The quantum phases of matter and gauge-gravity duality

Transcription

1 The quantum phases of matter and gauge-gravity duality University of Michigan, Ann Arbor, March 13, 2013 Subir Sachdev HARVARD

2 Sommerfeld-Bloch theory of metals, insulators, and superconductors: many-electron quantum states are adiabatically connected to independent electron states

3 Modern phases of quantum matter Not adiabatically connected to independent electron states: many-particle quantum entanglement

4 Quantum Entanglement: quantum superposition Hydrogen atom: with more than one particle Hydrogen molecule: = _ = 1 2 ( ) Superposition of two electron states leads to non-local correlations between spins

5 Quantum Entanglement: quantum superposition Hydrogen atom: with more than one particle Hydrogen molecule: = _ = 1 2 ( ) Einstein-Podolsky-Rosen paradox : Non-local correlations between observations arbitrarily far apart

6 Outline 1. Z2 Spin liquid in the kagome antiferromagnet 2. Superfluid-insulator transition of ultracold atoms in optical lattices: Quantum criticality and conformal field theories 3. Holography and the quasi-normal modes of black-hole horizons 4. Strange metals: What lies beyond the horizon?

7 Outline 1. Z2 Spin liquid in the kagome antiferromagnet 2. Superfluid-insulator transition of ultracold atoms in optical lattices: Quantum criticality and conformal field theories 3. Holography and the quasi-normal modes of black-hole horizons 4. Strange metals: What lies beyond the horizon?

8 ZnCu(OH) 6 Cl 2 herbertsmithite single crystals

9 Kagome antiferromagnet H = J ij S i S j

10 Kagome antiferromagnet H = J ij S i S j = P. Fazekas and P. W. Anderson, Philos. Mag. 30, 23 (1974).

11 Kagome antiferromagnet H = J ij S i S j = P. Fazekas and P. W. Anderson, Philos. Mag. 30, 23 (1974).

12 Kagome antiferromagnet H = J ij S i S j = P. Fazekas and P. W. Anderson, Philos. Mag. 30, 23 (1974).

13 Kagome antiferromagnet H = J ij S i S j = P. Fazekas and P. W. Anderson, Philos. Mag. 30, 23 (1974).

14 Kagome antiferromagnet H = J ij S i S j = P. Fazekas and P. W. Anderson, Philos. Mag. 30, 23 (1974).

15 Kagome antiferromagnet H = J ij S i S j = P. Fazekas and P. W. Anderson, Philos. Mag. 30, 23 (1974).

16 Kagome antiferromagnet H = J ij S i S j = P. Fazekas and P. W. Anderson, Philos. Mag. 30, 23 (1974).

17 Kagome antiferromagnet Alternative view Pick a reference configuration

18 Kagome antiferromagnet Alternative view A nearby configuration

19 Kagome antiferromagnet Alternative view Difference: a closed loop

20 Kagome antiferromagnet Alternative view Ground state: sum over closed loops

21 Kagome antiferromagnet Alternative view Ground state: sum over closed loops

22 Mott insulator: Kagome antiferromagnet Alternative view Ground state: sum over closed loops

23 Kagome antiferromagnet Alternative view Ground state: sum over closed loops

24 Entanglement entropy B A P i ) Ground state of entire system, = ih A =Tr B = density matrix of region A Entanglement entropy S E = Tr ( A ln A )

25 Entanglement entropy i ) Ground state of entire system, = ih Take i = 1 p 2 ( "i A #i B #i A "i B ) Then A =Tr B = density matrix of region A = 1 2 ( "i A h" A + #i A h# A ) Entanglement entropy S E = Tr ( A ln A ) =ln2

26 Entanglement entropy of a band insulator Band insulators E Insulator k An even number of electrons per unit cell

27 Entanglement entropy of a band insulator B A P S E = ap b exp( cp ) where P is the surface area (perimeter) of the boundary between A and B.

28 Entanglement in the Z2 spin liquid B A P The sum over closed loops is characteristic of the Z2 spin liquid, introduced in N. Read and S. Sachdev, Phys. Rev. Lett. 66, 1773 (1991), X.-G. Wen, Phys. Rev. B 44, 2664 (1991)

29 Entanglement in the Z2 spin liquid B A P Sum over closed loops: only an even number of links cross the boundary between A and B The sum over closed loops is characteristic of the Z2 spin liquid, introduced in N. Read and S. Sachdev, Phys. Rev. Lett. 66, 1773 (1991), X.-G. Wen, Phys. Rev. B 44, 2664 (1991)

