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1 Talk online at

2 Outline 1. CFT3s in condensed matter physics Superfluid-insulator and Neel-valence bond solid transitions 2. Quantum-critical transport Collisionless-t0-hydrodynamic crossover of CFT3s 3. SYM3 with N = 8 supersymmetry 4. Nernst effect in the cuprate superconductors Quantum criticality and dyonic black holes

3 Outline 1. CFT3s in condensed matter physics Superfluid-insulator and Neel-valence bond solid transitions 2. Quantum-critical transport Collisionless-t0-hydrodynamic crossover of CFT3s 3. SYM3 with N = 8 supersymmetry 4. Nernst effect in the cuprate superconductors Quantum criticality and dyonic black holes

4 Ultracold 87 Rb atoms - bosons M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, and I. Bloch, Nature 415, 39 (2002).

5 The insulator:

6 Excitations of the insulator:

7 Excitations of the insulator: S = d 2 rdτ [ τ ψ 2 + c 2 ψ 2 + s ψ 2 + u 2 ψ 4]

8 S = d 2 rdτ [ τ ψ 2 + c 2 ψ 2 + s ψ 2 + u 2 ψ 4]

9 S = d 2 rdτ [ τ ψ 2 + c 2 ψ 2 + s ψ 2 + u 2 ψ 4]

10 Antiferromagnetism in the cuprate superconductors Ca 1.90 Na 0.10 CuO 2 Cl 2 Bi 2.2 Sr 1.8 Ca 0.8 Dy 0.2 Cu 2 O y

11 S=1/2 Heisenberg antiferromagnets on the square lattice H = J S i S j Q For small Q ij ijkl ( Si S j 1 4 )( Sk S l 1 4) Order parameter Φ =( 1) i S i

12 S=1/2 Heisenberg antiferromagnets on the square lattice H = J S i S j Q ij ijkl Low energy field theory ( Si S j 1 4 )( Sk S l 1 4)

13 S=1/2 Heisenberg antiferromagnets on the square lattice H = J S i S j Q ij ijkl Low energy field theory ( Si S j 1 4 )( Sk S l 1 4) Incorrect: predicts a state for s>s c with no broken symmetries, a gap to all excitations, and a low energy S = 1 quasiparticle of Φ quanta. There is no such state for the S =1/2 antiferromagnet

14 S=1/2 Heisenberg antiferromagnets on the square lattice H = J S i S j Q ij ijkl Low energy field theory ( Si S j 1 4 )( Sk S l 1 4)

15 S=1/2 Heisenberg antiferromagnets on the square lattice H = J S i S j Q Phase diagram ij ijkl ( Si S j 1 4 )( Sk S l 1 4) Coulomb phase Gapless photon A μ Higgs phase: Néel order z α 0, Φ 0. No broken symmetry?

16 S=1/2 Heisenberg antiferromagnets on the square lattice H = J S i S j Q Phase diagram ij ijkl ( Si S j 1 4 )( Sk S l 1 4) Coulomb phase Gapless photon A μ Higgs phase: Néel order z α 0, Φ 0. No broken symmetry?

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18 N. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694 (1989).

19 S=1/2 Heisenberg antiferromagnets on the square lattice H = J S i S j Q ij ijkl ( Si S j 1 4 )( Sk S l 1 4) Phase diagram or Higgs phase: Néel order z α 0, Φ 0. Coulomb phase: VBS order, Monopole V 0

20 Outline 1. CFT3s in condensed matter physics Superfluid-insulator and Neel-valence bond solid transitions 2. Quantum-critical transport Collisionless-t0-hydrodynamic crossover of CFT3s 3. SYM3 with N = 8 supersymmetry 4. Nernst effect in the cuprate superconductors Quantum criticality and dyonic black holes

21 Outline 1. CFT3s in condensed matter physics Superfluid-insulator and Neel-valence bond solid transitions 2. Quantum-critical transport Collisionless-t0-hydrodynamic crossover of CFT3s 3. SYM3 with N = 8 supersymmetry 4. Nernst effect in the cuprate superconductors Quantum criticality and dyonic black holes

22 S = d 2 rdτ [ τ ψ 2 + c 2 ψ 2 + s ψ 2 + u 2 ψ 4]

23 T Quantum critical T KT 0 Superfluid g c Insulator g

24 T Wave oscillations of the condensate (classical Gross- Pitaevski equation) Quantum critical T KT 0 Superfluid g c Insulator g

