Disordered spacetimes in AdS/CMT. Andrew Lucas

Transcription

1 Disordered spacetimes in AdS/CMT Andrew Lucas Stanford Physics Disorder in Condensed Matter and Black Holes; Leiden January 9, 2017

2 Advertisement page review article on AdS/CMT, together with: Sean Hartnoll Stanford Physics Subir Sachdev Harvard Physics & Perimeter Institute

3 Advertisement page review article on AdS/CMT, together with: Sean Hartnoll Stanford Physics Subir Sachdev Harvard Physics & Perimeter Institute this talk loosely follows Sections 7.2, 5.6, 5.8, 5.9 and 5.10

4 Disordered Quantum Matter 3 Quenched Randomness (interacting) QFT in d spatial dimensions with H = H 0 d d x h(x)o(x), H 0 translation invariant and h(x) random: E [h(x)] = 0, E [h(x)h(y)] = ε 2 δ(x y).

5 Disordered Quantum Matter 3 Quenched Randomness (interacting) QFT in d spatial dimensions with H = H 0 d d x h(x)o(x), H 0 translation invariant and h(x) random: E [h(x)] = 0, E [h(x)h(y)] = ε 2 δ(x y). what is the phase diagram as a function of ε and T?

6 Disordered Quantum Matter 3 Quenched Randomness (interacting) QFT in d spatial dimensions with H = H 0 d d x h(x)o(x), H 0 translation invariant and h(x) random: E [h(x)] = 0, E [h(x)h(y)] = ε 2 δ(x y). what is the phase diagram as a function of ε and T? measurable properties such as conductivity?

7 Disordered Quantum Matter 3 Quenched Randomness (interacting) QFT in d spatial dimensions with H = H 0 d d x h(x)o(x), H 0 translation invariant and h(x) random: E [h(x)] = 0, E [h(x)h(y)] = ε 2 δ(x y). what is the phase diagram as a function of ε and T? measurable properties such as conductivity? non-perturbative results in ε?

8 Disordered Quantum Matter 4 The Holographic Setup holographic realization of this disorder is natural: for simplicity let us consider asymptotically AdS (z 0) spacetime: ds 2 = L2 z 2 [ dz 2 + dx µ dx µ ] +.

9 Disordered Quantum Matter 4 The Holographic Setup holographic realization of this disorder is natural: for simplicity let us consider asymptotically AdS (z 0) spacetime: ds 2 = L2 z 2 [ dz 2 + dx µ dx µ ] +. let φ be bulk scalar dual to O: φ(z 0) = h(x) O(x) L d/2 zd+1 + (2 d 1)L d/2 z +.

10 Disordered Quantum Matter 4 The Holographic Setup holographic realization of this disorder is natural: for simplicity let us consider asymptotically AdS (z 0) spacetime: ds 2 = L2 z 2 [ dz 2 + dx µ dx µ ] +. let φ be bulk scalar dual to O: φ(z 0) = h(x) O(x) L d/2 zd+1 + (2 d 1)L d/2 z +. solve nonlinear equations of motion from action S = d d+2 x g (R 2Λ 12 ) ( φ)2 m2 2 φ2 + with inhomogeneous boundary conditions h(x) on φ.

11 Disordered Quantum Matter 4 The Holographic Setup holographic realization of this disorder is natural: for simplicity let us consider asymptotically AdS (z 0) spacetime: ds 2 = L2 z 2 [ dz 2 + dx µ dx µ ] +. let φ be bulk scalar dual to O: φ(z 0) = h(x) O(x) L d/2 zd+1 + (2 d 1)L d/2 z +. solve nonlinear equations of motion from action S = d d+2 x g (R 2Λ 12 ) ( φ)2 m2 2 φ2 + with inhomogeneous boundary conditions h(x) on φ. fluctuations around solution = correlation functions.

12 Disordered Quantum Matter 5 Harris Criterion geometry (strongly) inhomogeneous at radial scale z = disorder significantly alters physics at scale z

13 Disordered Quantum Matter 5 Harris Criterion geometry (strongly) inhomogeneous at radial scale z = disorder significantly alters physics at scale z is disorder relevant in the IR? irrelevant: relevant:

14 Disordered Quantum Matter 5 Harris Criterion geometry (strongly) inhomogeneous at radial scale z = disorder significantly alters physics at scale z holographic analysis: homogeneous φ: backreaction cannot be neglected if which occurs at a scale R ab + Λg ab AdS 1 z 2 T ab[φ] φ2 z 2 1 h 2 z 2(d+1 )

15 Disordered Quantum Matter 5 Harris Criterion geometry (strongly) inhomogeneous at radial scale z = disorder significantly alters physics at scale z holographic analysis: homogeneous φ: backreaction cannot be neglected if which occurs at a scale R ab + Λg ab AdS 1 z 2 T ab[φ] φ2 z 2 1 h 2 z 2(d+1 ) low energy physics will be substantially modified if < d + 1

16 Disordered Quantum Matter 5 Harris Criterion geometry (strongly) inhomogeneous at radial scale z = disorder significantly alters physics at scale z holographic analysis: disordered φ: in AdS background, φ = d d k h(k)e ik x F (kz), F (z) z (d+1)/2 K (d+1 )/2 (z).

