States of quantum matter with long-range entanglement in d spatial dimensions. Gapped quantum matter Spin liquids, quantum Hall states

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1 States of quantum matter with long-range entanglement in d spatial dimensions Gapped quantum matter Spin liquids, quantum Hall states Conformal quantum matter Graphene, ultracold atoms, antiferromagnets Compressible quantum matter Graphene, strange metals in high temperature superconductors, spin liquids

2 Conformal quantum matter A. Field theory:graphene B. Field theory: superfluidinsulator transition C. Field theory: antiferromagnets D. Gauge-gravity duality

3 Conformal quantum matter A. Field theory:graphene B. Field theory: superfluidinsulator transition C. Field theory: antiferromagnets D. Gauge-gravity duality

4 Honeycomb lattice (describes graphene after adding long-range Coulomb interactions) H = t ij c iα c jα + U i n i 1 2 n i 1 2

5 Q 1 Semi-metal with massless Dirac fermions at small U/t Q 1 Brillouin zone

6 We define the Fourier transform of the fermions by c A (k) = r c A (r)e ik r (4) and similarly for c B. A and B are sublattice indices. The hopping Hamiltonian is H 0 = t ij c Aiα c Bjα + c Bjα c Aiα (5) where α is a spin index. If we introduce Pauli matrices τ a in sublattice space (a = x, y, z), this Hamiltonian can be written as H 0 = d 2 k 4π 2 c (k) t cos(k e 1 ) + cos(k e 2 ) + cos(k e 3 ) τ x + t sin(k e 1 )+sin(k e 2 )+sin(k e 3 ) τ y c(k) (6) The low energy excitations of this Hamiltonian are near k ±Q 1.

7 In terms of the fields near Q 1 and Q 1,wedefine Ψ A1α (k) = c Aα (Q 1 + k) Ψ A2α (k) = c Aα ( Q 1 + k) Ψ B1α (k) = c Bα (Q 1 + k) Ψ B2α (k) = c Bα ( Q 1 + k) (7) We consider Ψ to be a 8 component vector, and introduce Pauli matrices ρ a which act in the 1, 2 valley space. Then the Hamiltonian is d 2 k H 0 = 4π 2 Ψ (k) vτ y k x + vτ x ρ z k y Ψ(k), (8) where v =3t/2; below we set v = 1. Now define Ψ = Ψ ρ z τ z. Then we can write the imaginary time Lagrangian as L 0 = iψ (ωγ 0 + k x γ 1 + k y γ 2 ) Ψ (9) where γ 0 = ρ z τ z γ 1 = ρ z τ x γ 2 = τ y (10)

8 Then we decouple the interaction via Exercise: Observe that L 0 is invariant under the scaling transformation x = xe and τ = τe.writethehubbard interaction U in terms of the Dirac fermions, and show that it has the tree-level scaling transformation U = Ue. So argue that all short-range interactions are irrelevant in the Dirac semi-metal phase. Antiferromagnetism We use the operator equation (valid on each site i): U n 1 n 1 = 2U S a2 + U 4 (11)

9 Q 1 Q 1 Brillouin zone

10 Q 1 The theory of free Dirac fermions is invariant under conformal transformations of spacetime. This is a realization of a simple conformal field theory in 2+1 dimensions: a CFT3 Q 1 Brillouin zone

11 H = i,j The Hubbard Model at large U t ij c iα c jα + U i n i 1 2 n i 1 2 µ i c iα c iα In the limit of large U, and at a density of one particle per site, this maps onto the Heisenberg antiferromagnet H AF = i<j J ij S a i S a j where a = x, y, z, S a i = 1 2 ca iα σa αβc iβ, with σ a the Pauli matrices and J ij = 4t2 ij U

