Quantum critical transport, duality, and M-theory

Transcription

1 Quantum critical transport, duality, and M-theory hep-th/ Christopher Herzog (Washington) Pavel Kovtun (UCSB) Subir Sachdev (Harvard) Dam Thanh Son (Washington) Talks online at

2 Conductivity σ σ T 0 = Superconductor ( ) σ Insulator T 0 = 0 ( ) σ Quantum critical point T 0 ( ) 4e h 2 D. B. Haviland, Y. Liu, and A. M. Goldman, Phys. Rev. Lett. 62, 2180 (1989)

3 Trap for ultracold 87 Rb atoms

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

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

7 Outline 1. Boson Hubbard model conformal field theory (CFT) 2. Hydrodynamics of CFTs 3. Duality 4. SYM 3 with N=8 supersymmetry

8 Outline 1. Boson Hubbard model conformal field theory (CFT) 2. Hydrodynamics of CFTs 3. Duality 4. SYM 3 with N=8 supersymmetry

9 Degrees of freedom: Bosons, i j j j j, hopping between the sites, j, of a lattice, with short-range repulsive interactions. U H = t bb- μ n + nj ( n j 1) + 2 ij Boson Hubbard model b n b b j j j j M.PA. Fisher, P.B. Weichmann, G. Grinstein, and D.S. Fisher Phys. Rev. B 40, 546 (1989). For small U, superfluid t For large U, insulator t

10 The insulator:

11 Excitations of the insulator: Particles ~ ψ Holes ~ ψ

12 Excitations of the insulator: Particles ~ ψ Holes ~ ψ

13 Superfluid ψ 0 σ = Insulator ψ = 0 σ = 0 s c s

14 Conformal field theory: Wilson-Fisher fixed point Superfluid ψ 0 σ = Insulator ψ = 0 σ = 0 s c s

15 Outline 1. Boson Hubbard model conformal field theory (CFT) 2. Hydrodynamics of CFTs 3. Duality 4. SYM 3 with N=8 supersymmetry

16 CFT correlator of U ( 1 ) current J μ in 2+1 dimensions 2 ( ) ( ) = J p J p K p η μ ν μν p p μ ν 2 p K: a universal number analogous to the level number of the Kac-Moody algebra in 1+1 dimensions Application of Kubo formula shows that σ = 4e h 2 2πK M. P. A. Fisher, Phys. Rev. Lett. 65, 923 (1990)

17 However: computation is at T = 0, with ω 0, while experimental measurements are for ω kt B

18 However: computation is at T = 0, with ω 0, while experimental measurements are for ω kt B Does this matter?

19 CFT correlator of U( 1 ) current J μ in 1+1 dimensions Charge density correlation at T = 0 : J x, τ J 0 ~ R ( ) ( ) J k, ω J k, ω ~ t R ( ) ( ) t 1 τ + ix ( ) k 2 k ω 2 2 2

20 CFT correlator of U( 1 ) current J μ in 1+1 dimensions Charge density correlation at T 0 : J (, ) ( 0 ) R x τ JR ~ sin 2 π T 2 2 ( πt( τ + ix) ) J k, iω J k, iω ~ ( ) ( ) t n t n k 2 k + ω 2 2 n Conformal mapping of plane to cylinder with circumference 1/T

21 CFT correlator of U( 1 ) current J μ in 1+1 dimensions Charge density correlation at T 0 : J (, ) ( 0 ) R x τ JR ~ sin J k, iω J k, iω ~ ( ) ( ) t n t n J k, ω J k, ω ~ t ( ) ( ) t 2 π T 2 2 ( πt( τ + ix) ) k k 2 k + ω 2 2 n 2 k ω 2 2 Conformal mapping of plane to cylinder with circumference 1/T

22 However: computation is at T = 0, with ω 0, while experimental measurements are for ω kt B Does this matter?

23 However: computation is at T = 0, with ω 0, while experimental measurements are for ω kt B Does this matter? For CFTs in 1+1 dimensions: NO Correlators of conserved charges are independent of the ratio ω kt B No diffusion of charge, and no hydrodynamics

24 However: computation is at T = 0, with ω 0, while experimental measurements are for ω kt B Does this matter? For CFTs in 2+1 dimensions: YES

25 However: computation is at T = 0, with ω 0, while experimental measurements are for ω kt B k B T ω Does this matter? For CFTs in 2+1 dimensions: YES is a characteristic decoherence or collision tme i k ω B T kt B : Collisionless physics : Hydrodynamic, collision-dominated transport K. Damle and S. Sachdev, Phys. Rev. B 56, 8714 (1997).