17 Disordered Quantum Matter 5 Harris Criterion geometry (strongly) inhomogeneous at radial scale z = disorder significantly alters physics at scale z holographic analysis: disordered φ: in AdS background, φ = d d k h(k)e ik x F (kz), F (z) z (d+1)/2 K (d+1 )/2 (z). the geometry is substantially modified if [ ] 1 E d d k h(k) 2 F (kz) 2 ε 2 z d+2 2

18 Disordered Quantum Matter 5 Harris Criterion geometry (strongly) inhomogeneous at radial scale z = disorder significantly alters physics at scale z holographic analysis: disordered φ: in AdS background, φ = d d k h(k)e ik x F (kz), F (z) z (d+1)/2 K (d+1 )/2 (z). the geometry is substantially modified if [ ] 1 E d d k h(k) 2 F (kz) 2 ε 2 z d+2 2 disorder relevant (in the IR) if < d (Harris criterion)

19 Ground States 6 Marginal Disorder ground state when disorder is relevant in IR?

20 Ground States 6 Marginal Disorder ground state when disorder is relevant in IR? no perturbatively accessible fixed point: β(ε) = (d uc d)ε cε 2, (c > 0).

21 Ground States 6 Marginal Disorder ground state when disorder is relevant in IR? no perturbatively accessible fixed point: β(ε) = (d uc d)ε cε 2, (c > 0). let us consider the holographic model S = d 3 x g (R 2Λ 12 ) ( φ)2 m2 2 φ2 with d = 1, = 3/2 (Harris marginal: ε is dimensionless) [Hartnoll, Santos: ]

22 Ground States 7 Marginal Disorder: Lifshitz Scaling ansatz: (disorder averaged) homogeneous metric: ds 2 = A ε (z) ( dz 2 + dx 2) B ε (z)dt 2,

23 Ground States 7 Marginal Disorder: Lifshitz Scaling ansatz: (disorder averaged) homogeneous metric: ds 2 = A ε (z) ( dz 2 + dx 2) B ε (z)dt 2, EOMs with slick perturbation theory: [review: Section 7.2] R ab [A ε, B ε ] + = E[T ab [φ 0, A 0, B 0 ]] ε2 z 2

24 Ground States 7 Marginal Disorder: Lifshitz Scaling ansatz: (disorder averaged) homogeneous metric: ds 2 = A ε (z) ( dz 2 + dx 2) B ε (z)dt 2, EOMs with slick perturbation theory: [review: Section 7.2] R ab [A ε, B ε ] + = E[T ab [φ 0, A 0, B 0 ]] ε2 z 2 pair of ODEs for A ε, B ε with solution: A ε (z) = a ε z 2, B ε(z) = b ε z 2+ε2 /4

25 Ground States 7 Marginal Disorder: Lifshitz Scaling ansatz: (disorder averaged) homogeneous metric: ds 2 = A ε (z) ( dz 2 + dx 2) B ε (z)dt 2, EOMs with slick perturbation theory: [review: Section 7.2] R ab [A ε, B ε ] + = E[T ab [φ 0, A 0, B 0 ]] ε2 z 2 pair of ODEs for A ε, B ε with solution: A ε (z) = a ε z 2, B ε(z) = b ε z 2+ε2 /4 physical interpretation: dynamical critical exponent! ω k z, z = 1 + ε2 8 + O ( ε 4).

26 Ground States 8 Marginal Disorder: Numerics numerical analysis reveals: [Hartnoll, Santos: ]; [Hartnoll, Ramirez, Santos: , ] inhomogeneous ground states:

27 Ground States 8 Marginal Disorder: Numerics numerical analysis reveals: [Hartnoll, Santos: ]; [Hartnoll, Ramirez, Santos: , ] inhomogeneous ground states: finite T entropy has expected scaling s T 1/z

28 Ground States 8 Marginal Disorder: Numerics numerical analysis reveals: [Hartnoll, Santos: ]; [Hartnoll, Ramirez, Santos: , ] inhomogeneous ground states: finite T entropy has expected scaling s T 1/z thermal conductivity: ( κ(t ) Re T a+ib). complex scaling dimensions?