30 Entanglement in the Z2 spin liquid B A P S E = ap ln(2) where P is the surface area (perimeter) of the boundary between A and B. A. Hamma, R. Ionicioiu, and P. Zanardi, Phys. Rev. A 71, (2005) M. Levin and X.-G. Wen, Phys. Rev. Lett. 96, (2006); A. Kitaev and J. Preskill, Phys. Rev. Lett. 96, (2006) Y. Zhang, T. Grover, and A. Vishwanath, Phys. Rev. B 84, (2011)

31 Mott insulator: Kagome antiferromagnet Strong numerical evidence for a Z2 spin liquid Simeng Yan, D. A. Huse, and S. R. White, Science 332, 1173 (2011). Hong-Chen Jiang, Z. Wang, and L. Balents, Nature Physics 8, 902 (2012) S. Depenbrock, I. P. McCulloch, and U. Schollwoeck, Physical Review Letters 109, (2012)

32 Fractionalized excitations in the spin-liquid state of a kagome-lattice antiferromagnet Tian-Heng Han 1, Joel S. Helton 2, Shaoyan Chu 3, Daniel G. Nocera 4, Jose A. Rodriguez-Rivera 2,5, Collin Broholm 2,6 & Young S. Lee 1 Nature 492, 406 (2012)

33 Outline 1. Z2 Spin liquid in the kagome antiferromagnet 2. Superfluid-insulator transition of ultracold atoms in optical lattices: Quantum criticality and conformal field theories 3. Holography and the quasi-normal modes of black-hole horizons 4. Strange metals: What lies beyond the horizon?

34 Outline 1. Z2 Spin liquid in the kagome antiferromagnet 2. Superfluid-insulator transition of ultracold atoms in optical lattices: Quantum criticality and conformal field theories 3. Holography and the quasi-normal modes of black-hole horizons 4. Strange metals: What lies beyond the horizon?

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

36 T Quantum critical T KT h i6=0 h i =0 Superfluid Insulator 0 g c c g

37 T! a complex field representing the Bose-Einstein condensate of the superfluid T KT Quantum critical h i6=0 h i =0 Superfluid Insulator 0 g c c g

38 S = Z d 2 t 2 c 2 r r 2 V ( ) V ( ) = ( c) 2 + u 2 2 T T KT Quantum critical h i6=0 h i =0 Superfluid Insulator 0 g c c g

39 S = Z d 2 t 2 c 2 r r 2 V ( ) V ( ) = ( c) 2 + u 2 2 T V Particles and holes correspond to the 2 normal modes in the oscillation of about = 0. Quantum critical T KT Re( ) Ψ Im( ) Ψ h i6=0 h i =0 Superfluid Insulator 0 g c c g

40 Insulator (the vacuum) at large repulsion between bosons

41 Excitations of the insulator: Particles

42 Excitations of the insulator: Holes

43 S = Z d 2 t 2 c 2 r r 2 V ( ) V ( ) = ( c) 2 + u 2 2 T V Particles and holes correspond to the 2 normal modes in the oscillation of about = 0. Quantum critical T KT Re( ) Ψ Im( ) Ψ h i6=0 h i =0 Superfluid Insulator 0 g c c g

44 S = Z d 2 t 2 c 2 r r 2 V ( ) V ( ) = ( c) 2 + u 2 2 T 1 V Nambu Goldstone mode Re( Ψ) T KT Quantum Higgs mode Im( Ψ) j/j c 1 critical Nambu-Goldstone mode is the oscillation in the phase at a constant non-zero. h i6=0 h i =0 Superfluid Insulator 0 g c c g

45 S = Z d 2 t 2 c 2 r r 2 V ( ) V ( ) = ( c) 2 + u 2 2 T A conformal field theory in 2+1 spacetime dimensions: T KT a CFT3 Quantum critical h i6=0 h i =0 Superfluid Insulator 0 g c c g

46 S = Z d 2 t 2 c 2 r r 2 V ( ) V ( ) = ( c) 2 + u 2 2 T Quantum state with complex, many-body, long-range quantum Quantum entanglement critical T KT h i6=0 h i =0 Superfluid Insulator 0 g c c g

47 S = Z d 2 t 2 c 2 r r 2 V ( ) V ( ) = ( c) 2 + u 2 2 T No well-defined normal modes, or particle-like excitations Quantum critical T KT h i6=0 h i =0 Superfluid Insulator 0 g c c g

48 S = Z d 2 t 2 c 2 r r 2 V ( ) V ( ) = ( c) 2 + u 2 2 T 1 V Nambu Goldstone mode Re( Ψ) T KT Quantum Higgs mode Im( Ψ) j/j c 1 critical Higgs mode is the oscillation in the amplitude. Thisdecays rapidly by emitting pairs of Nambu-Goldstone modes. h i6=0 h i =0 Superfluid Insulator 0 g c c g

