1. Electrical and thermal transport near quantum phase transitions in condensed matter, and in dyonic black holes
Sean Hartnoll (KITP) Pavel Kovtun (KITP) Marcus Müller (Harvard) Subir Sachdev (Harvard)

2. Transport near strongly interacting quantum critical points
Outline
Transport near strongly interacting quantum critical points
The superfluid-insulator transition in the boson Hubbard model: Hydrodynamic-collisionless crossover of a CFT
Exact solutions of CFTs in 1+1 dimensions No hydrodynamics
Exact solution of a CFT in 2+1 dimensions - Yang-Mills theory with N=8 supersymmetry: Black holes in AdS4
General hydrodynamic theory in the presence of a magnetic field, chemical potential and impurities: Nernst effect in the cuprate superconductors; Dyonic black holes in AdS4

3. Transport near strongly interacting quantum critical points
Outline
Transport near strongly interacting quantum critical points
The superfluid-insulator transition in the boson Hubbard model: Hydrodynamic-collisionless crossover of a CFT
Exact solutions of CFTs in 1+1 dimensions No hydrodynamics
Exact solution of a CFT in 2+1 dimensions - Yang-Mills theory with N=8 supersymmetry: Black holes in AdS4
General hydrodynamic theory in the presence of a magnetic field, chemical potential and impurities: Nernst effect in the cuprate superconductors; Dyonic black holes in AdS4

4. Trap for ultracold 87Rb atoms

5. M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, and I
M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, and I. Bloch, Nature 415, 39 (2002).

6. Boson Hubbard model
M.PA. Fisher, P.B. Weichmann, G. Grinstein, and D.S. Fisher Phys. Rev. B 40, 546 (1989).

7. M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, and I
M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, and I. Bloch, Nature 415, 39 (2002).

8. Velocity distribution of 87Rb atoms
Superfliud
M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, and I. Bloch, Nature 415, 39 (2002).

9. Velocity distribution of 87Rb atoms
Insulator
M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, and I. Bloch, Nature 415, 39 (2002).

10. The insulator:

11. Excitations of the insulator:

12. Excitations of the insulator:

13. Excitations of the insulator:

16. Depth of periodic potential
Non-zero temperature phase diagram
Superfluid
Insulator
Depth of periodic potential

17. Depth of periodic potential
Non-zero temperature phase diagram
Dynamics of the classical Gross-Pitaevski equation
Superfluid
Insulator
Depth of periodic potential

18. Depth of periodic potential
Non-zero temperature phase diagram
Dilute Boltzmann gas of particle and holes
Superfluid
Insulator
Depth of periodic potential

19. Depth of periodic potential
Non-zero temperature phase diagram
No wave or quasiparticle description
Superfluid
Insulator
Depth of periodic potential

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

21. Depth of periodic potential
Non-zero temperature phase diagram
Superfluid
Insulator
Depth of periodic potential

22. Depth of periodic potential
Non-zero temperature phase diagram
Collisionless-to hydrodynamic crossover of a conformal field theory (CFT)
Superfluid
Insulator
Depth of periodic potential
K. Damle and S. Sachdev, Phys. Rev. B 56, 8714 (1997).

23. Collisionless-to-hydrodynamic crossover of a CFT in 2+1 dimensions
K. Damle and S. Sachdev, Phys. Rev. B 56, 8714 (1997).

24. Collisionless-to-hydrodynamic crossover of a CFT in 2+1 dimensions
K. Damle and S. Sachdev, Phys. Rev. B 56, 8714 (1997).

25. Hydrodynamics of a conformal field theory (CFT)
The scattering cross-section of the thermal excitations is universal and so transport co-efficients are universally determined by kBT
Charge diffusion constant
Conductivity
K. Damle and S. Sachdev, Phys. Rev. B 56, 8714 (1997).

26. Transport near strongly interacting quantum critical points
Outline
Transport near strongly interacting quantum critical points
The superfluid-insulator transition in the boson Hubbard model: Hydrodynamic-collisionless crossover of a CFT
Exact solutions of CFTs in 1+1 dimensions No hydrodynamics
Exact solution of a CFT in 2+1 dimensions - Yang-Mills theory with N=8 supersymmetry: Black holes in AdS4
General hydrodynamic theory in the presence of a magnetic field, chemical potential and impurities: Nernst effect in the cuprate superconductors; Dyonic black holes in AdS4

27. Transport near strongly interacting quantum critical points
Outline
Transport near strongly interacting quantum critical points
The superfluid-insulator transition in the boson Hubbard model: Hydrodynamic-collisionless crossover of a CFT
Exact solutions of CFTs in 1+1 dimensions No hydrodynamics
Exact solution of a CFT in 2+1 dimensions - Yang-Mills theory with N=8 supersymmetry: Black holes in AdS4
General hydrodynamic theory in the presence of a magnetic field, chemical potential and impurities: Nernst effect in the cuprate superconductors; Dyonic black holes in AdS4

