanglement

slide31

Quantum

superposition and entanglement

Black holes

String theory

Quantum critical points of electrons

in crystals

slide32

Quantum

superposition and entanglement

Black holes

String theory

Quantum critical points of electrons

in crystals

slide33

Spinning electrons localized on a square lattice

S=1/2

spins

J

J/

Examine ground state as a function of

slide34

Spinning electrons localized on a square lattice

J

J/

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

slide39

Pressure in TlCuCl3

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

slide40

TlCuCl3

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

slide41

TlCuCl3

Quantum paramagnet at ambient pressure

slide42

TlCuCl3

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).

slide49

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)

slide50

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)

slide51

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)

slide52

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)

slide55

Tensor network representation of entanglement

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)

slide57

Entanglement entropy

d

A

Most links describe entanglement within A

slide58

Entanglement entropy

d

A

Links overestimate entanglement between A and B

slide59

Entanglement entropy

d

A

Entanglement entropy =

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

slide60

Long-range entanglement in a CFT3

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

slide61

Characteristics of

quantum critical point

slide62

Characteristics of

quantum critical point

slide63

Characteristics of

quantum critical point

slide64

Characteristics of

quantum critical point

slide65

Quantum

superposition and entanglement

Black holes

String theory

Quantum critical points of electrons

in crystals

slide66

Quantum

superposition and entanglement

Black holes

String theory

Quantum critical points of electrons

in crystals

slide67

Quantum

superposition and entanglement

Black holes

String theory

Quantum critical points of electrons

in crystals

slide74

String theory near

a D-brane

d

Emergent direction

of AdS4

slide75

Tensor network representation of entanglement

at quantum critical point

d

Emergent direction

of AdS4

Brian Swingle, arXiv:0905.1317

slide76

Entanglement entropy

d

A

Entanglement entropy =

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

Emergent direction

of AdS4

slide77

Quantum matter with

long-range entanglement

r

Emergent holographic direction

J. McGreevy, arXiv0909.0518

slide78

Quantum matter with

long-range entanglement

A

r

Emergent holographic direction

slide79

Quantum matter with

long-range entanglement

A

Area measures

entanglement

entropy

r

Emergent holographic direction

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

slide80

Quantum

superposition and entanglement

Black holes

String theory

Quantum critical points of electrons

in crystals

slide81

Quantum

superposition and entanglement

Black holes

String theory

Quantum critical points of electrons

in crystals

CFT3

slide82

Quantum

superposition and entanglement

Black holes

String theory

Quantum critical points of electrons

in crystals

slide86

Thermally excited

spin waves

Thermally excited

triplon particles

slide87

Thermally excited

spin waves

Thermally excited

triplon particles

Short-range entanglement

Short-range entanglement

slide88

Excitations of a ground state with long-range entanglement

Thermally excited

spin waves

Thermally excited

triplon particles

slide89

Excitations of a ground state with long-range entanglement

Needed: Accurate theory of quantum critical spin dynamics

Thermally excited

spin waves

Thermally excited

triplon particles

slide90

String theory at non-zero temperatures

A 2+1 dimensional system at its quantum critical point

slide91

String theory at non-zero temperatures

A 2+1 dimensional system at its quantum critical point

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

slide92

Black Holes

Objects so massive that light is gravitationally bound to them.

slide93

Black Holes

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.

slide94

Black Holes + Quantum theory

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

slide99

Quantum Entanglement across a black hole horizon

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

Black hole

horizon

slide100

Quantum Entanglement across a black hole horizon

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

Black hole

horizon

slide101

Quantum Entanglement across a black hole horizon

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)

slide102

String theory at non-zero temperatures

A 2+1 dimensional system at its quantum critical point

A “horizon”,

whose temperature and entropy equal those of the quantum critical point

slide103

String theory at non-zero temperatures

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

slide104

String theory at non-zero temperatures

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

slide105

Quantum

superposition and entanglement

Black holes

String theory

Quantum critical points of electrons

in crystals

slide106

Quantum

superposition and entanglement

Black holes

String theory

Quantum critical points of electrons

in crystals

slide107

Metals, “strange metals”, and high temperature superconductors

Insights from gravitational “duals”

slide109

Iron pnictides:

a new class of high temperature superconductors

Ishida, Nakai, and Hosono

arXiv:0906.2045v1

slide110

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)

slide111

Short-range entanglement

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)

slide112

AF

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)

slide113

AF

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)

slide114

Sommerfeld-Bloch theory of ordinary metals

Momenta with

electron states

occupied

Momenta with

electron states

empty

slide115

Sommerfeld-Bloch theory of ordinary metals

Key feature of the theory:

the Fermi surface

115

slide116

AF

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)

slide117

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)

slide120

Strange

Metal

Ordinary

Metal

slide121

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)

slide122

Excitations of a ground state with long-range entanglement

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)

slide123

Key (difficult) problem:

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

+

slide124

Challenge to string theory:

Describe quantum critical points

and phases of metals

slide125

Challenge to string theory:

Describe quantum critical points

and phases of metals

Can we obtain gravitational theories

of superconductors and

ordinary Sommerfeld-Bloch metals ?

slide126

Challenge to string theory:

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)

slide127

Challenge to string theory:

Describe quantum critical points

and phases of metals

Do the “holographic” gravitational theories

also yield metals distinct from

ordinary Sommerfeld-Bloch metals ?

slide128

Challenge to string theory:

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)

slide129

Challenge to string theory:

Describe quantum critical points

and phases of metals

How do we discard artifacts, and choose the

holographic theories applicable to condensed matter physics ?

slide130

Challenge to string theory:

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)

slide131

Entanglement entropy of Fermi surfaces

B

A

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

slide132

Quantum matter with

long-range entanglement

r

Emergent holographic direction

J. McGreevy, arXiv0909.0518

slide133

Quantum matter with

long-range entanglement

r

Emergent holographic direction

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

slide140

Conclusions

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

slide141

Conclusions

Simplest examples of long-range entanglement are in

insulating antiferromagnets:

Z2 spin liquids and

quantum critical points

slide142

Conclusions

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

slide143

Conclusions

String theory and gravity in emergent dimensions

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

slide144

Conclusions

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”

slide145

anti-de Sitter space

Emergent holographic direction

r

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