49 S = Z d 2 t 2 c 2 r r 2 V ( ) V ( ) = ( c) 2 + u 2 2 T 1 V Nambu Goldstone mode Re( Ψ) T KT Quantum Higgs mode Im( Ψ) j/j c 1 critical Despite rapid decay, there is a well-defined Higgs quasi-normal mode. This is associated with a pole in the lower-half of the complex frequency plane. h i6=0 h i =0 Superfluid Insulator 0 A. V. Chubukov, S. Sachdev, and J. Ye, PRB 49, g (1994). S. Sachdev, PRB 59, (1999). c W. Zwerger, PRL 92, (2004). D. Podolsky, A. Auerbach, and D. P. Arovas, PRB 84, (2011). D. Podolsky and S. Sachdev, PRB 86, (2012). L. Pollet and N. Prokof ev, PRL 109, (2012). g

50 S = Z d 2 t 2 c 2 r r 2 V ( ) V ( ) = ( c) 2 + u 2 2! 1 V j/j c 1 Nambu Goldstone mode Re( Ψ) Higgs mode Im( Ψ) D. Podolsky and S. Sachdev, Phys. Rev. B 86, (2012). The Higgs quasi-normal mode is at the frequency! pole = i N p 2 log 3 2 p i! + O 1 N 2 where is the particle gap at the complementary point in the paramagnetic state with the same value of c, and N = 2 is the number of vector components of. The universal answer is a consequence of the strong interactions in the CFT3

51 Observation of Higgs quasi-normal mode across the superfluid-insulator transition of ultracold atoms in a 2-dimensional optical lattice: Response to modulation of lattice depth scales as expected from the LHP pole a h 0 /U j Mott Superfluid 1 1 Insulator j/j c 2 b Lattice depth (E r ) Lattice loading Modulation Hold time Ramp to atomic limit 0 T mod = k B T/U V T tot = 200 ms Time (ms) mod (Hz) Temperature measurement A = 0.03V 0 Figure 1 Illustration of the Higgs mode and experimental sequence. a, Classical energy b density 0.17V as a function of the order parameter Y. Within the ordered (superfluid) phase, Nambu Goldstone 1 and Higgs modes arise from phase and amplitude0.15 modulations (blue and red arrows in panel 1). As the coupling j 5 J/U (see main text) approaches 0 the critical value j c, the energy density transforms into a function with a minimum at Y 5 0 (panels 2 and 3) Simultaneously, the curvature in the radial direction decreases, leading to a characteristic reduction of the excitation frequency for the Higgs mode. In the 0.11 disordered (Mott insulating) phase, two gapped modes exist, respectively corresponding to particle and hole excitations in our case (red and blue arrow in 0.18 panel 3). b, The Higgs mode can 2 be excited with a periodic modulation of the coupling j, which amounts to a shaking of the classical energy density 0.16 potential. In the experimental sequence, 0 this is realized by a modulation of the optical lattice potential 0.14 (see main text for details). t 5 1/n mod ; E r, lattice recoil energy. V 0 = 8E r j/j c = 2.2 V 0 = 9E r j/j c = 1.6 system with a recently developed scheme based on single-atomresolved detection It is 3the high sensitivity of this method that allowed us to reduce the modulation amplitude by almost an order of magnitude compared 0.16 with earlier experiments 20,21 and to stay well within the linear response regime 0 (Supplementary Information). V 0 = 10E r j/j c = 1.2 Macmillan Publishers Lim e Higgs Manuel mode. Endres, a, The Takeshi fittedfukuhara, gap valuesdavid hn 0 /UPekker, Marc region. Cheneau, b, Temperature Peter Schaub, response Christian to lattice Gross, modulati stic softening Eugene close Demler, to the Stefan critical Kuhr, point and inimmanuel quantitative Bloch, blue Nature line) 487, and454 fit with (2012). an error function (solid black