12 Dirac semi-metal Insulating antiferromagnet with Neel order U/t

13 Antiferromagnetism We use the operator equation (valid on each site i): U n 1 n 1 = 2U S a2 + U 4 (11) Then we decouple the interaction via 2U exp dτsi a2 = DJi a (τ)exp 3 dτ 3 8U Ja2 i Ji a Si a i i (12) We now integrate out the fermions, and look for the saddle point of the resulting effective action for Ji a. At the saddle-point we find Long wavelength fluctuations about this saddle point are described by a field theory of the Néel order parameter, ϕ a, coupled to the Dirac fermions in the Gross-Neveu model. L = Ψγ µ µ Ψ ( µ ϕ a ) 2 + sϕ a2 + u 24 ϕ a2 2 λϕ a Ψρ z σ a Ψ (15) I.F. Herbut, V. Juricic, and B. Roy, Phys. Rev. B 79, (2009).

14 ε k ε k Insulating Dirac antiferromagnet semi-metal with Neel order ϕ a =0 k ϕ a =0 k s

15 ε k ε k Dirac semi-metal Insulating antiferromagnet with Neel order ϕ a =0 s k ϕ a =0 At the quantum critical point, the non-linear couplings λ and u in the Gross- Neveu model reach non-zero fixed-point values under the renormalization group flow. The critical theory is an interacting CFT3 k

16 ε k ε k Insulating Dirac antiferromagnet semi-metal with Neel order ϕ a =0 k ϕ a =0 k s Free CFT3 Interacting CFT3 with long-range entanglement

17 Long-range entanglement in a CFT3 Long-range entanglement: entanglement entropy obeys S EE = al γ, whereγ is a universal number associated with the CFT3. B A 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:

18 Electron Green s function for the interacting CFT3 G(k, ω) = Ψ(k, ω); Ψ (k, ω) iω + vk xτ y + vk y τ x ρ z (ω 2 + v 2 kx 2 + v 2 ky 2 (17) ) 1 η/2 where η > 0 is the anomalous dimension of the fermion. Note that this leads to a fermion spectral density which has no quasiparticle pole: thus the quantum critical point has no well-defined quasiparticle excitations.

19 ImG(k, ω) v k

20 ε k ε k Insulating Dirac antiferromagnet semi-metal with Neel order ϕ a =0 k ϕ a =0 k s Quantum phase transition described by a strongly-coupled conformal field theory without well-defined quasiparticles

21 Electrical transport The conserved electrical current is J µ = iψγ µ Ψ. (1) Let us compute its two-point correlator, K µν (k) ataspacetime momentum k µ at T = 0. At leading order, this is given by a one fermion loop diagram which evaluates to d 3 p Tr [γ µ (iγ λ p λ + mρ z σ z )γ ν (iγ δ (k δ + p δ )+mρ z σ z )] K µν (k) = 8π 3 (p 2 + m 2 )((p + k) 2 + m 2 ) = 2 δ µν k 1 µk ν k 2 x(1 x) π k 2 dx m 2 + k 2 x(1 x), (2) where the mass m =0inthesemi-metalandatthequantum critical point, while m = λn 0 in the insulator. Note that the current correlation is purely transverse, and this follows from the requirement of current conservation 0 k µ K µν =0. (3)

22 Of particular interest to us is the K 00 component, after analytic continuation to Minkowski space where the spacetime momentum k µ is replaced by (ω, k). The conductivity is obtained from this correlator via the Kubo formula σ(ω) = lim k 0 iω k 2 K 00(ω, k). (4) In the insulator, where m > 0, analysis of the integrand in Eq. (2) shows that that the spectral weight of the density correlator has a gap of 2m at k = 0, and the conductivity in Eq. (4) vanishes. These properties are as expected in any insulator. In the metal, and at the critical point, where m = 0, the fermionic spectrum is gapless, and so is that of the charge correlator. The density correlator in Eq. (2) and the conductivity in Eq. (4) evaluate to the simple universal results K 00 (ω, k) = 1 4 k 2 k 2 ω 2 σ(ω) = 1/4. (5) Going beyond one-loop, we find no change in these results in the

23 semi-metal to all orders in perturbation theory. At the quantum critical point, there are no anomalous dimensions for the conserved current, but the amplitude does change yielding K 00 (ω, k) = K k 2 k 2 ω 2 σ(ω) = K, (6) where K is a universal number dependent only upon the universality class of the quantum critical point. The value of the K for the Gross-Neveu model is not known exactly, but can be estimated by computations in the (3 d) or 1/N expansions.