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

27 CFT correlator of U ( 1 ) current J μ in 2+1 dimensions 2 ( ) ( ) = J p J p K p η μ ν μν p p μ ν 2 p K: a universal number analogous to the level number of the Kac-Moody algebra in 1+1 dimensions Application of Kubo formula shows that σ = 4e h 2 2πK

28 ( ) CFT correlator of U 1 current J at T 0 μ = 2 ( ) ( ) = J p J p K p η μ ν μν p p μ ν 2 p K: a universal number analogous to the level number of the Kac-Moody algebra in 1+1 dimensions Application of Kubo formula shows that σ ( ω ) T 4e = = h 2 2πK

29 ( ) CFT correlator of U 1 current J at T 0 μ > ( ) ( ) ( ) 2 2 T T,, = (, ) L L ω ω ω ω + (, ω) J k J k k P K k P K k μ ν μν μν The projectors are defined by kk i j p T L μpν T Pij = δij and P P ; p ( k, ) 2 μν = ημν 2 μν = ω k p ( k ω) LT, while, are universal functions o K f ω and T k T Application of Kubo formula shows that 2 2 4e 4e σ π ω π ω T h h ( ω ) T L = 2 K ( 0, ) = 2 K ( 0, )

30 Superfluid Insulator

31 CFT at T > 0 Superfluid Insulator

32 Outline 1. Boson Hubbard model conformal field theory (CFT) 2. Hydrodynamics of CFTs 3. Duality 4. SYM 3 with N=8 supersymmetry

33 Excitations of the insulator: Particles ~ ψ Holes ~ ψ

34 Approaching the transition from the superfluid Excitations of the superfluid: (A) Spin waves

35 Approaching the transition from the superfluid Excitations of the superfluid: (B) Vortices vortex

36 Approaching the transition from the superfluid Excitations of the superfluid: (B) Vortices E vortex

37 Approaching the transition from the superfluid Excitations of the superfluid: Spin wave and vortices

38 Superfluid ψ 0 σ = Conformal field theory: Wilson-Fisher fixed point Insulator ψ = 0 σ = 0 s c s C. Dasgupta and B.I. Halperin, Phys. Rev. Lett. 47, 1556 (1981)

39 Superfluid ψ 0 ϕ = 0 σ = Conformal field theory: Wilson-Fisher fixed point Insulator ψ = 0 ϕ 0 σ = 0 s s, c s c s C. Dasgupta and B.I. Halperin, Phys. Rev. Lett. 47, 1556 (1981)

40 Consequences of duality on CFT correlators of U( 1 ) currents Application of Kubo formula shows that 2 2 4e 4e σ π ω π ω T h h ( ω ) T L = 2 K ( 0, ) = 2 K ( 0, ) C. Herzog, P. Kovtun, S. Sachdev, and D.T. Son, hep-th/

41 Outline 1. Boson Hubbard model conformal field theory (CFT) 2. Hydrodynamics of CFTs 3. Duality 4. SYM 3 with N=8 supersymmetry

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46 ImC tt /k 2 CFT at T=0 3k 4πT 3ω 4πT

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48 ImC tt /k 2 diffusion peak 3k 4πT 3ω 4πT

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51 ab 2 2 (, ω) (, ω) = δ ω (, ω) + (, ω) ( ) a b T T L L Jμ k Jν k k Pμν K k Pμν K k The self-duality of the 4D SO(8) gauge fields leads to C. Herzog, P. Kovtun, S. Sachdev, and D.T. Son, hep-th/

52 ab 2 2 (, ω) (, ω) = δ ω (, ω) + (, ω) ( ) a b T T L L Jμ k Jν k k Pμν K k Pμν K k The self-duality of the 4D SO(8) gauge fields leads to C. Herzog, P. Kovtun, S. Sachdev, and D.T. Son, hep-th/

53 ab 2 2 (, ω) (, ω) = δ ω (, ω) + (, ω) ( ) a b T T L L Jμ k Jν k k Pμν K k Pμν K k The self-duality of the 4D SO(8) gauge fields leads to Holographic self-duality C. Herzog, P. Kovtun, S. Sachdev, and D.T. Son, hep-th/

54 Open questions 1. Does K L K T = constant (i.e. holographic self-duality) hold for SYM 3 SCFT at finite N? 2. Is there any CFT 3 with an Abelian U(1) current whose conductivity can be determined by self-duality? (unlikely, because global and topological U(1) currents are interchanged under duality). 3. Is there any CFT 3 solvable by AdS/CFT which is not (holographically) self-dual? 4. Is there an AdS 4 description of the hydrodynamics of the O(N) Wilson-Fisher CFT 3? (can use 1/N expansion to control strongly-coupled gravity theory).