52 h mod /U 3 4 Observation of Higgs quasinormal mode across the diagonalization of the trapped discrete superfluidinsulator spectrum of amplitudethe strength of the response to a e not shown (Methods) and n 0,G r approximation. transition a.u., of arbitrary n for the four amplitude modes ultracold atoms in contrast to the superfluid lmost a 2-dimensional constant (not shown). us couplings j/j c. Each red circle optical lattice: ntensity of the colour being uency Response of the lowest-lying to mode g (solid line; same as in Fig. 2a) modulation of ical point due to the finite size of al response lattice atdepth V E r (blue e.m.) with a cluster meand a heuristic model (dashed line; scales as expected on was from donethe for V 0 LHP 5 9.5E r (grey grey area), to account for the andpole predicts the energy (1 j/j c ) 2 k B ΔT/U k B ΔT/U mod /U mod /U Figure 4 Scaling of the low-frequency response. The low-frequency response in the superfluid regime shows a scaling compatible with the prediction (1 2 j/j c ) 22 n 3 (Methods). Shown is the temperature response rescaled with (1 2 j/j c ) 2 for V E r (grey), 9.5E r (black), 9E r (green), 8.5E r (blue) and 8E r (red) as a function of the modulation frequency. The black line is a fit of the form an b with a fitted exponent b 5 2.9(5). The inset shows the same data points without rescaling, for comparison. Error bars, s.e.m Macmillan Publishers Limited. All rights reserved D. Podolsky and S. Sachdev, Phy. Rev. B 86, (2012). Manuel Endres, Takeshi Fukuhara, David Pekker, Marc Cheneau, Peter Schaub, Christian Gross, Eugene Demler, Stefan Kuhr, and Immanuel Bloch, Nature 487, 454 (2012).

53 S = Z d 2 t 2 c 2 r r 2 V ( ) V ( ) = ( c) 2 + u 2 2 T A conformal field theory in 2+1 spacetime dimensions: T KT a CFT3 Quantum critical h i6=0 h i =0 Superfluid Insulator 0 g c c g

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

55 CFT3 at T>0 T Quantum critical T KT Superfluid Insulator 0 g c c g

56 Quantum critical dynamics Quantum nearly perfect fluid with shortest possible local equilibration time, eq eq = C ~ k B T where C is a universal constant. Response functions are characterized by poles in LHP with! k B T/~. These poles (quasi-normal modes) appear naturally in the holographic theory. (Analogs of Higgs quasi-normal mode.) S. Sachdev, Quantum Phase Transitions, Cambridge (1999).

57 Quantum critical dynamics Transport co-oe cients not determined by collision rate of quasiparticles, but by fundamental constants of nature Conductivity = Q2 h [Universal constant O(1) ] (Q is the charge of one boson) M.P.A. Fisher, G. Grinstein, and S.M. Girvin, Phys. Rev. Lett. 64, 587 (1990) K. Damle and S. Sachdev, Phys. Rev. B 56, 8714 (1997).

58 CFT3 at T>0 T Quantum critical T KT Superfluid Insulator 0 g c c g

59 CFT3 at T>0 T T KT Quantum Boltzmann theory of particles/holes/vortices critical does not apply Superfluid Insulator 0 g c c g

60 CFT3 at T>0 T T KT Quantum Needed: Accurate critical theory of quantum critical dynamics Superfluid Insulator 0 g c c g

61 Outline 1. Z2 Spin liquid in the kagome antiferromagnet 2. Superfluid-insulator transition of ultracold atoms in optical lattices: Quantum criticality and conformal field theories 3. Holography and the quasi-normal modes of black-hole horizons 4. Strange metals: What lies beyond the horizon?

62 Outline 1. Z2 Spin liquid in the kagome antiferromagnet 2. Superfluid-insulator transition of ultracold atoms in optical lattices: Quantum criticality and conformal field theories 3. Holography and the quasi-normal modes of black-hole horizons 4. Strange metals: What lies beyond the horizon?

63 Renormalization group: ) Follow coupling constants of quantum many body theory as a function of length scale r r x i J. McGreevy, arxiv

64 Renormalization group: ) Follow coupling constants of quantum many body theory as a function of length scale r r x i Key idea: ) Implement r as an extra dimension, and map to a local theory in d + 2 spacetime dimensions. J. McGreevy, arxiv

65 Holography x i r For a relativistic CFT in d spatial dimensions, the metric in the holographic space is fixed by demanding the scale transformation (i =1...d) x i! x i, t! t, ds! ds

66 Holography x i r This gives the unique metric ds 2 = 1 r 2 dt 2 + dr 2 + dx 2 i This is the metric of anti-de Sitter space AdS d+2.

67 AdS/CFT correspondence AdS 4 R 2,1 Minkowski CFT3 x i zr

68 Holography and Entanglement AdS 4 R 2,1 Minkowski CFT3 A x i zr

69 Holography and Entanglement AdS 4 R 2,1 Minkowski CFT3 A x i zr Associate entanglement entropy with an observer in the enclosed spacetime region, who cannot observe outside : i.e. the region is surrounded by an imaginary horizon. S. Ryu and T. Takayanagi, Phys. Rev. Lett. 96, (2006).