24 ε k ε k Insulating Dirac antiferromagnet semi-metal with Neel order ϕ a =0 k ϕ a =0 k s Free CFT3 Interacting CFT3 with long-range entanglement

25 ε k ε k Insulating Dirac antiferromagnet semi-metal with Neel order ϕ a =0 k ϕ a =0 k s σ(ω) = πe2 2h σ(ω) = Ke2

26 Phase diagram at non-zero temperatures Quantum critical Insulator with thermally Semi-metal excited spin waves

27 Phase diagram at non-zero temperatures Quantum critical Insulator with thermally excited spin waves Semi-metal σ(ω T )= πe2 2h

28 Phase diagram at non-zero temperatures Quantum critical Insulator with thermally excited spin waves Semi-metal σ(ω T )= πe2 2h σ(ω T )= Ke2

29 Optical conductivity of graphene Z. Q. Li, E. A. Henriksen, Z. Jiang, Z. Hao, M. C. Martin, P. Kim, H. L. Stormer, and D. N. Basov, Nature Physics 4, 532 (2008).

30 Optical conductivity of graphene Undoped graphene Z. Q. Li, E. A. Henriksen, Z. Jiang, Z. Hao, M. C. Martin, P. Kim, H. L. Stormer, and D. N. Basov, Nature Physics 4, 532 (2008).

31 Non-zero temperatures At the quantum-critical point at one-loop order, we can set m = 0, and then repeat the computation in Eq. (2) at T > 0. This only requires replacing the integral over the loop frequency by a summation over the Matsubara frequencies, which are quantized by odd multiples of πt. Such a computation, via Eq. (4) leads to the conductivity Re[σ(ω)] = (2T ln 2) δ(ω)+ 1 4 tanh ω 4T ; (7) the imaginary part of σ(ω) is the Hilbert transform of Re[σ(ω)] 1/4. Note that this reduces to Eq. (5) in the limit ω T. However, the most important new feature of Eq. (7) arises for ω T, where we find a delta function at zero frequency in the real part. Thus the d.c. conductivity is infinite at this order, arising from the collisionless transport of thermally excited carriers.

32 Electrical transport in a free CFT3 for T > 0 σ Tδ(ω) ω/t

33 Particles Holes Momentum Momentum Current ε k Current ε k k Particle hole symmetry: current carrying state has zero momentum, and collisions can relax current to zero k

34 Electrical transport for a (weakly) interacting CFT3 σ(ω, T) = e2 h Σ ω k B T ; Σ a universal function Re[σ(ω)] K. Damle and S. Sachdev, Phys. Rev. B 56, 8714 (1997). 1 ω k B T

35 Electrical transport for a (weakly) interacting CFT3 σ(ω, T) = e2 h Σ ω k B T ; Σ a universal function Re[σ(ω)] O((u ) 2 ), where u is the fixed point interaction K. Damle and S. Sachdev, Phys. Rev. B 56, 8714 (1997). 1 ω k B T

36 Electrical transport for a (weakly) interacting CFT3 σ(ω, T) = e2 h Σ ω k B T ; Σ a universal function Re[σ(ω)] O(1/(u ) 2 ) O((u ) 2 ), where u is the fixed point interaction K. Damle and S. Sachdev, Phys. Rev. B 56, 8714 (1997). 1 ω k B T