28. Exact solutions of CFTs in 1+1 dimensions

29. Exact solutions of CFTs in 1+1 dimensions

30. Transport near strongly interacting quantum critical points
Outline
Transport near strongly interacting quantum critical points
The superfluid-insulator transition in the boson Hubbard model: Hydrodynamic-collisionless crossover of a CFT
Exact solutions of CFTs in 1+1 dimensions No hydrodynamics
Exact solution of a CFT in 2+1 dimensions - Yang-Mills theory with N=8 supersymmetry: Black holes in AdS4
General hydrodynamic theory in the presence of a magnetic field, chemical potential and impurities: Nernst effect in the cuprate superconductors; Dyonic black holes in AdS4

31. Transport near strongly interacting quantum critical points
Outline
Transport near strongly interacting quantum critical points
The superfluid-insulator transition in the boson Hubbard model: Hydrodynamic-collisionless crossover of a CFT
Exact solutions of CFTs in 1+1 dimensions No hydrodynamics
Exact solution of a CFT in 2+1 dimensions - Yang-Mills theory with N=8 supersymmetry: Black holes in AdS4
General hydrodynamic theory in the presence of a magnetic field, chemical potential and impurities: Nernst effect in the cuprate superconductors; Dyonic black holes in AdS4

32. Hydrodynamics of a conformal field theory (CFT)
The AdS/CFT correspondence (Maldacena, Polyakov) relates the hydrodynamics of CFTs to the quantum gravity theory of the horizon of a black hole in Anti-de Sitter space.

33. Hydrodynamics of a conformal field theory (CFT)
The AdS/CFT correspondence (Maldacena, Polyakov) relates the hydrodynamics of CFTs to the quantum gravity theory of the horizon of a black hole in Anti-de Sitter space.
Holographic representation of black hole physics in a 2+1 dimensional CFT at a temperature equal to the Hawking temperature of the black hole.
3+1 dimensional AdS space
Black hole

34. Hydrodynamics of a conformal field theory (CFT)
Hydrodynamics of a CFT
Waves of gauge fields in a curved background

35. Hydrodynamics of a conformal field theory (CFT)
For the (unique) CFT with a SU(N) gauge field and 16 supercharges, we know the exact diffusion constant associated with a global SO(8) symmetry:
Spin diffusion constant
Spin conductivity
P. Kovtun, C. Herzog, S. Sachdev, and D.T. Son, Phys. Rev. D 75, (2007)

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

37. ImC/k2
CFT at T=0
P. Kovtun, C. Herzog, S. Sachdev, and D.T. Son, Phys. Rev. D 75, (2007)

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

39. ImC/k2
diffusion peak
P. Kovtun, C. Herzog, S. Sachdev, and D.T. Son, Phys. Rev. D 75, (2007)

40. Transport near strongly interacting quantum critical points
Outline
Transport near strongly interacting quantum critical points
The superfluid-insulator transition in the boson Hubbard model: Hydrodynamic-collisionless crossover of a CFT
Exact solutions of CFTs in 1+1 dimensions No hydrodynamics
Exact solution of a CFT in 2+1 dimensions - Yang-Mills theory with N=8 supersymmetry: Black holes in AdS4
General hydrodynamic theory in the presence of a magnetic field, chemical potential and impurities: Nernst effect in the cuprate superconductors; Dyonic black holes in AdS4

41. Transport near strongly interacting quantum critical points
Outline
Transport near strongly interacting quantum critical points
The superfluid-insulator transition in the boson Hubbard model: Hydrodynamic-collisionless crossover of a CFT
Exact solutions of CFTs in 1+1 dimensions No hydrodynamics
Exact solution of a CFT in 2+1 dimensions - Yang-Mills theory with N=8 supersymmetry: Black holes in AdS4
General hydrodynamic theory in the presence of a magnetic field, chemical potential and impurities: Nernst effect in the cuprate superconductors; Dyonic black holes in AdS4

42. For experimental applications, we must move away from the ideal CFT
e.g.

43. For experimental applications, we must move away from the ideal CFT
A chemical potential 
e.g.

44. For experimental applications, we must move away from the ideal CFT
A chemical potential 
CFT
e.g.

45. For experimental applications, we must move away from the ideal CFT
A chemical potential 
e.g.

46. For experimental applications, we must move away from the ideal CFT
A chemical potential 
CFT
e.g.

47. For experimental applications, we must move away from the ideal CFT
A chemical potential 
A magnetic field B
CFT
e.g.

