1. and the phases of matter
Quantum entanglement
and the phases of matter
Colloquium Ehrenfestii
Leiden University
May 9, 2012
HARVARD
sachdev.physics.harvard.edu

2. Max Metlitski
Liza Huijse
Brian Swingle

3. Sommerfeld-Bloch theory of metals, insulators, and superconductors:
many-electron quantum states are adiabatically connected to independent electron states

4. Band insulators 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

5. Metals 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

6. Superconductors Sommerfeld-Bloch theory of
metals, insulators, and superconductors:
many-electron quantum states are adiabatically connected to independent electron states
Superconductors

7. Modern phases of quantum matter
Not adiabatically connected
to independent electron states:

8. many-particle, long-range quantum entanglement
Modern phases of quantum matter
Not adiabatically connected
to independent electron states:
many-particle, long-range
quantum entanglement

9. superposition and entanglement
Quantum
superposition and entanglement

10. Quantum Entanglement: quantum superposition with more than one particle

11. Quantum Entanglement: quantum superposition with more than one particle
Hydrogen atom:

12. 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

13. Quantum Entanglement: quantum superposition with more than one particle_

14. Quantum Entanglement: quantum superposition with more than one particle_

15. Quantum Entanglement: quantum superposition with more than one particle_

16. Quantum Entanglement: quantum superposition with more than one particle_
Einstein-Podolsky-Rosen “paradox”: Non-local correlations between observations arbitrarily far apart

17. Mott insulator: Triangular lattice antiferromagnet
Nearest-neighbor model has non-collinear Neel order

18. Mott insulator: Triangular lattice antiferromagnet

19. 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)

20. 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).

21. 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).

22. 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).

23. 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).

24. 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).

25. 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).

26. Topological order in the Z2 spin liquid ground stateB A

27. Topological order in the Z2 spin liquid ground stateB A

28. B A Topological order in the Z2 spin liquid ground state
M. Levin and X.-G. Wen, Phys. Rev. Lett. 96, (2006); A. Kitaev and J. Preskill, Phys. Rev. Lett. 96, (2006);
Y. Zhang, T. Grover, and A. Vishwanath, Phys. Rev. B 84, (2011).

29. 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).
Young Lee,
APS meeting, March 2012

30. superposition and entanglement
Quantum
superposition and entanglement

31. superposition and entanglement Quantum critical points of electrons
in crystals
String theory
Black holes

32. superposition and entanglement Quantum critical points of electrons
in crystals
String theory
Black holes

33. Examine ground state as a function of
Spinning electrons localized on a square lattice
S=1/2
spins J J/
Examine ground state as a function of

34. Spinning electrons localized on a square latticeJ J/
At large ground state is a “quantum paramagnet” with spins locked in valence bond singlets

35. Spinning electrons localized on a square latticeJ J/

36. Spinning electrons localized on a square latticeJ J/

37. Spinning electrons localized on a square latticeJ J/
No EPR pairs

39. Pressure in TlCuCl3
A. Oosawa, K. Kakurai, T. Osakabe, M. Nakamura, M. Takeda, and H. Tanaka, Journal of the Physical Society of Japan, 73, (2004).

40. TlCuCl3
An insulator whose spin susceptibility vanishes exponentially as the temperature T tends to zero.

41. Quantum paramagnet at ambient pressure
TlCuCl3
Quantum paramagnet at ambient pressure

42. Neel order under pressure
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, (2004).

46. Spin waves

47. Spin waves

48. Spin waves

49. 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, (2008)

50. 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, (2008)

51. 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, (2008)

52. Excitations of TlCuCl3 with varying pressure
S. Sachdev,
arXiv:
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, (2008)

54. Quantum critical point with non-local entanglement in spin wavefunction

55. Tensor network representation of entanglement
at quantum critical point
M. Levin and C. P. Nave, Phys. Rev. Lett. 99, (2007)
G. Vidal, Phys. Rev. Lett. 99, (2007)
F. Verstraete, M. M. Wolf, D. Perez-Garcia, and J. I. Cirac, Phys. Rev. Lett. 96, (2006)

56. Entanglement entropy d A

57. Most links describe entanglement within A
Entanglement entropy d A
Most links describe entanglement within A

58. Links overestimate entanglement between A and B
Entanglement entropy d A
Links overestimate entanglement between A and B

59. Entanglement entropy =d A
Entanglement entropy =
Number of links on optimal surface intersecting minimal number of links.

60. Long-range entanglement in a CFT3B A
M. A. Metlitski, C. A. Fuertes, and S. Sachdev, Physical Review B 80, (2009).
H. Casini, M. Huerta, and R. Myers, JHEP 1105:036, (2011)
I. Klebanov, S. Pufu, and B. Safdi, arXiv:

61. quantum critical point
Characteristics of
quantum critical point

62. quantum critical point
Characteristics of
quantum critical point

63. quantum critical point
Characteristics of
quantum critical point

64. quantum critical point
Characteristics of
quantum critical point

65. superposition and entanglement Quantum critical points of electrons
in crystals
String theory
Black holes

66. superposition and entanglement Quantum critical points of electrons
in crystals
String theory
Black holes

67. superposition and entanglement Quantum critical points of electrons
in crystals
String theory
Black holes

68. String theory

73. d Tensor network representation of entanglement
at quantum critical point d

74. String theory near
a D-brane d
Emergent direction
of AdS4

75. d Tensor network representation of entanglement
at quantum critical point d
Emergent direction
of AdS4
Brian Swingle, arXiv:

76. Entanglement entropy =d A
Entanglement entropy =
Number of links on optimal surface intersecting minimal number of links.
Emergent direction
of AdS4

77. r Quantum matter with long-range entanglement
Emergent holographic direction
J. McGreevy, arXiv

78. r A Quantum matter with long-range entanglement
Emergent holographic direction

79. r A Area measures entanglement entropy Quantum matter with
long-range entanglement A
Area measures
entanglement
entropy r
Emergent holographic direction
S. Ryu and T. Takayanagi, Phys. Rev. Lett. 96, (2006).

