1. Strong coupling problems in condensed matter and the AdS/CFT correspondence HARVARD arXiv:0910.1139 Reviews: Talk online: sachdev.physics.harvard.edu arXiv:0901.4103

2. Frederik Denef, Harvard Sean Hartnoll, Harvard Christopher Herzog, Princeton Pavel Kovtun, Victoria Dam Son, Washington Frederik Denef, Harvard Sean Hartnoll, Harvard Christopher Herzog, Princeton Pavel Kovtun, Victoria Dam Son, Washington Yejin Huh, Harvard Max Metlitski, Harvard Yejin Huh, Harvard Max Metlitski, Harvard HARVARD

3. 1. Quantum-critical transport Collisionless-t0-hydrodynamic crossover of CFT3s 2. Exact solution from AdS/CFT Constraints from duality relations 3. Quantum criticality of Dirac fermions “Vector” 1/N expansion 4. Quantum criticality of Fermi surfaces The genus expansion

4. 1. Quantum-critical transport Collisionless-t0-hydrodynamic crossover of CFT3s 2. Exact solution from AdS/CFT Constraints from duality relations 3. Quantum criticality of Dirac fermions “Vector” 1/N expansion 4. Quantum criticality of Fermi surfaces The genus expansion

5. The Superfluid-Insulator transition Boson Hubbard model M.P. A. Fisher, P.B. Weichmann, G. Grinstein, and D.S. Fisher, Phys. Rev. B 40, 546 (1989).

6. M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, and I. Bloch, Nature 415, 39 (2002). Superfluid-insulator transition

7. Insulator (the vacuum) at large U

8. Excitations:

10. Excitations of the insulator:

12. CFT3

14. Classical vortices and wave oscillations of the condensate Dilute Boltzmann/Landau gas of particle and holes

15. CFT at T>0

16. D. B. Haviland, Y. Liu, and A. M. Goldman, Phys. Rev. Lett. 62, 2180 (1989) Resistivity of Bi films M. P. A. Fisher, Phys. Rev. Lett. 65, 923 (1990)

17. Quantum critical transport S. Sachdev, Quantum Phase Transitions, Cambridge (1999).

18. Quantum critical transport K. Damle and S. Sachdev, Phys. Rev. B 56, 8714 (1997).

19. Quantum critical transport P. Kovtun, D. T. Son, and A. Starinets, Phys. Rev. Lett. 94, 11601 (2005), 8714 (1997).

20. Quantum critical transport

23. Density correlations in CFTs at T >0

25. Conformal mapping of plane to cylinder with circumference 1/T

27. Density correlations in CFTs at T >0 No hydrodynamics in CFT2s.

28. Density correlations in CFTs at T >0

29. K. Damle and S. Sachdev, Phys. Rev. B 56, 8714 (1997).

30. Density correlations in CFTs at T >0 K. Damle and S. Sachdev, Phys. Rev. B 56, 8714 (1997).

31. 1. Quantum-critical transport Collisionless-t0-hydrodynamic crossover of CFT3s 2. Exact solution from AdS/CFT Constraints from duality relations 3. Quantum criticality of Dirac fermions “Vector” 1/N expansion 4. Quantum criticality of Fermi surfaces The genus expansion

32. 1. Quantum-critical transport Collisionless-t0-hydrodynamic crossover of CFT3s 2. Exact solution from AdS/CFT Constraints from duality relations 3. Quantum criticality of Dirac fermions “Vector” 1/N expansion 4. Quantum criticality of Fermi surfaces The genus expansion

35. P. Kovtun, C. Herzog, S. Sachdev, and D.T. Son, Phys. Rev. D 75, 085020 (2007) Collisionless to hydrodynamic crossover of SYM3

36. P. Kovtun, C. Herzog, S. Sachdev, and D.T. Son, Phys. Rev. D 75, 085020 (2007) Collisionless to hydrodynamic crossover of SYM3

37. Universal constants of SYM3 P. Kovtun, C. Herzog, S. Sachdev, and D.T. Son, Phys. Rev. D 75, 085020 (2007) C. Herzog, JHEP 0212, 026 (2002)

38. Electromagnetic self-duality

39. K: a universal number analogous to the level number of the Kac-Moody algebra in 1+1 dimensions

41. C. Dasgupta and B.I. Halperin, Phys. Rev. Lett. 47, 1556 (1981)

43. C. Herzog, P. Kovtun, S. Sachdev, and D.T. Son, hep-th/0701036

44. The self-duality of the 4D abelian gauge fields leads to C. Herzog, P. Kovtun, S. Sachdev, and D.T. Son, hep-th/0701036