25 T Dilute Boltzmann gas of particle and holes Quantum critical T KT 0 Superfluid g c Insulator g

26 CFT at T>0 T Quantum critical T KT 0 Superfluid g c Insulator g

27 Resistivity of Bi films Conductivity σ σ Superconductor (T 0) = σ Insulator (T 0) = 0 σ Quantum critical point (T 0) 4e2 h 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)

28 Density correlations in CFTs at T >0 Two-point density correlator, χ(k, ω) Kubo formula for conductivity σ(ω) = lim k 0 iω χ(k, ω) k2

29 Density correlations in CFTs at T >0 Two-point density correlator, χ(k, ω) Kubo formula for conductivity σ(ω) = lim k 0 iω χ(k, ω) k2

30 Density correlations in CFTs at T >0 Two-point density correlator, χ(k, ω) Kubo formula for conductivity σ(ω) = lim k 0 iω χ(k, ω) k2 K. Damle and S. Sachdev, Phys. Rev. B 56, 8714 (1997).

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

32 Outline 1. CFT3s in condensed matter physics Superfluid-insulator and Neel-valence bond solid transitions 2. Quantum-critical transport Collisionless-t0-hydrodynamic crossover of CFT3s 3. SYM3 with N = 8 supersymmetry 4. Nernst effect in the cuprate superconductors Quantum criticality and dyonic black holes

33 Outline 1. CFT3s in condensed matter physics Superfluid-insulator and Neel-valence bond solid transitions 2. Quantum-critical transport Collisionless-t0-hydrodynamic crossover of CFT3s 3. SYM3 with N = 8 supersymmetry 4. Nernst effect in the cuprate superconductors Quantum criticality and dyonic black holes

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35 Collisionless to hydrodynamic crossover of SYM3 Imχ(k, ω)/k 2 Im K k2 ω 2 P. Kovtun, C. Herzog, S. Sachdev, and D.T. Son, Phys. Rev. D 75, (2007)

36 Collisionless to hydrodynamic crossover of SYM3 Imχ(k, ω)/k 2 Im Dχ c Dk 2 iω P. Kovtun, C. Herzog, S. Sachdev, and D.T. Son, Phys. Rev. D 75, (2007)

37 Universal constants of SYM3 χ c = k BT (hv) 2 Θ 1 D = hv2 k B T Θ 2 σ(ω) = 4e 2 4e 2 2N 3/2 K = 3 Θ 1 = 8π2 2N 3/2 Θ 2 = 3 8π 2 h K, ω k BT h Θ 1Θ 2, ω k B T C. Herzog, JHEP 0212, 026 (2002) P. Kovtun, C. Herzog, S. Sachdev, and D.T. Son, Phys. Rev. D 75, (2007) 9

38 Electromagnetic self-duality

39 Outline 1. CFT3s in condensed matter physics Superfluid-insulator and Neel-valence bond solid transitions 2. Quantum-critical transport Collisionless-t0-hydrodynamic crossover of CFT3s 3. SYM3 with N = 8 supersymmetry 4. Nernst effect in the cuprate superconductors Quantum criticality and dyonic black holes

40 Outline 1. CFT3s in condensed matter physics Superfluid-insulator and Neel-valence bond solid transitions 2. Quantum-critical transport Collisionless-t0-hydrodynamic crossover of CFT3s 3. SYM3 with N = 8 supersymmetry 4. Nernst effect in the cuprate superconductors Quantum criticality and dyonic black holes

41 Dope the antiferomagnets with charge carriers of density x by applying a chemical potential μ Ca 1.90 Na 0.10 CuO 2 Cl 2 Bi 2.2 Sr 1.8 Ca 0.8 Dy 0.2 Cu 2 O y

42 Superconductor T μ

43 T Superconductor Nernst measurements μ

44 Nernst experiment e y H m H

45 T Superconductor Nernst measurements μ

46 T Nernst measurements Superconductor μ Scanning tunnelling microscopy

47 STM studies of the underdoped superconductor Ca 1.90 Na 0.10 CuO 2 Cl 2 Bi 2.2 Sr 1.8 Ca 0.8 Dy 0.2 Cu 2 O y

48 Topograph Ca 1.90 Na 0.10 CuO 2 Cl 2 Bi 2.2 Sr 1.8 Ca 0.8 Dy 0.2 Cu 2 O y 12 nm Y. Kohsaka et al. Science 315, 1380 (2007)

49 di/dv Spectra Ca 1.90 Na 0.10 CuO 2 Cl 2 Bi 2.2 Sr 1.8 Ca 0.8 Dy 0.2 Cu 2 O y Intense Tunneling-Asymmetry (TA) variation are highly similar Y. Kohsaka et al. Science 315, 1380 (2007)