29 Ground States 9 Hovering Black Holes AdS-Einstein-Maxwell theory: hovering black holes [Horowitz, Iqbal, Santos, Way: ] + +

30 Ground States 9 Hovering Black Holes AdS-Einstein-Maxwell theory: hovering black holes [Horowitz, Iqbal, Santos, Way: ] + + point charge can float above charged black hole (static geodesic) grow into black hole

31 Ground States 9 Hovering Black Holes AdS-Einstein-Maxwell theory: hovering black holes [Horowitz, Iqbal, Santos, Way: ] + + point charge can float above charged black hole (static geodesic) grow into black hole glassy geometries with many floating black holes? [Anninos, Anous, Denef, Peeters: ]

32 Transport 10 Review of Transport Ohm s law the simplest experiment: I V = IR R 1 σ V

33 Transport 10 Review of Transport Ohm s law the simplest experiment: I V = IR R 1 σ V more generally, thermoelectric transport: ( J Q ) ( σ α = T ᾱ κ ) ( E T )

34 Transport 11 Metals or Insulators? we ll classify disordered systems crudely: metal: lim T 0 σ(t ) > 0, insulator: lim T 0 σ(t ) = 0.

35 Transport 11 Metals or Insulators? we ll classify disordered systems crudely: metal: lim σ(t ) > 0, insulator: lim σ(t ) = 0. T 0 T 0 phase diagram of a metal? (2+1 spacetime dimensions) disorder strength interaction strength

36 Transport 11 Metals or Insulators? we ll classify disordered systems crudely: metal: lim σ(t ) > 0, insulator: lim σ(t ) = 0. T 0 T 0 phase diagram of a metal? (2+1 spacetime dimensions) disorder strength interaction strength coherent metal

37 Transport 11 Metals or Insulators? we ll classify disordered systems crudely: metal: lim σ(t ) > 0, insulator: lim σ(t ) = 0. T 0 T 0 phase diagram of a metal? (2+1 spacetime dimensions) disorder strength Anderson insulator interaction strength coherent metal

38 Transport 11 Metals or Insulators? we ll classify disordered systems crudely: metal: lim σ(t ) > 0, insulator: lim σ(t ) = 0. T 0 T 0 phase diagram of a metal? (2+1 spacetime dimensions) disorder strength Anderson insulator interaction strength (many-body) localized? coherent metal

39 Transport 11 Metals or Insulators? we ll classify disordered systems crudely: metal: lim σ(t ) > 0, insulator: lim σ(t ) = 0. T 0 T 0 phase diagram of a metal? (2+1 spacetime dimensions) disorder strength Anderson insulator (diffusion-limited) incoherent metal? interaction strength coherent metal

40 Transport 11 Metals or Insulators? we ll classify disordered systems crudely: metal: lim σ(t ) > 0, insulator: lim σ(t ) = 0. T 0 T 0 phase diagram of a metal? (2+1 spacetime dimensions) disorder strength Anderson insulator (diffusion-limited) incoherent metal? interaction strength coherent metal holography essentially discovers this cartoon

41 Transport 12 Conductivity of a Clean Metal J =0 J = Qv and E =0

42 Transport J =0 12 Conductivity of a Clean Metal J =0 J = nv and E =0 J = Qv and E =0

43 Transport J =0 12 Conductivity of a Clean Metal J =0 J = nv and E =0 J = Qv and E =0 σ sensitive to how translational symmetry broken

44 Transport 13 Mean Field: Drude Model momentum conservation equation: t T ti = iωt ti = T ti + ne }{{ τ }{{} i } Lorentz force momentum relaxation

45 Transport 13 Mean Field: Drude Model momentum conservation equation: t T ti = iωt ti = T ti }{{} momentum density T ti + ne }{{ τ }{{} i } Lorentz force momentum relaxation = (ɛ + P )v i, }{{} J i = nv i, charge current

46 Transport 13 Mean Field: Drude Model momentum conservation equation: Drude peak: t T ti = iωt ti = T ti }{{} momentum density T ti + ne }{{ τ }{{} i } Lorentz force momentum relaxation = (ɛ + P )v i, }{{} J i = nv i, charge current σ(ω) = J i E i = n2 ɛ + P 1. 1 τ iω

47 Transport 13 Mean Field: Drude Model momentum conservation equation: Drude peak: t T ti = iωt ti = T ti }{{} momentum density T ti + ne }{{ τ }{{} i } Lorentz force momentum relaxation = (ɛ + P )v i, }{{} J i = nv i, charge current σ(ω) = J i E i = n2 ɛ + P 1. 1 τ iω transport dominated by slow momentum relaxation [Hartnoll, Kovtun, Müller, Sachdev, ]