70 Holography and Entanglement AdS 4 R 2,1 Minkowski CFT3 Minimal surface area A measures entanglement x i entropy zr S. Ryu and T. Takayanagi, Phys. Rev. Lett. 96, (2006).

71 Holography and Entanglement AdS 4 R 2,1 Minkowski CFT3 A x i zr Computation of minimal surface area yields S E = ap, where is a shape-dependent universal number. M. A. Metlitski, C. A. Fuertes, and S. Sachdev, Physical Review B 80, (2009); H. Casini, M. Huerta, and R. Myers, JHEP 1105:036, (2011); I. Klebanov, S. Pufu, and B. Safdi, arxiv:

72 AdS/CFT correspondence AdS 4 R 2,1 Minkowski CFT3 x i zr This emergent spacetime is a solution of Einstein gravity with a negative cosmological constant Z S E = d 4 x p apple R 1 g 2apple 2 + 6L 2

73 AdS/CFT correspondence at non-zero temperatures AdS4-Schwarzschild black-brane r R h There is a family of solutions of Einstein gravity which describe non-zero S = Z d 4 x p g apple 1 2apple 2 R + 6L 2 temperatures

74 AdS/CFT correspondence at non-zero temperatures AdS4-Schwarzschild black-brane r R h Black-brane (horizon) at temperature of 2+1 dimensional quantum critical system 2 apple L dr ds 2 2 = f(r)dt 2 + dx 2 + dy 2 r f(r) with f(r) =1 (r/r h ) 3

75 AdS/CFT correspondence at non-zero temperatures AdS4-Schwarzschild black-brane r R h A 2+1 dimensional system at its quantum critical point: k B T = 3~ 4 R h. Black-brane (horizon) at temperature of 2+1 dimensional quantum critical system 2 apple L dr ds 2 2 = f(r)dt 2 + dx 2 + dy 2 r f(r) with f(r) =1 (r/r h ) 3

76 AdS/CFT correspondence at non-zero temperatures AdS4-Schwarzschild black-brane r R h A 2+1 dimensional system at its quantum critical point: k B T = 3~ 4 R h. Black-brane (horizon) at temperature of 2+1 dimensional quantum critical system Friction of quantum criticality = waves falling past the horizon D. T. Son

77 AdS/CFT correspondence at non-zero temperatures AdS4-Schwarzschild black-brane r R h A 2+1 dimensional system at its quantum critical point: k B T = 3~ 4 R h. Black-brane (horizon) at temperature of 2+1 dimensional quantum critical system Quasi-normal modes of waves near horizon -- quasi-normal modes of quantum criticality (and Higgs)

78 AdS4 theory of quantum criticality Most general e ective holographic theory for linear charge transport with 4 spatial derivatives: S bulk = 1 + Z g 2 M Z d 4 x p g d 4 x p g apple 1 2apple 2 apple 1 4 F abf ab + L 2 C abcd F ab F cd R + 6L 2 Here This action F is characterized by 3 dimensionless parameters, is dual whichto can a conserved be linkedu(1) to data current of the of the CFT ab is a 4-dimensional gauge field strength, which CFT. (OPECcoe cients): 2-point correlators of the conserved current J µ and the stress energy tensor T µ abcd is the Weyl tensor., and a 3-point T, J, J correlator. R. C. Myers, S. Sachdev, and A. Singh, Phys. Rev. D 83, (2011) D. Chowdhury, S. Raju, S. Sachdev, A. Singh, and P. Strack, Phys. Rev. B 87, (2013),

79 AdS4 theory of quantum criticality Most general e ective holographic theory for linear charge transport with 4 spatial derivatives: S bulk = 1 + Z g 2 M Z d 4 x p g d 4 x p g apple 1 2apple 2 apple 1 4 F abf ab + L 2 C abcd F ab F cd R + 6L 2 This action is characterized by 3 dimensionless parameters, which can be linked to data of the CFT (OPE coe cients): 2-point correlators of the conserved current J µ and the stress energy tensor T µ, and a 3-point T, J, J correlator. R. C. Myers, S. Sachdev, and A. Singh, Phys. Rev. D 83, (2011) D. Chowdhury, S. Raju, S. Sachdev, A. Singh, and P. Strack, Phys. Rev. B 87, (2013),

