1. Quantum criticality, the cuprate superconductors, and the AdS/CFT correspondence HARVARD Talk online: sachdev.physics.harvard.edu

2. Max Metlitski, Harvard HARVARD arXiv:1001.1153 Eun Gook Moon, Harvard

3. 1. Coupled dimer antiferromagnets Introduction to quantum criticality 2. Theory of Ising-nematic ordering in the cuprate metals Strongly-coupled field theory 3. The AdS/CFT correspondence Phases of finite density quantum matter at strong coupling 4. Phase diagram of the cuprates Quantum criticality of the competition between antiferromagnetism and superconductivity Outline

4. 1. Coupled dimer antiferromagnets Introduction to quantum criticality 2. Theory of Ising-nematic ordering in the cuprate metals Strongly-coupled field theory 3. The AdS/CFT correspondence Phases of finite density quantum matter at strong coupling 4. Phase diagram of the cuprates Quantum criticality of the competition between antiferromagnetism and superconductivity Outline

5. The cuprate superconductors

6. Ground state has long-range Néel order Square lattice antiferromagnet

7. TlCuCl 3

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

9. Ground state has long-range Néel order Square lattice antiferromagnet

10. J J/ Weaken some bonds to induce spin entanglement in a new quantum phase

11. Square lattice antiferromagnet J J/ Ground state is a “quantum paramagnet” with spins locked in valence bond singlets

12. Pressure in TlCuCl 3

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

14. CFT3

20. N. Cavadini, G. Heigold, W. Henggeler, A. Furrer, H.-U. Güdel, K. Krämer and H. Mutka, Phys. Rev. B 63 172414 (2001). TlCuCl 3 at ambient pressure

21. N. Cavadini, G. Heigold, W. Henggeler, A. Furrer, H.-U. Güdel, K. Krämer and H. Mutka, Phys. Rev. B 63 172414 (2001). Sharp spin 1 particle excitation above an energy gap (spin gap) TlCuCl 3 at ambient pressure

23. Spin waves

26. 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) TlCuCl 3 with varying pressure

27. Prediction of quantum field theory

29. S. Sachdev, arXiv:0901.4103

30. CFT3

31. Pressure in TlCuCl 3 S. Sachdev and J. Ye, Phys. Rev. Lett. 69, 2411 (1992).

32. CFT3 at T>0 Pressure in TlCuCl 3 S. Sachdev and J. Ye, Phys. Rev. Lett. 69, 2411 (1992).

33. CFT3 at T>0 Pressure in TlCuCl 3 S. Sachdev and J. Ye, Phys. Rev. Lett. 69, 2411 (1992).

34. 1. Coupled dimer antiferromagnets Introduction to quantum criticality 2. Theory of Ising-nematic ordering in the cuprate metals Strongly-coupled field theory 3. The AdS/CFT correspondence Phases of finite density quantum matter at strong coupling 4. Phase diagram of the cuprates Quantum criticality of the competition between antiferromagnetism and superconductivity Outline

35. 1. Coupled dimer antiferromagnets Introduction to quantum criticality 2. Theory of Ising-nematic ordering in the cuprate metals Strongly-coupled field theory 3. The AdS/CFT correspondence Phases of finite density quantum matter at strong coupling 4. Phase diagram of the cuprates Quantum criticality of the competition between antiferromagnetism and superconductivity Outline

36. Central ingredients in cuprate phase diagram: antiferromagnetism, superconductivity, and change in Fermi surface

37. Strange Metal

38. Quantum criticality of Pomeranchuk instability x y

39. x y

40. x y

41. Nematic order in YBCO V. Hinkov, D. Haug, B. Fauqué, P. Bourges, Y. Sidis, A. Ivanov, C. Bernhard, C. T. Lin, and B. Keimer, Science 319, 597 (2008)

42. Broken rotational symmetry in the pseudogap phase of a high-Tc superconductor R. DaouR. Daou, J. Chang, David LeBoeuf, Olivier Cyr-Choiniere, Francis Laliberte, Nicolas Doiron-Leyraud, B. J. Ramshaw, Ruixing Liang,g, DavidOlivier Cyr-Cis Laliberte, NicolasRamshaw, Ruixing D. A. Bonn, W. N. Hardy, and Louis Taillefer W. N. Hardy, and Lou Nature, 463, 519 (2010)., 519 (2010).