37 Electrical transport for a (weakly) interacting CFT3 σ(ω, T) = e2 h Σ ω k B T ; Σ a universal function Re[σ(ω)] O(1/(u ) 2 ) O((u ) 2 ), where u is the fixed point interaction K. Damle and S. Sachdev, Phys. Rev. B 56, 8714 (1997). 1 ω k B T

38 Electrical transport for a (weakly) interacting CFT3 σ(ω, T) = e2 h Σ ω k B T ; Σ a universal function Re[σ(ω)] O(1/(u ) 2 ) O((u ) 2 ), where u is the fixed point interaction Needed: a method for computing the d.c. conductivity of interacting CFT3s (including that of pure graphene!) K. Damle and S. Sachdev, Phys. Rev. B 56, 8714 (1997). 1 ω k B T

39 Conformal quantum matter A. Field theory:graphene B. Field theory: superfluidinsulator transition C. Field theory: antiferromagnets D. Gauge-gravity duality

40 Conformal quantum matter A. Field theory:graphene B. Field theory: superfluidinsulator transition C. Field theory: antiferromagnets D. Gauge-gravity duality

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

42 The Superfluid-Insulator transition Boson Hubbard model [b j,b k ]=δ jk M.P. A. Fisher, P.B. Weichmann, G. Grinstein, and D.S. Fisher, Phys. Rev. B 40, 546 (1989).

43 Excitations of the insulator: S = d 2 rdτ τ ψ 2 + v 2 ψ 2 +(g g c ) ψ 2 + u 2 ψ 4 M.P. A. Fisher, P.B. Weichmann, G. Grinstein, and D.S. Fisher, Phys. Rev. B 40, 546 (1989).

44 T Quantum critical T KT ψ =0 ψ =0 Superfluid Insulator 0 g c g

45 T Quantum critical T KT ψ =0 ψ =0 Superfluid Insulator 0 g c CFT3 g

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

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

48 Quantum critical transport Quantum nearly perfect fluid with shortest possible equilibration time, τ eq τ eq = C k B T where C is a universal constant S. Sachdev, Quantum Phase Transitions, Cambridge (1999).

49 Quantum critical transport Quantum nearly perfect fluid with shortest possible equilibration time, τ eq τ eq = C k B T where C is a universal constant S. Sachdev, Quantum Phase Transitions, Cambridge (1999).

50 Quantum critical transport Quantum nearly perfect fluid with shortest possible equilibration time, τ eq τ eq = C k B T where C is a universal constant Zaanen: Planckian dissipation S. Sachdev, Quantum Phase Transitions, Cambridge (1999).

51 Quantum critical transport Transport co-oefficients not determined by collision rate, but by universal constants of nature Conductivity σ = 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).

52 Quantum critical transport Transport co-oefficients not determined by collision rate, but by universal constants of nature Momentum transport η s viscosity entropy density = k B [Universal constant O(1) ] P. Kovtun, D. T. Son, and A. Starinets, Phys. Rev. Lett. 94, (2005)

53 Quantum critical transport Describe charge transport using Boltzmann theory of interacting bosons: dv dt + v = F. τ c This gives a frequency (ω) dependent conductivity σ(ω) = σ 0 1 i ωτ c where τ c /(k B T ) is the time between boson collisions. Also, we have σ(ω )=σ, associated with the density of states for particle-hole creation (the optical conductivity ) in the CFT3.

54 Quantum critical transport Describe charge transport using Boltzmann theory of interacting bosons: dv dt + v = F. τ c This gives a frequency (ω) dependent conductivity σ(ω) = σ 0 1 i ωτ c where τ c /(k B T ) is the time between boson collisions. Also, we have σ(ω )=σ, associated with the density of states for particle-hole creation (the optical conductivity ) in the CFT3.