80. superposition and entanglement Quantum critical points of electrons
in crystals
String theory
Black holes

81. superposition and entanglement Quantum critical points of electrons
CFT3
Quantum critical points of electrons
in crystals
String theory
Black holes

82. superposition and entanglement Quantum critical points of electrons
in crystals
String theory
Black holes

83. Quantum critical point with non-local entanglement in spin wavefunction

86. Thermally excited
spin waves
Thermally excited
triplon particles

87. Thermally excited Thermally excited spin waves triplon particles
Short-range entanglement
Short-range entanglement

88. Excitations of a ground state with long-range entanglement
Thermally excited
spin waves
Thermally excited
triplon particles

89. Needed: Accurate theory of quantum critical spin dynamics
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

90. A 2+1 dimensional system at its quantum critical point
String theory at non-zero temperatures
A 2+1 dimensional system at its quantum critical point

91. 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 !

92. Objects so massive that light is gravitationally bound to them.
Black Holes
Objects so massive that light is gravitationally bound to them.

93. Objects so massive that light is gravitationally bound to them.
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.

94. 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

95. Quantum Entanglement across a black hole horizon_

96. Quantum Entanglement across a black hole horizon_

97. Quantum Entanglement across a black hole horizon_
Black hole
horizon

98. Quantum Entanglement across a black hole horizon_
Black hole
horizon

99. 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

100. 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

101. This entanglement leads to a black hole temperature
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)

102. A 2+1 dimensional system at its quantum critical point
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

103. Friction of quantum criticality = waves falling into black brane
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

104. A 2+1 dimensional system at its quantum critical point
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

105. superposition and entanglement Quantum critical points of electrons
in crystals
String theory
Black holes

106. superposition and entanglement Quantum critical points of electrons
in crystals
String theory
Black holes

107. Metals, “strange metals”, and high temperature superconductors
Insights from gravitational “duals”

108. High temperature superconductors

109. a new class of high temperature superconductors
Iron pnictides:
a new class of high temperature superconductors
Ishida, Nakai, and Hosono
arXiv: v1

110. H. Ikeda, H. Takeya, K. Hirata, T. Terashima, and Y. Matsuda,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 B 81, (2010)

111. AF Short-range entanglement in state with Neel (AF) order
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 B 81, (2010)

112. 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 B 81, (2010)

113. H. Ikeda, H. Takeya, K. Hirata, T. Terashima, and Y. Matsuda,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 B 81, (2010)

114. Sommerfeld-Bloch theory of ordinary metals
Momenta with
electron states
occupied
Momenta with
electron states
empty

115. Key feature of the theory:
Sommerfeld-Bloch theory of ordinary metals
Key feature of the theory:
the Fermi surface
115

116. H. Ikeda, H. Takeya, K. Hirata, T. Terashima, and Y. Matsuda,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 B 81, (2010)

117. H. Ikeda, H. Takeya, K. Hirata, T. Terashima, and Y. Matsuda,
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 B 81, (2010)

119. Ordinary
Metal

120. Strange
Metal
Ordinary
Metal

121. H. Ikeda, H. Takeya, K. Hirata, T. Terashima, and Y. Matsuda,
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 B 81, (2010)

122. 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 B 81, (2010)

123. Key (difficult) problem:
Describe quantum critical points and phases of systems with Fermi surfaces leading to metals with novel types of long-range entanglement +

124. Challenge to string theory: Describe quantum critical points
and phases of metals

125. Challenge to string theory: Describe quantum critical points
and phases of metals
Can we obtain gravitational theories
of superconductors and
ordinary Sommerfeld-Bloch metals ?

126. 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, (2011)

127. 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 ?

128. 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. D 79, (2009);
M. Cubrovic, J. Zaanen, and K. Schalm, Science 325, 439 (2009);
T. Faulkner, H. Liu, J. McGreevy, and D. Vegh, arXiv: ;
F. Denef, S. A. Hartnoll, and S. Sachdev, Phys. Rev. D 80, (2009)

129. 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 ?

130. 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:
L. Huijse, S. Sachdev, B. Swingle, Physical Review B 85, (2012)

131. Entanglement entropy of Fermi surfacesB A
Y. Zhang, T. Grover, and A. Vishwanath, Physical Review Letters 107, (2011)

132. r Quantum matter with long-range entanglement
Emergent holographic direction
J. McGreevy, arXiv

133. Quantum matter with
long-range entanglement r
Emergent holographic direction
Abandon conformal invariance, and only require scale invariance at long lengths and times.....

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

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

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

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

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

141. Simplest examples of long-range entanglement are in
Conclusions
Simplest examples of long-range entanglement are in
insulating antiferromagnets:
Z2 spin liquids and
quantum critical points

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

143. String theory and gravity in emergent dimensions
Conclusions
String theory and gravity in emergent dimensions
offer a remarkable new approach to describing states with long-range quantum entanglement.

144. String theory and gravity in emergent dimensions
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”

145. Emergent holographic direction
anti-de Sitter space
Emergent holographic direction r

146. anti-de Sitter space