80 AdS4 theory of quantum criticality Most general e ective holographic theory for linear charge transport with 4 spatial derivatives: S bulk = 1 + Z g 2 M Z d 4 x p g d 4 x p g apple 1 2apple 2 apple 1 4 F abf ab + L 2 C abcd F ab F cd R + 6L 2 This Boundary actionand is characterized bulk methods byboth 3 dimensionless show that parameters, apple 1/12, which andcan thebe bound linkedisto saturated data of the bycft free (OPE fields. coe cients): 2-point correlators of the conserved current J µ and the stress energy tensor T µ, and a 3-point T, J, J correlator. R. C. Myers, S. Sachdev, and A. Singh, Phys. Rev. D 83, (2011) D. Chowdhury, S. Raju, S. Sachdev, A. Singh, and P. Strack, Phys. Rev. B 87, (2013),

81 AdS4 theory of quantum criticality h(!) Q (1) 1.0 = 1 12 =0 0.5 = T R. C. Myers, S. Sachdev, and A. Singh, Physical Review D 83, (2011)

82 AdS4 theory of quantum criticality h(!) Q (1) 1.0 = 1 12 =0 0.5 = 1 12 The > 0 result has similarities to the quantum-boltzmann result for transport of particle-like excitations T R. C. Myers, S. Sachdev, and A. Singh, Physical Review D 83, (2011)

83 AdS4 theory of quantum criticality h(!) Q (1) 1.0 = 1 12 =0 0.5 = 1 12 The < 0 result can be interpreted as the transport of vortex-like excitations T R. C. Myers, S. Sachdev, and A. Singh, Physical Review D 83, (2011)

84 AdS4 theory of quantum criticality h(!) Q (1) 1.0 = 1 12 = = 1 12 Stability constraints on the e ective theory ( < 1/12) allow only a limited!-dependence in the conductivity. This contrasts with the Boltzmann theory in which (!)/ 1 becomes very large in the regime of its validity T R. C. Myers, S. Sachdev, and A. Singh, Physical Review D 83, (2011)

85 AdS4 theory of quantum criticality Poles in LHP of conductivity at! k B T/~; analog of Higgs quasinormal mode quasinormal modes of black brane (a)<{ (w; =1/12)} (b)<{ˆ(w; =1/12)} (c)<{ (w; = 1/12)} (d)<{ˆ(w; = 1/12)} 00 W. Witzack-Krempa and S. Sachdev, Physical Review= D apple 86, (2012)

86 AdS4 theory of quantum criticality Zeros in LHP of conductivity quasinormal modes of S-dual theory (a)<{ (w; =1/12)} (b)<{ˆ(w; =1/12)} (c)<{ (w; = 1/12)} (d)<{ˆ(w; = 1/12)} 00 W. Witzack-Krempa and S. Sachdev, Physical Review= D apple 86, (2012)

87 AdS4 theory of quantum criticality It can be shown that the conductivity of any CFT3 must satisfy two sum rules Z 1 0 Z 1 0 d!re [ (!) (1)] = 0 apple 1 1 d!re (!) (1) = 0 The AdS 4 theory satisfies both sum rules exactly. The Boltzmann theory must make a choice between the particle or vortex basis, and so satisfies only one of the sum rules. W. Witzack-Krempa and S. Sachdev, Physical Review D 86, (2012)

88 Traditional CMT Identify quasiparticles and their dispersions Compute scattering matrix elements of quasiparticles (or of collective modes) These parameters are input into a quantum Boltzmann equation Deduce dissipative and dynamic properties at nonzero temperatures

89 Traditional CMT Identify quasiparticles and their dispersions Compute scattering matrix elements of quasiparticles (or of collective modes) These parameters are input into a quantum Boltzmann equation Deduce dissipative and dynamic properties at nonzero temperatures Holography and black-branes Start with strongly interacting CFT without particle- or wave-like excitations Compute OPE co-efficients of operators of the CFT Relate OPE co-efficients to couplings of an effective graviational theory on AdS Solve Einsten-Maxwell equations. Dynamics of quasinormal modes of black branes.

90 Traditional CMT Identify quasiparticles and their dispersions Compute scattering matrix elements of quasiparticles (or of collective modes) These parameters are input into a quantum Boltzmann equation Deduce dissipative and dynamic properties at nonzero temperatures Holography and black-branes Start with strongly interacting CFT without particle- or wave-like excitations Compute OPE co-efficients of operators of the CFT Relate OPE co-efficients to couplings of an effective graviational theory on AdS Solve Einsten-Maxwell equations. Dynamics of quasinormal modes of black branes.

91 Traditional CMT Identify quasiparticles and their dispersions Compute scattering matrix elements of quasiparticles (or of collective modes) These parameters are input into a quantum Boltzmann equation Deduce dissipative and dynamic properties at nonzero temperatures Holography and black-branes Start with strongly interacting CFT without particle- or wave-like excitations Compute OPE co-efficients of operators of the CFT Relate OPE co-efficients to couplings of an effective graviational theory on AdS Solve Einsten-Maxwell equations. Dynamics of quasinormal modes of black branes.