55 Electrical transport in a free-field theory for T > 0 σ Tδ(ω) ω/t

56 Boltzmann theory of bosons Re[σ(ω)] σ 0 1/τ c σ ω

57 So far, we have described the quantum critical point using the boson particle and hole excitations of the insulator. T Quantum critical T KT ψ =0 ψ =0 Superfluid Insulator 0 g c g

58 However, we could equally well describe the conductivity using the excitations of the superfluid, which are vortices. T These are quantum particles (in 2+1 dimensions) which described by a (mirror/e.m.) dual CFT3 with an emergent U(1) gauge field. Their T>0 dynamics can also be described by a Boltzmann equation: Quantum critical Conductivity = Resistivity of vortices T KT ψ =0 ψ =0 Superfluid Insulator 0 g c g

59 However, we could equally well describe the conductivity using the excitations of the superfluid, which are vortices. T These are quantum particles (in 2+1 dimensions) which described by a (mirror/e.m.) dual CFT3 with an emergent U(1) gauge field. Their T>0 dynamics can also be described by a Boltzmann equation: Quantum critical Conductivity = Resistivity of vortices T KT ψ =0 ψ =0 Superfluid Insulator 0 g c g

60 Boltzmann theory of bosons Re[σ(ω)] σ 0 1/τ c σ ω

61 Boltzmann theory of vortices Re[σ(ω)] 1/τ cv σ 1/σ 0v ω

62 Boltzmann theory of bosons Re[σ(ω)] σ 0 1/τ c σ ω

63 Boltzmann theory of bosons Re[σ(ω)] σ 0 1/τ c σ ω

64 Boltzmann theory of bosons Re[σ(ω)] σ 0 1/τ c Needed: a method for computing the d.c. conductivity σ of interacting CFT3s (including that of the boson Hubbard model!) ω

65 Conformal quantum matter A. Field theory:graphene B. Field theory: superfluidinsulator transition C. Field theory: antiferromagnets D. Gauge-gravity duality

66 Conformal quantum matter A. Field theory:graphene B. Field theory: superfluidinsulator transition C. Field theory: antiferromagnets D. Gauge-gravity duality

67 Quantum critical point in a frustrated square lattice antiferromagnet ) or Néel state s c Valence bond solid (VBS) state with a nearly gapless, emergent photon s Long-range entanglement described by a CFT3 with an emergent U(1) photon O.I. Motrunich and A. Vishwanath, Phys. Rev. B 70, (2004). T. Senthil, A. Vishwanath, L. Balents, S. Sachdev and M.P.A. Fisher, Science 303, 1490 (2004).

68 Quantum critical point in a frustrated square lattice antiferromagnet ) or Néel state s c Valence bond solid (VBS) state with a nearly gapless, emergent photon s Critical theory for photons and deconfined spinons: S z = d 2 rdτ ( µ ia µ )z α 2 +s z α 2 +u( z α 2 ) e 2 ( µνλ ν A λ ) 2 0 O.I. Motrunich and A. Vishwanath, Phys. Rev. B 70, (2004). T. Senthil, A. Vishwanath, L. Balents, S. Sachdev and M.P.A. Fisher, Science 303, 1490 (2004).

69 Im[Ψ vbs ] Distribution of VBS order Ψ vbs at large Q Re[Ψ vbs ] Circular symmetry is evidence for emergent U(1) photon A.W. Sandvik, Phys. Rev. Lett. 98, (2007).

70 Conformal quantum matter A. Field theory:graphene B. Field theory: superfluidinsulator transition C. Field theory: antiferromagnets D. Gauge-gravity duality

71 Conformal quantum matter A. Field theory:graphene B. Field theory: superfluidinsulator transition C. Field theory: antiferromagnets D. Gauge-gravity duality

72 Field theories in d + 1 spacetime dimensions are characterized by couplings g which obey the renormalization group equation u dg du = β(g) where u is the energy scale. The RG equation is local in energy scale, i.e. the RHS does not depend upon u.