44. Quantum criticality of Pomeranchuk instability x y

45. x y

47. Quantum critical T I-n

48. Quantum criticality of Pomeranchuk instability Quantum critical T I-n Strange Metal ?

55. 1. Coupled dimer antiferromagnets Introduction to quantum criticality 2. Theory of Ising-nematic ordering in the cuprate metals Strongly-coupled field theory 3. The AdS/CFT correspondence Phases of finite density quantum matter at strong coupling 4. Phase diagram of the cuprates Quantum criticality of the competition between antiferromagnetism and superconductivity Outline

56. 1. Coupled dimer antiferromagnets Introduction to quantum criticality 2. Theory of Ising-nematic ordering in the cuprate metals Strongly-coupled field theory 3. The AdS/CFT correspondence Phases of finite density quantum matter at strong coupling 4. Phase diagram of the cuprates Quantum criticality of the competition between antiferromagnetism and superconductivity Outline

57. e.g. Graphene at zero bias

59. e.g. Graphene at non-zero bias

60. AdS/CFT correspondence The quantum theory of a black hole in a 3+1- dimensional negatively curved AdS universe is holographically represented by a CFT (the theory of a quantum critical point) in 2+1 dimensions Maldacena, Gubser, Klebanov, Polyakov, Witten 3+1 dimensional AdS space

61. AdS/CFT correspondence The quantum theory of a black hole in a 3+1- dimensional negatively curved AdS universe is holographically represented by a CFT (the theory of a quantum critical point) in 2+1 dimensions Maldacena, Gubser, Klebanov, Polyakov, Witten 3+1 dimensional AdS space A 2+1 dimensional system at its quantum critical point

62. AdS/CFT correspondence The quantum theory of a black hole in a 3+1- dimensional negatively curved AdS universe is holographically represented by a CFT (the theory of a quantum critical point) in 2+1 dimensions Maldacena, Gubser, Klebanov, Polyakov, Witten 3+1 dimensional AdS space Quantum criticality in 2+1 dimensions Black hole temperature = temperature of quantum criticality

63. AdS/CFT correspondence The quantum theory of a black hole in a 3+1- dimensional negatively curved AdS universe is holographically represented by a CFT (the theory of a quantum critical point) in 2+1 dimensions Strominger, Vafa 3+1 dimensional AdS space Quantum criticality in 2+1 dimensions Black hole entropy = entropy of quantum criticality

64. AdS/CFT correspondence The quantum theory of a black hole in a 3+1- dimensional negatively curved AdS universe is holographically represented by a CFT (the theory of a quantum critical point) in 2+1 dimensions 3+1 dimensional AdS space Quantum criticality in 2+1 dimensions Quantum critical dynamics = waves in curved space Maldacena, Gubser, Klebanov, Polyakov, Witten

65. AdS/CFT correspondence The quantum theory of a black hole in a 3+1- dimensional negatively curved AdS universe is holographically represented by a CFT (the theory of a quantum critical point) in 2+1 dimensions 3+1 dimensional AdS space Quantum criticality in 2+1 dimensions Friction of quantum criticality = waves falling into black hole Kovtun, Policastro, Son

66. AdS/CFT correspondence The quantum theory of a black hole in a 3+1- dimensional negatively curved AdS universe is holographically represented by a CFT (the theory of a quantum critical point) in 2+1 dimensions 3+1 dimensional AdS space Quantum criticality in 2+1 dimensions Friction of quantum criticality = waves falling into black hole Kovtun, Policastro, Son

68. T. Faulkner, H. Liu, J. McGreevy, and D. Vegh, arXiv:0907.2694 F. Denef, S. Hartnoll, and S. Sachdev, arXiv:0908.1788 Examine free energy and Green’s function of a probe particle

69. Short time behavior depends upon conformal AdS 4 geometry near boundary T. Faulkner, H. Liu, J. McGreevy, and D. Vegh, arXiv:0907.2694 F. Denef, S. Hartnoll, and S. Sachdev, arXiv:0908.1788

70. Long time behavior depends upon near-horizon geometry of black hole T. Faulkner, H. Liu, J. McGreevy, and D. Vegh, arXiv:0907.2694 F. Denef, S. Hartnoll, and S. Sachdev, arXiv:0908.1788

71. Radial direction of gravity theory is measure of energy scale in CFT T. Faulkner, H. Liu, J. McGreevy, and D. Vegh, arXiv:0907.2694 F. Denef, S. Hartnoll, and S. Sachdev, arXiv:0908.1788

72. J. McGreevy, arXiv0909.0518

73. T. Faulkner, H. Liu, J. McGreevy, and D. Vegh, arXiv:0907.2694 Infrared physics of Fermi surface is linked to the near horizon AdS 2 geometry of Reissner-Nordstrom black hole

74. T. Faulkner, H. Liu, J. McGreevy, and D. Vegh, arXiv:0907.2694 AdS 4 Geometric interpretation of RG flow

75. T. Faulkner, H. Liu, J. McGreevy, and D. Vegh, arXiv:0907.2694 AdS 2 x R 2 Geometric interpretation of RG flow

76. Green’s function of a fermion T. Faulkner, H. Liu, J. McGreevy, and D. Vegh, arXiv:0907.2694 See also S.-S. Lee, Phys. Rev. D 79, 086006 (2009); M. Cubrovic, J. Zaanen, and K. Schalm, Science 325, 439 (2009);F. Denef, S. A. Hartnoll, and S. Sachdev, Phys. Rev. D 80, 126016 (2009)