92 Traditional CMT Identify quasiparticles and their dispersions Compute scattering matrix elements of quasiparticles (or of collective modes) These parameters are input into a quantum Boltzmann equation Deduce dissipative and dynamic properties at nonzero temperatures Holography and black-branes Start with strongly interacting CFT without particle- or wave-like excitations Compute OPE co-efficients of operators of the CFT Relate OPE co-efficients to couplings of an effective graviational theory on AdS Solve Einsten-Maxwell equations. Dynamics of quasinormal modes of black branes.

93 AdS4 theory of quantum criticality PRL 95, (2005) P H Y S I C A L R E V I E W L E T T E R S week ending 28 OCTOBER 2005 Universal Scaling of the Conductivity at the Superfluid-Insulator Phase Transition Jurij Šmakov and Erik Sørensen Department of Physics and Astronomy, McMaster University, Hamilton, Ontario L8S 4M1, Canada (Received 30 May 2005; published 27 October 2005) The scaling of the conductivity at the superfluid-insulator quantum phase transition in two dimensions is studied by numerical simulations of the Bose-Hubbard model. In contrast to previous studies, we focus on properties of this model in the experimentally relevant thermodynamic limit at finite temperature T.We find clear evidence for deviations from! k scaling of the conductivity towards! k =T scaling at low Matsubara frequencies! k. By careful analytic continuation using Padé approximants we show that this behavior carries over to the real frequency axis where the conductivity scales with!=t at small frequencies and low temperatures. We estimate the universal dc conductivity to be 0:45 5 Q 2 =h, distinct from previous estimates in the T 0,!=T 1 limit. σ ( ω /T ) (b) L τ =32 L τ =40 L τ =48 L τ =64 SSE σ(iω k ) (c) β=2 β=4 β=6 β=8 β=10 ω k /(2π ω/t ω k 0

94 AdS4 theory of quantum criticality PRL 95, (2005) P H Y S I C A L R E V I E W L E T T E R S week ending 28 OCTOBER 2005 Universal Scaling of the Conductivity at the Superfluid-Insulator Phase Transition Jurij Šmakov and Erik Sørensen Department of Physics and Astronomy, McMaster University, Hamilton, Ontario L8S 4M1, Canada (Received 30 May 2005; published 27 October 2005) The scaling of the conductivity at the superfluid-insulator quantum phase transition in two dimensions is studied by numerical simulations of the Bose-Hubbard model. In contrast to previous studies, we focus on properties of this model in the experimentally relevant thermodynamic limit at finite temperature T.We find clear evidence for deviations from! k scaling of the conductivity towards! k =T scaling at low Matsubara frequencies! k. By careful analytic continuation using Padé approximants we show that this behavior carries over to the real frequency axis where the conductivity scales with!=t at small frequencies and low temperatures. We estimate the universal dc conductivity to be 0:45 5 Q 2 =h, distinct from previous estimates in the T 0,!=T 1 limit. QMC yields (0)/ Holography yields (0)/ 1 =1+4 with apple 1/12. Maximum possible holographic value (0)/ 1 =1.33 W. Witzack-Krempa and S. Sachdev, arxiv:

95 Outline 1. Z2 Spin liquid in the kagome antiferromagnet 2. Superfluid-insulator transition of ultracold atoms in optical lattices: Quantum criticality and conformal field theories 3. Holography and the quasi-normal modes of black-hole horizons 4. Strange metals: What lies beyond the horizon?

96 Outline 1. Z2 Spin liquid in the kagome antiferromagnet 2. Superfluid-insulator transition of ultracold atoms in optical lattices: Quantum criticality and conformal field theories 3. Holography and the quasi-normal modes of black-hole horizons 4. Strange metals: What lies beyond the horizon?

97 Resistivity 0 + AT n BaFe 2 (As 1 x P x ) 2 FL K. Hashimoto, K. Cho, T. Shibauchi, S. Kasahara, Y. Mizukami, R. Katsumata, Y. Tsuruhara, T. Terashima, H. Ikeda, M. A. Tanatar, H. Kitano, N. Salovich, R. W. Giannetta, P. Walmsley, A. Carrington, R. Prozorov, and Y. Matsuda, Science 336, 1554 (2012).