49. S.A. Hartnoll, P.K. Kovtun, M. Müller, and S. Sachdev, arXiv:0706.3215

50. Conservation laws/equations of motion
S.A. Hartnoll, P.K. Kovtun, M. Müller, and S. Sachdev, arXiv:

51. S.A. Hartnoll, P.K. Kovtun, M. Müller, and S. Sachdev, arXiv:0706.3215
Constitutive relations which follow from Lorentz transformation to moving frame
S.A. Hartnoll, P.K. Kovtun, M. Müller, and S. Sachdev, arXiv:

52. S.A. Hartnoll, P.K. Kovtun, M. Müller, and S. Sachdev, arXiv:0706.3215
Single dissipative term allowed by requirement of positive entropy production. There is only one independent transport co-efficient
S.A. Hartnoll, P.K. Kovtun, M. Müller, and S. Sachdev, arXiv:

53. For experimental applications, we must move away from the ideal CFT
A chemical potential 
A magnetic field B
CFT
e.g.

54. For experimental applications, we must move away from the ideal CFT
A chemical potential 
A magnetic field B
An impurity scattering rate 1/imp (its T dependence follows from scaling arguments)
CFT
e.g.

55. S.A. Hartnoll, P.K. Kovtun, M. Müller, and S. Sachdev, arXiv:0706.3215

56. S.A. Hartnoll, P.K. Kovtun, M. Müller, and S. Sachdev, arXiv:0706.3215
From these relations, we obtained results for the transport co-efficients, expressed in terms of a “cyclotron” frequency and damping:
S.A. Hartnoll, P.K. Kovtun, M. Müller, and S. Sachdev, arXiv:

57. S.A. Hartnoll, P.K. Kovtun, M. Müller, and S. Sachdev, arXiv:0706.3215
From these relations, we obtained results for the transport co-efficients, expressed in terms of a “cyclotron” frequency and damping:
S.A. Hartnoll, P.K. Kovtun, M. Müller, and S. Sachdev, arXiv:

58. S.A. Hartnoll, P.K. Kovtun, M. Müller, and S. Sachdev, arXiv:0706.3215
From these relations, we obtained results for the transport co-efficients, expressed in terms of a “cyclotron” frequency and damping:
S.A. Hartnoll, P.K. Kovtun, M. Müller, and S. Sachdev, arXiv:

59. S.A. Hartnoll, P.K. Kovtun, M. Müller, and S. Sachdev, arXiv:0706.3215
From these relations, we obtained results for the transport co-efficients, expressed in terms of a “cyclotron” frequency and damping:
S.A. Hartnoll, P.K. Kovtun, M. Müller, and S. Sachdev, arXiv:

60. S.A. Hartnoll, P.K. Kovtun, M. Müller, and S. Sachdev, arXiv:0706.3215
From these relations, we obtained results for the transport co-efficients, expressed in terms of a “cyclotron” frequency and damping:
S.A. Hartnoll, P.K. Kovtun, M. Müller, and S. Sachdev, arXiv:

61. S.A. Hartnoll, P.K. Kovtun, M. Müller, and S. Sachdev, arXiv:0706.3215
From these relations, we obtained results for the transport co-efficients, expressed in terms of a “cyclotron” frequency and damping:
S.A. Hartnoll, P.K. Kovtun, M. Müller, and S. Sachdev, arXiv:

62. S.A. Hartnoll, P.K. Kovtun, M. Müller, and S. Sachdev, arXiv:0706.3215
From these relations, we obtained results for the transport co-efficients, expressed in terms of a “cyclotron” frequency and damping:
S.A. Hartnoll, P.K. Kovtun, M. Müller, and S. Sachdev, arXiv:

63. Transport near strongly interacting quantum critical points
Outline
Transport near strongly interacting quantum critical points
The superfluid-insulator transition in the boson Hubbard model: Hydrodynamic-collisionless crossover of a CFT
Exact solutions of CFTs in 1+1 dimensions No hydrodynamics
Exact solution of a CFT in 2+1 dimensions - Yang-Mills theory with N=8 supersymmetry: Black holes in AdS4
General hydrodynamic theory in the presence of a magnetic field, chemical potential and impurities: Nernst effect in the cuprate superconductors; Dyonic black holes in AdS4

64. Transport near strongly interacting quantum critical points
Outline
Transport near strongly interacting quantum critical points
The superfluid-insulator transition in the boson Hubbard model: Hydrodynamic-collisionless crossover of a CFT
Exact solutions of CFTs in 1+1 dimensions No hydrodynamics
Exact solution of a CFT in 2+1 dimensions - Yang-Mills theory with N=8 supersymmetry: Black holes in AdS4
General hydrodynamic theory in the presence of a magnetic field, chemical potential and impurities: Nernst effect in the cuprate superconductors; Dyonic black holes in AdS4