73 u J. McGreevy, arxiv

74 r J. McGreevy, arxiv

75 r

76 r Key idea: Implement r as an extra dimension, and map to a local theory in d + 2 spacetime dimensions.

77 For a relativistic CFT in d spatial dimensions, the metric in the holographic space is uniquely fixed by demanding the following scale transformaion (i =1...d) x i ζx i, t ζt, ds ds This gives the unique metric ds 2 = 1 r 2 dt 2 + dr 2 + dx 2 i Reparametrization invariance in r has been used to the prefactor of dx 2 equal to 1/r i 2. This fixes r ζr under the scale transformation. This is the metric of the space AdS d+2.

78 For a relativistic CFT in d spatial dimensions, the metric in the holographic space is uniquely fixed by demanding the following scale transformaion (i =1...d) x i ζx i, t ζt, ds ds This gives the unique metric ds 2 = 1 r 2 dt 2 + dr 2 + dx 2 i Reparametrization invariance in r has been used to the prefactor of dx 2 equal to 1/r i 2. This fixes r ζr under the scale transformation. This is the metric of the space AdS d+2.

79 AdS/CFT correspondence AdS d+2 R d,1 Minkowski CFT d+1 zr

80 AdS/CFT correspondence AdS 4 R 2,1 Minkowski CFT3 zr

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

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

83 AdS/CFT correspondence at non-zero temperatures AdS4-Schwarzschild black-brane r A 2+1 dimensional system at its quantum critical point: k B T = 3 4πR. There is a family of solutions of Einstein gravity which describe non-zero temperatures 2 L dr ds 2 2 = r f(r) f(r)dt2 + dx 2 + dy 2 with f(r) =1 (r/r) 3

84 AdS/CFT correspondence at non-zero temperatures AdS4-Schwarzschild black-brane r A 2+1 dimensional system at its quantum critical point: k B T = 3 4πR. Black-brane at temperature of 2+1 dimensional quantum critical system 2 L dr ds 2 2 = r f(r) f(r)dt2 + dx 2 + dy 2 with f(r) =1 (r/r) 3

85 AdS/CFT correspondence at non-zero temperatures AdS4-Schwarzschild black-brane r Black-brane at temperature of 2+1 dimensional quantum critical system Beckenstein-Hawking entropy of black brane = entropy of CFT3 A 2+1 dimensional system at its quantum critical point: k B T = 3 4πR.

86 AdS/CFT correspondence at non-zero temperatures AdS4-Schwarzschild black-brane A 2+1 dimensional system at its quantum critical point: k B T = 3 4πR. Black-brane at temperature of 2+1 dimensional quantum critical system Friction of quantum criticality = waves falling into black brane

87 AdS4 theory of nearly perfect fluids To leading order in a gradient expansion, charge transport in an infinite set of strongly-interacting CFT3s can be described by Einstein-Maxwell gravity/electrodynamics on AdS 4 -Schwarzschild S EM = d 4 x g 1 4g4 2 F ab F ab. C. P. Herzog, P. K. Kovtun, S. Sachdev, and D. T. Son, Phys. Rev. D 75, (2007).

88 AdS4 theory of nearly perfect fluids To leading order in a gradient expansion, charge transport in an infinite set of strongly-interacting CFT3s can be described by Einstein-Maxwell gravity/electrodynamics on AdS 4 -Schwarzschild S EM = d 4 x g 1 4g4 2 F ab F ab. This is to be solved subject to the constraint A µ (r 0,x,y,t)=A µ (x, y, t) where A µ is a source coupling to a conserved U(1) current J µ of the CFT3 S = S CFT + i dxdydta µ J µ C. P. Herzog, P. K. Kovtun, S. Sachdev, and D. T. Son, Phys. Rev. D 75, (2007).