77. Green’s function of a fermion Similar to our theory of the singular Fermi surface near the Ising-nematic quantum critical point T. Faulkner, H. Liu, J. McGreevy, and D. Vegh, arXiv:0907.2694

78. Green’s function of a fermion Similar to our theory of the singular Fermi surface near the Ising-nematic quantum critical point T. Faulkner, H. Liu, J. McGreevy, and D. Vegh, arXiv:0907.2694

79. 1. Coupled dimer antiferromagnets Introduction to quantum criticality 2. Theory of Ising-nematic ordering in the cuprate metals Strongly-coupled field theory 3. The AdS/CFT correspondence Phases of finite density quantum matter at strong coupling 4. Phase diagram of the cuprates Quantum criticality of the competition between antiferromagnetism and superconductivity Outline

80. 1. Coupled dimer antiferromagnets Introduction to quantum criticality 2. Theory of Ising-nematic ordering in the cuprate metals Strongly-coupled field theory 3. The AdS/CFT correspondence Phases of finite density quantum matter at strong coupling 4. Phase diagram of the cuprates Quantum criticality of the competition between antiferromagnetism and superconductivity Outline

81. Central ingredients in cuprate phase diagram: antiferromagnetism, superconductivity, and change in Fermi surface Strange Metal

82. Antiferro- magnetism Fermi surface d-wave supercon- ductivity

83. Antiferro- magnetism Fermi surface d-wave supercon- ductivity

84. Pressure in TlCuCl 3 Christian Ruegg et al., Phys. Rev. Lett. 100, 205701 (2008) S. Sachdev and J. Ye, Phys. Rev. Lett. 69, 2411 (1992). Canonical quantum critical phase diagram of coupled-dimer antiferromagnet

85. Antiferro- magnetism Fermi surface d-wave supercon- ductivity

86. Antiferro- magnetism Fermi surface d-wave supercon- ductivity

87. Fermi surface+antiferromagnetism Hole states occupied Electron states occupied +

88. S. Sachdev, A. V. Chubukov, and A. Sokol, Phys. Rev. B 51, 14874 (1995). A. V. Chubukov and D. K. Morr, Physics Reports 288, 355 (1997). Hole pockets Electron pockets Hole-doped cuprates

89. S. Sachdev, A. V. Chubukov, and A. Sokol, Phys. Rev. B 51, 14874 (1995). A. V. Chubukov and D. K. Morr, Physics Reports 288, 355 (1997). Hole pockets Electron pockets Hole-doped cuprates

90. S. Sachdev, A. V. Chubukov, and A. Sokol, Phys. Rev. B 51, 14874 (1995). A. V. Chubukov and D. K. Morr, Physics Reports 288, 355 (1997). Hole pockets Electron pockets Hole-doped cuprates Hot spots

91. S. Sachdev, A. V. Chubukov, and A. Sokol, Phys. Rev. B 51, 14874 (1995). A. V. Chubukov and D. K. Morr, Physics Reports 288, 355 (1997). Hole pockets Electron pockets Hole-doped cuprates Fermi surface breaks up at hot spots into electron and hole “pockets” Hole pockets Hot spots

92. S. Sachdev, A. V. Chubukov, and A. Sokol, Phys. Rev. B 51, 14874 (1995). A. V. Chubukov and D. K. Morr, Physics Reports 288, 355 (1997). Hole pockets Electron pockets Hole-doped cuprates Fermi surface breaks up at hot spots into electron and hole “pockets” Hot spots

93. Theory of quantum criticality in the cuprates T*T*

94. Antiferro- magnetism Fermi surface d-wave supercon- ductivity

95. Antiferro- magnetism Fermi surface d-wave supercon- ductivity

96. Theory of quantum criticality in the cuprates

100. Criticality of the coupled dimer antiferromagnet at x=x s

101. Theory of quantum criticality in the cuprates Criticality of the topological change in Fermi surface at x=x m

102. Theory of quantum criticality in the cuprates

104. Similar phase diagram for the pnictides Ishida, Nakai, and Hosono arXiv:0906.2045v1 S. Nandi, M. G. Kim, A. Kreyssig, R. M. Fernandes, D. K. Pratt, A. Thaler, N. Ni, S. L. Bud'ko, P. C. Canfield, J. Schmalian, R. J. McQueeney, A. I. Goldman, arXiv:0911.3136.

105. H c2 T*T*

106. G. Knebel, D. Aoki, and J. Flouquet, arXiv:0911.5223 Similar phase diagram for CeRhIn 5

107. Identified quantum criticality in cuprate superconductors with a critical point at optimal doping associated with onset of spin density wave order in a metal Conclusions Elusive optimal doping quantum critical point has been “hiding in plain sight”. It is shifted to lower doping by the onset of superconductivity

108. Conclusions Theories for the onset of spin density wave and Ising- nematic order in metals are strongly coupled in two dimensions

109. The AdS/CFT offers promise in providing a new understanding of strongly interacting quantum matter at non-zero density Conclusions