50 Topograph Ca 1.90 Na 0.10 CuO 2 Cl 2 Bi 2.2 Sr 1.8 Ca 0.8 Dy 0.2 Cu 2 O y 12 nm Y. Kohsaka et al. Science 315, 1380 (2007)

51 Tunneling Asymmetry (TA)-map at E=150meV Ca 1.90 Na 0.10 CuO 2 Cl 2 Bi 2.2 Sr 1.8 Ca 0.8 Dy 0.2 Cu 2 O y 12 nm Y. Kohsaka et al. Science 315, 1380 (2007)

52 Tunneling Asymmetry (TA)-map at E=150meV Ca 1.90 Na 0.10 CuO 2 Cl 2 Bi 2.2 Sr 1.8 Ca 0.8 Dy 0.2 Cu 2 O y 12 nm Y. Kohsaka et al. Science 315, 1380 (2007)

53 Tunneling Asymmetry (TA)-map at E=150meV Ca 1.90 Na 0.10 CuO 2 Cl 2 Bi 2.2 Sr 1.8 Ca 0.8 Dy 0.2 Cu 2 O y 12 nm Indistinguishable bond-centered TA contrast with disperse 4a 0 -wide nanodomains Y. Kohsaka et al. Science 315, 1380 (2007)

54 TA Contrast is at oxygen site (Cu-O-Cu bond-centered) R map (150 mv) Ca 1.88 Na 0.12 CuO 2 Cl 2, 4 K 4a 0 12 nm Y. Kohsaka et al. Science 315, 1380 (2007)

55 TA Contrast is at oxygen site (Cu-O-Cu bond-centered) R map (150 mv) Ca 1.88 Na 0.12 CuO 2 Cl 2, 4 K 4a 0 12 nm Evidence for VBS order - a valence bond supersolid S. Sachdev and N. Read, Int. J. Mod. Phys. B 5, 219 (1991).

56 Superconductor T μ

57 T g Superconductor μ Insulator x =1/8

58 T g Superconductor μ Insulator x =1/8

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

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61 S.A. Hartnoll, P.K. Kovtun, M. Müller, and S. Sachdev, Phys. Rev. B (2007)

62 Conservation laws/equations of motion S.A. Hartnoll, P.K. Kovtun, M. Müller, and S. Sachdev, Phys. Rev. B (2007)

63 Constitutive relations which follow from Lorentz transformation to moving frame S.A. Hartnoll, P.K. Kovtun, M. Müller, and S. Sachdev, Phys. Rev. B (2007)

64 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, Phys. Rev. B (2007)

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

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

67 S.A. Hartnoll, P.K. Kovtun, M. Müller, and S. Sachdev, Phys. Rev. B (2007)

68 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, Phys. Rev. B (2007)

69 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, Phys. Rev. B (2007)

70 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, Phys. Rev. B (2007)

71 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, Phys. Rev. B (2007)

72 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, Phys. Rev. B (2007)

73 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, Phys. Rev. B (2007)

74 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, Phys. Rev. B (2007)

75 From these relations, we obtained results for the transport co-efficients, expressed in terms of a cyclotron frequency and damping: Transverse thermoelectric co-efficient ( ) ( ) 2 h α xy =Φ s B (k B T ) 2 2πτimp ρ 2 +Φ σ Φ ε+p (k B T ) 3 /2πτ imp 2ek B Φ 2 ε+p (k BT ) 6 + B 2 ρ 2 (2πτ imp / ), 2 where B = Bφ 0 /( v) 2 ; ρ = ρ/( v) 2. S.A. Hartnoll, P.K. Kovtun, M. Müller, and S. Sachdev, Phys. Rev. B (2007)

76 LSCO - Theory S.A. Hartnoll, P.K. Kovtun, M. Müller, and S. Sachdev, Phys. Rev. B (2007)

77 LSCO - Experiments N. P. Ong et al.

78 LSCO - Theory Only input parameters v = 47 mev Å τ imp s Output ω c =6.2GHz B ( 35K 1T T ) 3 S.A. Hartnoll, P.K. Kovtun, M. Müller, and S. Sachdev, Phys. Rev. B (2007)

79 LSCO - Theory Only input parameters v = 47 mev Å τ imp s Output ω c =6.2GHz B ( 35K 1T T Similar to velocity estimates by A.V. Balatsky and Z-X. Shen, Science 284, 1137 (1999). S.A. Hartnoll, P.K. Kovtun, M. Müller, and S. Sachdev, Phys. Rev. B (2007) ) 3

80 To the solvable supersymmetric, Yang-Mills theory CFT, we add A chemical potential μ 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, Phys. Rev. B (2007)

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