98 Resistivity 0 + AT n BaFe 2 (As 1 x P x ) 2 Strange Metal Metal FL K. Hashimoto, K. Cho, T. Shibauchi, S. Kasahara, Y. Mizukami, R. Katsumata, Y. Tsuruhara, T. Terashima, H. Ikeda, M. A. Tanatar, H. Kitano, N. Salovich, R. W. Giannetta, P. Walmsley, A. Carrington, R. Prozorov, and Y. Matsuda, Science 336, 1554 (2012).

99 The Metal k y kf! k x Electrons (fermions) occupy states inside a Fermi surface (circle) of radius k F which is determined by the density of electrons, Q.

100 A Strange Metal Can bosons form a metal? Yes, if each boson, b, fractionalizes b = f 1 f 2! into 2 fermions ( quarks ) Each quark is charged under an emergent gauge force, which encapsulates the entanglement in the ground state. The quarks have hidden Fermi surfaces of radius k F.

101 A Strange Metal O. I. Motrunich and M. P. A. Fisher, Phys. Rev. B 75, (2007) L. Huijse and S. Sachdev, Phys. Rev. D 84, (2011) S. Sachdev, arxiv: Can bosons form a metal? Yes, if each boson, b, fractionalizes b = f 1 f 2! into 2 fermions ( quarks ) Each quark is charged under an emergent gauge force, which encapsulates the entanglement in the ground state. The quarks have hidden Fermi surfaces of radius k F.

102 A Strange Metal O. I. Motrunich and M. P. A. Fisher, Phys. Rev. B 75, (2007) L. Huijse and S. Sachdev, Phys. Rev. D 84, (2011) S. Sachdev, arxiv: Can bosons form a metal? Yes, if each boson, b, fractionalizes b = f 1 f 2! into 2 fermions ( quarks ) Each quark is charged under an emergent gauge force, which encapsulates the entanglement in the ground state. The quarks have hidden Fermi surfaces of radius k F.

103 A Strange Metal k y kf! Can bosons form a metal? k x Yes, if each boson, b, fractionalizes b = f 1 f 2! into 2 fermions ( quarks ) Each quark is charged under an emergent gauge force, which encapsulates the entanglement in the ground state. The quarks have hidden Fermi surfaces of radius k F.

104 Holographic theory of a strange metal Electric flux The density of particles Q creates an electric flux E r which modifies the metric of the emergent spacetime.

105 Holographic theory of a strange metal Hidden Fermi surfaces of quarks? Electric flux The density of particles Q creates an electric flux E r which modifies the metric of the emergent spacetime.

106 Holographic theory of a strange metal Hidden Fermi surfaces of quarks? Electric flux The general metric transforms under rescaling as x i! x i, t! z t, ds! /d ds. Recall: conformal matter has = 0, z =1, and the metric is anti-de Sitter

107 Holographic theory of a strange metal Hidden Fermi surfaces of quarks? Electric flux The general metric transforms under rescaling as N. Ogawa, T. Takayanagi, and T. Ugajin, JHEP 1201, 125 (2012). L. Huijse, S. Sachdev, B. Swingle, Physical Review B 85, (2012) x i! x i, t! z t, ds! /d ds. The value = d 1 reproduces all the essential characteristics Recall: conformal matter has = 0, z =1, and the metric is of the entropy and entanglement entropy of a strange metal. anti-de Sitter

108 Holographic theory of a strange metal Hidden Fermi surfaces of quarks? Electric flux The general metric transforms under rescaling as N. Ogawa, T. Takayanagi, and T. Ugajin, JHEP 1201, 125 (2012). L. Huijse, S. Sachdev, B. Swingle, Physical Review B 85, (2012) x i! x i, t! z t, ds! /d ds. The null-energy condition of gravity yields z 1 + /d. Ind = 2, this leadsrecall: to z 3/2. conformal Field theory matteronhas strange = 0, metal z =1, yields andz the =3/2 metric to 3 is loops! anti-de Sitter M. A. Metlitski and S. Sachdev, Phys. Rev. B 82, (2010)

109 Conclusions Realizations of many-particle entanglement: Z2 spin liquids and conformal quantum critical points

110 Conclusions Conformal quantum matter New insights and solvable models for diffusion and transport of strongly interacting systems near quantum critical points The description is far removed from, and complementary to, that of the quantum Boltzmann equation which builds on the quasiparticle/vortex picture. Good prospects for experimental tests of frequencydependent, non-linear, and non-equilibrium transport

111 Conclusions More complex examples in metallic states are experimentally ubiquitous, but pose difficult strong-coupling problems to conventional methods of field theory

112 Conclusions String theory and gravity in emergent dimensions offer a remarkable new approach to describing states with manyparticle quantum entanglement. Much recent progress offers hope of a holographic description of strange metals