45. The self-duality of the 4D abelian gauge fields leads to C. Herzog, P. Kovtun, S. Sachdev, and D.T. Son, hep-th/0701036

46. Electromagnetic self-duality

47. 1. Quantum-critical transport Collisionless-t0-hydrodynamic crossover of CFT3s 2. Exact solution from AdS/CFT Constraints from duality relations 3. Quantum criticality of Dirac fermions “Vector” 1/N expansion 4. Quantum criticality of Fermi surfaces The genus expansion

48. 1. Quantum-critical transport Collisionless-t0-hydrodynamic crossover of CFT3s 2. Exact solution from AdS/CFT Constraints from duality relations 3. Quantum criticality of Dirac fermions “Vector” 1/N expansion 4. Quantum criticality of Fermi surfaces The genus expansion

49. d-wave superconductivity in cuprates Electron states occupied

50. d-wave superconductivity in cuprates + + - -

51. + + - -

52. 4 two-component Dirac fermions

53. d-wave superconductivity in cuprates

54. Time-reversal symmetry breaking

55. Lattice rotation symmetry breaking

56. M. Vojta, Y. Zhang, and S. Sachdev, Phys. Rev. Lett. 85, 4940 (2000) E.-A. Kim, M. J. Lawler, P. Oreto, S. Sachdev, E. Fradkin, S.A. Kivelson,.-A. Kim,er, P. OretoE. Fradkivelson, Phys. Rev. BPhys. Rev. B 77, 184514 (2008).

57. M. Vojta, Y. Zhang, and S. Sachdev, Physical Review Letters 85, 4940 (2000) Ising order and Dirac fermions couple via a “Yukawa” term. Nematic ordering Time reversal symmetry breaking

58. M. Vojta, Y. Zhang, and S. Sachdev, Physical Review Letters 85, 4940 (2000) Ising order and Dirac fermions couple via a “Yukawa” term. Nematic ordering Time reversal symmetry breaking

59. Integrating out the fermions yields an effective action for the scalar order parameter Expansion in number of fermion spin components N f Y. Huh and S. Sachdev, Physical Review B 78, 064512 (2008).

60. Integrating out the fermions yields an effective action for the scalar order parameter Expansion in number of fermion spin components N f Y. Huh and S. Sachdev, Physical Review B 78, 064512 (2008).

61. 1. Quantum-critical transport Collisionless-t0-hydrodynamic crossover of CFT3s 2. Exact solution from AdS/CFT Constraints from duality relations 3. Quantum criticality of Dirac fermions “Vector” 1/N expansion 4. Quantum criticality of Fermi surfaces The genus expansion

62. 1. Quantum-critical transport Collisionless-t0-hydrodynamic crossover of CFT3s 2. Exact solution from AdS/CFT Constraints from duality relations 3. Quantum criticality of Dirac fermions “Vector” 1/N expansion 4. Quantum criticality of Fermi surfaces The genus expansion

63. Quantum criticality of Pomeranchuk instability x y

64. Electron Green’s function in Fermi liquid (T=0)

66. Quantum criticality of Pomeranchuk instability x y

67. x y

68. x y

70. Quantum critical

71. Quantum criticality of Pomeranchuk instability Quantum critical Classical d=2 Ising criticality

72. Quantum criticality of Pomeranchuk instability Quantum critical D=2+1 Ising criticality ?

73. Quantum criticality of Pomeranchuk instability

78. Hertz theory

79. Quantum criticality of Pomeranchuk instability Hertz theory

80. Quantum criticality of Pomeranchuk instability Sung-Sik Lee, Physical Review B 80, 165102 (2009)

81. Quantum criticality of Pomeranchuk instability A string theory for the Fermi surface ?

85. AdS 4 -Reissner-Nordstrom black hole

86. 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

87. 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

88. 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

89. 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

90. AdS 4 -Reissner-Nordstrom black hole

91. AdS 2 x R 2 near-horizon geometry

92. 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

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

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

95. Green’s function of a fermion T. Faulkner, H. Liu, J. McGreevy, and D. Vegh, arXiv:0907.2694 See also M. Cubrovic, J Zaanen, and K. Schalm, arXiv:0904.1993

96. Green’s function of a fermion T. Faulkner, H. Liu, J. McGreevy, and D. Vegh, arXiv:0907.2694 Similar to non-Fermi liquid theories of Fermi surfaces coupled to gauge fields, and at quantum critical points

97. Free energy from gravity theory F. Denef, S. Hartnoll, and S. Sachdev, arXiv:0908.1788

98. General theory of finite temperature dynamics and transport near quantum critical points, with applications to antiferromagnets, graphene, and superconductors Conclusions

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