89 AdS4 theory of electrical transport in a strongly interacting CFT3 for T > 0 σ 1 g 2 4 Conductivity is independent of ω/t. ω/t

90 AdS4 theory of electrical transport in a strongly interacting CFT3 for T > 0 σ 1 g 2 4 Conductivity is independent of ω/t. Consequence of self-duality of Maxwell theory in 3+1 dimensions C. P. Herzog, P. K. Kovtun, S. Sachdev, and D. T. Son, Phys. Rev. D 75, (2007). ω/t

91 Electrical transport in a free CFT3 for T > 0 σ Tδ(ω) Complementary ω-dependent conductivity in the free theory ω/t

92 Improving the AdS4 theory of nearly perfect fluids To leading order in a gradient expansion, charge transport in an infinite set of strongly-interacting CFT3s can be described by Einstein-Maxwell gravity/electrodynamics on AdS 4 -Schwarzschild S EM = d 4 x g 1 4e 2 F abf ab. R. C. Myers, S. Sachdev, and A. Singh, Physical Review D 83, (2011)

93 Improving the AdS4 theory of nearly perfect fluids To leading order in a gradient expansion, charge transport in an infinite set of strongly-interacting CFT3s can be described by Einstein-Maxwell gravity/electrodynamics on AdS 4 -Schwarzschild S EM = d 4 x g 1 We include all possible 4-derivative terms: 4e 2 F abf ab after. suitable field redefinitions, the required theory has only one dimensionless constant γ (L is the radius of AdS 4 ): S EM = d 4 x g 1 4g 2 4 F ab F ab + γl2 g 2 4 C abcd F ab F cd where C abcd is the Weyl curvature tensor. Stability and causality constraints restrict γ < 1/12., R. C. Myers, S. Sachdev, and A. Singh, Physical Review D 83, (2011)

94 Improving the AdS4 theory of nearly perfect fluids 1.5 σ(ω) h σ( ) Q 2 σ 1.0 γ = 1 12 γ =0 0.5 γ = ω 4πT R. C. Myers, S. Sachdev, and A. Singh, Physical Review D 83, (2011)

95 Improving the AdS4 theory of nearly perfect fluids 1.5 σ(ω) h σ( ) Q 2 σ 1.0 γ = 1 12 γ =0 0.5 γ = 1 12 The γ > 0 result has similarities to the quantum-boltzmann result for transport of particle-like excitations ω 4πT R. C. Myers, S. Sachdev, and A. Singh, Physical Review D 83, (2011)

96 Improving the AdS4 theory of nearly perfect fluids 1.5 σ(ω) h σ( ) Q 2 σ 1.0 γ = 1 12 γ =0 0.5 γ = 1 12 The γ < 0 result can be interpreted as the transport of vortex-like excitations ω 4πT R. C. Myers, S. Sachdev, and A. Singh, Physical Review D 83, (2011)

97 Improving the AdS4 theory of nearly perfect fluids 1.5 σ(ω) h σ( ) Q 2 σ 1.0 γ = 1 12 γ = γ = 1 12 The γ = 0 case is the exact result for the large N limit of SU(N) gauge theory with N =8supersymmetry(the ABJM model). The ω-independence is a consequence of self-duality under particle-vortex duality (S-duality) ω 4πT R. C. Myers, S. Sachdev, and A. Singh, Physical Review D 83, (2011)

98 Improving the AdS4 theory of nearly perfect fluids 1.5 σ(ω) h σ( ) Q 2 σ 1.0 γ = 1 12 γ =0 0.5 γ = 1 12 Stability constraints on the effective theory ( γ < 1/12) allow only a limited ω-dependence in the conductivity ω 4πT R. C. Myers, S. Sachdev, and A. Singh, Physical Review D 83, (2011)

99 Conclusions Conformal quantum matter

100 Conclusions Conformal quantum matter New insights and solvable models for diffusion and transport of strongly interacting systems near quantum critical points

101 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.

102 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. Prospects for experimental tests of frequency-dependent, non-linear, and non-equilibrium transport