65. Temperature-doping phase diagram of the cuprate superconductors
STM in zero field

66. Y. Kohsaka, C. Taylor, K. Fujita, A. Schmidt, C. Lupien, T. Hanaguri, M. Azuma,
M. Takano, H. Eisaki, H. Takagi, S. Uchida, and J. C. Davis, Science 315, 1380 (2007)

67. Y. Kohsaka, C. Taylor, K. Fujita, A. Schmidt, C. Lupien, T. Hanaguri, M. Azuma,
M. Takano, H. Eisaki, H. Takagi, S. Uchida, and J. C. Davis, Science 315, 1380 (2007)

68. Y. Kohsaka, C. Taylor, K. Fujita, A. Schmidt, C. Lupien, T. Hanaguri, M. Azuma,
M. Takano, H. Eisaki, H. Takagi, S. Uchida, and J. C. Davis, Science 315, 1380 (2007)

69. Y. Kohsaka, C. Taylor, K. Fujita, A. Schmidt, C. Lupien, T. Hanaguri, M. Azuma,
M. Takano, H. Eisaki, H. Takagi, S. Uchida, and J. C. Davis, Science 315, 1380 (2007)

70. Y. Kohsaka, C. Taylor, K. Fujita, A. Schmidt, C. Lupien, T. Hanaguri, M. Azuma,
M. Takano, H. Eisaki, H. Takagi, S. Uchida, and J. C. Davis, Science 315, 1380 (2007)

71. “Glassy” Valence Bond Solid (VBS) ?
Y. Kohsaka, C. Taylor, K. Fujita, A. Schmidt, C. Lupien, T. Hanaguri, M. Azuma,
M. Takano, H. Eisaki, H. Takagi, S. Uchida, and J. C. Davis, Science 315, 1380 (2007)

72. Temperature-doping phase diagram of the cuprate superconductors
“Glassy” Valence Bond Solid (VBS) ?

73. Dip in Tc near x=1/8 indicates proximity of insulator
LSCO Phase diagram
Dip in Tc near x=1/8 indicates proximity of insulator

74. For experimental applications, we must move away from the ideal CFT
A chemical potential 
CFT
e.g.

75.  For experimental applications, we must move away from the ideal CFT
A chemical potential 
Doping route in cuprate
For the cuprates, the Mott insulator is realized e.g. at x=1/8
CFT
e.g.

76. S.A. Hartnoll, P.K. Kovtun, M. Müller, and S. Sachdev, arXiv:0706.3215
LSCO - Theory
Only input parameters
Output
S.A. Hartnoll, P.K. Kovtun, M. Müller, and S. Sachdev, arXiv:

77. LSCO - Experiments

78. Transport near strongly interacting quantum critical points
Outline
Transport near strongly interacting quantum critical points
The superfluid-insulator transition in the boson Hubbard model: Hydrodynamic-collisionless crossover of a CFT
Exact solutions of CFTs in 1+1 dimensions No hydrodynamics
Exact solution of a CFT in 2+1 dimensions - Yang-Mills theory with N=8 supersymmetry: Black holes in AdS4
General hydrodynamic theory in the presence of a magnetic field, chemical potential and impurities: Nernst effect in the cuprate superconductors; Dyonic black holes in AdS4

79. Transport near strongly interacting quantum critical points
Outline
Transport near strongly interacting quantum critical points
The superfluid-insulator transition in the boson Hubbard model: Hydrodynamic-collisionless crossover of a CFT
Exact solutions of CFTs in 1+1 dimensions No hydrodynamics
Exact solution of a CFT in 2+1 dimensions - Yang-Mills theory with N=8 supersymmetry: Black holes in AdS4
General hydrodynamic theory in the presence of a magnetic field, chemical potential and impurities: Nernst effect in the cuprate superconductors; Dyonic black holes in AdS4

80. To the solvable supersymmetric, Yang-Mills theory CFT, we add
A chemical potential 
A magnetic field B
After the AdS/CFT mapping, we obtain the Einstein-Maxwell theory of a black hole with
An electric charge
A magnetic charge
The exact results are found to be in precise accord with all hydrodynamic results presented earlier
S.A. Hartnoll, P.K. Kovtun, M. Müller, and S. Sachdev, arXiv:

81. Conclusions
General theory of transport in a weakly disordered ``vortex liquid’’ state.
“Relativistic” magnetohydrodynamics offers an efficient approach to disentangling momentum and charge transport
Exact solutions via black hole mapping have yielded first exact results for transport co-efficients in interacting many-body systems, and were valuable in determining general structure of hydrodynamics.
Simplest model reproduces many trends of the Nernst measurements in cuprates.