Theory of the Nernst effect near the superfluid-insulator transition

Transcription

1 Theory of the Nernst effect near the superfluid-insulator transition Sean Hartnoll (KITP), Christopher Herzog (Washington), Pavel Kovtun (KITP), Marcus Mueller (Harvard), Subir Sachdev (Harvard), Dam Son (Washington)

2 Outline 1. Superfluid/supersolid/insulator quantum transitions Insulators at integer and commensurate densities 2. Theory of quantum-critical transport Collisionless-t0-hydrodynamic crossover of conformal field theories 3. Hydrodynamics at incommensurate densities with impurities and a magnetic field Exact relations between thermoelectric co-efficients 4. Nernst effect in the cuprate superconductors

3 Outline 1. Superfluid/supersolid/insulator quantum transitions Insulators at integer and commensurate densities 2. Theory of quantum-critical transport Collisionless-t0-hydrodynamic crossover of conformal field theories 3. Hydrodynamics at incommensurate densities with impurities and a magnetic field Exact relations between thermoelectric co-efficients 4. Nernst effect in the cuprate superconductors

4 Trap for ultracold 87 Rb atoms

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

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

7 Phase diagram of doped antiferromagnets g = ring exchange (Sandvik) La 2 CuO 4

8 Phase diagram of doped antiferromagnets g La 2 CuO 4 N. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694 (1989). T. Senthil, A. Vishwanath, L. Balents, S. Sachdev and M.P.A. Fisher, Science 303, 1490 (2004).

9 Phase diagram of doped antiferromagnets g La 2 CuO 4 S. Sachdev and N. Read, Int. J. Mod. Phys. B 5, 219 (1991). Hole density

10 Phase diagram of doped antiferromagnets g La 2 CuO 4 S. Sachdev and N. Read, Int. J. Mod. Phys. B 5, 219 (1991). Hole density

11 Phase diagram of doped antiferromagnets g VBS supersolid d-wave superconductor La 2 CuO 4 VBS insulator S. Sachdev and N. Read, Int. J. Mod. Phys. B 5, 219 (1991). Hole density

12 p 4 g X q VBS supersolid A 1 d-wave broken A 2 0 Doping M. Vojta and S. Sachdev, Phys. Rev. Lett. 83, 3916 (1999)

13 p 4 q g A 1 X VBS supersolid pair density wave d-wave broken A 2 0 Doping M. Vojta and S. Sachdev, Phys. Rev. Lett. 83, 3916 (1999)

14 VBS supersolids d-wave V nn / t Stripes p x 1, p Insulators t / J M. Vojta and S. Sachdev, Phys. Rev. Lett. 83, 3916 (1999)

15 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)

16 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)

17 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)

18 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)

19 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)

20 Glassy Valence Bond Supersolid 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)

21 Outline 1. Superfluid/supersolid/insulator quantum transitions Insulators at integer and commensurate densities 2. Theory of quantum-critical transport Collisionless-t0-hydrodynamic crossover of conformal field theories 3. Hydrodynamics at incommensurate densities with impurities and a magnetic field Exact relations between thermoelectric co-efficients 4. Nernst effect in the cuprate superconductors

22 Outline 1. Superfluid/supersolid/insulator quantum transitions Insulators at integer and commensurate densities 2. Theory of quantum-critical transport Collisionless-t0-hydrodynamic crossover of conformal field theories 3. Hydrodynamics at incommensurate densities with impurities and a magnetic field Exact relations between thermoelectric co-efficients 4. Nernst effect in the cuprate superconductors

23 The insulator:

24 Excitations of the insulator:

25 Excitations of the insulator:

26

27

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

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

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

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

32 Resistivity of Bi films 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)

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

34 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).

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

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

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

38 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.

39 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. 3+1 dimensional AdS 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. Black hole

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

41 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)

42 Collisionless-to-hydrodynamic crossover of solvable SYM 3 P. Kovtun, C. Herzog, S. Sachdev, and D.T. Son, Phys. Rev. D 75, (2007)

43 ImC/k 2 CFT at T=0 P. Kovtun, C. Herzog, S. Sachdev, and D.T. Son, Phys. Rev. D 75, (2007)

44 Collisionless-to-hydrodynamic crossover of solvable SYM 3 P. Kovtun, C. Herzog, S. Sachdev, and D.T. Son, Phys. Rev. D 75, (2007)

45 ImC/k 2 diffusion peak P. Kovtun, C. Herzog, S. Sachdev, and D.T. Son, Phys. Rev. D 75, (2007)

46 Outline 1. Superfluid/supersolid/insulator quantum transitions Insulators at integer and commensurate densities 2. Theory of quantum-critical transport Collisionless-t0-hydrodynamic crossover of conformal field theories 3. Hydrodynamics at incommensurate densities with impurities and a magnetic field Exact relations between thermoelectric co-efficients 4. Nernst effect in the cuprate superconductors

47 Outline 1. Superfluid/supersolid/insulator quantum transitions Insulators at integer and commensurate densities 2. Theory of quantum-critical transport Collisionless-t0-hydrodynamic crossover of conformal field theories 3. Hydrodynamics at incommensurate densities with impurities and a magnetic field Exact relations between thermoelectric co-efficients 4. Nernst effect in the cuprate superconductors

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

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

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

51 For experimental applications, we must move away from the ideal CFT A chemical potential μ CFT Supersolid e.g.

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

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

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

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

56

57 S.A. Hartnoll, P.K. Kovtun, M. Müller, and S. Sachdev, arxiv:

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

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

60 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:

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

62 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.

63 S.A. Hartnoll, P.K. Kovtun, M. Müller, and S. Sachdev, arxiv:

64 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:

65 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:

66 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:

67 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:

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, arxiv:

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, arxiv:

70 Outline 1. Superfluid/supersolid/insulator quantum transitions Insulators at integer and commensurate densities 2. Theory of quantum-critical transport Collisionless-t0-hydrodynamic crossover of conformal field theories 3. Hydrodynamics at incommensurate densities with impurities and a magnetic field Exact relations between thermoelectric co-efficients 4. Nernst effect in the cuprate superconductors

71 Outline 1. Superfluid/supersolid/insulator quantum transitions Insulators at integer and commensurate densities 2. Theory of quantum-critical transport Collisionless-t0-hydrodynamic crossover of conformal field theories 3. Hydrodynamics at incommensurate densities with impurities and a magnetic field Exact relations between thermoelectric co-efficients 4. Nernst effect in the cuprate superconductors

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, arxiv:

73 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, arxiv:

74 LSCO - Theory S.A. Hartnoll, P.K. Kovtun, M. Müller, and S. Sachdev, arxiv:

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

76 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, arxiv:

77 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, arxiv: ) 3

78 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, arxiv:

79 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. Simple model reproduces many trends of the Nernst measurements in cuprates.