48 Transport 14 Holographic Derivation of Drude Formula: Setup S = d d+2 x ( g R 2( Φ) 2 V (Φ) Z(Φ) ) 4 F 2 }{{} support homogeneous black hole background 1 d d+2 x g ( ( ψ) 2 + B(Φ)ψ 2) 2 }{{} inhomogeneous scalar hair ψ

49 Transport 14 Holographic Derivation of Drude Formula: Setup S = d d+2 x ( g R 2( Φ) 2 V (Φ) Z(Φ) ) 4 F 2 }{{} support homogeneous black hole background 1 d d+2 x g ( ( ψ) 2 + B(Φ)ψ 2) 2 }{{} inhomogeneous scalar hair ψ background: Φ, A t, g aa homogeneous at leading order; ψ ε inhomogeneous at leading order

50 Transport 15 Holographic Derivation of Drude Formula: Intuition v i ds 2 v i = A = p(dt v i dx i ) f(r) r 2 (dt v idx i ) 2 + n E i + P 1 i! bulk equations of motion are schematically: δg tx nδa x δ x ψ, iωδ x ψ δg tx, (ψ EOM) (zx-einstein) bulk modes must take Galilean boost form δa x δ x ψ + δg tx (think J = nv)

51 Transport 15 Holographic Derivation of Drude Formula: Intuition v i ds 2 v i = A = p(dt v i dx i ) f(r) r 2 (dt v idx i ) 2 + n E i + P 1 i! bulk equations of motion are schematically: δg tx nδa x δ x ψ, iωδ x ψ δg tx, (ψ EOM) (zx-einstein) bulk modes must take Galilean boost form δa x δ x ψ + δg tx (think J = nv) exactly recover hydrodynamic Drude formula [Lucas, ]

52 Transport 16 Holographic Derivation of Drude Formula: Technical Steps in holography we find [Lucas: ], [review: Section 5.6.5] ɛ + P τ k k 2 xψ(k, r h ) 2 γ(r h )

53 Transport 16 Holographic Derivation of Drude Formula: Technical Steps in holography we find [Lucas: ], [review: Section 5.6.5] ɛ + P τ k k 2 xψ(k, r h ) 2 γ(r h ) many-body memory matrix prediction: [Hartnoll, Hofman: ], [review: Section 5.6] ɛ + P τ ( ) Im G Ṙ P P = lim (ω) ω 0 ω k Im ( G R OO lim ω 0 (k, ω)) ω k 2 x h(k) 2

54 Transport 16 Holographic Derivation of Drude Formula: Technical Steps in holography we find [Lucas: ], [review: Section 5.6.5] ɛ + P τ k k 2 xψ(k, r h ) 2 γ(r h ) many-body memory matrix prediction: [Hartnoll, Hofman: ], [review: Section 5.6] ɛ + P τ ( ) Im G Ṙ P P = lim (ω) ω 0 ω k Im ( G R OO lim ω 0 square of field on horizon spectral weight: [Lucas: ], [review: Section 4.3.2] (k, ω)) ω ( ψ(k, r h ) 2 Im G R γ(r h ) = lim OO (k, ω) ) h(k) 2 ω 0 ω k 2 x h(k) 2

55 Transport 17 Mean Field Disorder break translation symmetry without breaking homogeneity: Bianchi spacetimes: [Iizuka et al, ] ds 2 = A(z)dz 2 B(z)dt 2 + C(z)dx D(z) ( ω2 2 + ω3) 2, ω 2 + iω 3 = e ikx 1 (dx 2 + dx 3)

56 Transport 17 Mean Field Disorder break translation symmetry without breaking homogeneity: Bianchi spacetimes: [Iizuka et al, ] ds 2 = A(z)dz 2 B(z)dt 2 + C(z)dx D(z) ( ω2 2 + ω3) 2, ω 2 + iω 3 = e ikx 1 (dx 2 + dx 3) Q-lattices: [Donos, Gauntlett, ] S dis = 1 d d+2 x g χ 2, χ e ikx. 2

57 Transport 17 Mean Field Disorder break translation symmetry without breaking homogeneity: Bianchi spacetimes: [Iizuka et al, ] ds 2 = A(z)dz 2 B(z)dt 2 + C(z)dx D(z) ( ω2 2 + ω3) 2, ω 2 + iω 3 = e ikx 1 (dx 2 + dx 3) Q-lattices: [Donos, Gauntlett, ] S dis = 1 d d+2 x g χ 2, χ e ikx. 2 linear axion model: [Andrade, Withers, ] S dis = 1 d d+2 x g ( χ I) 2, χ I mx i δ I i 2 I

58 Transport 17 Mean Field Disorder break translation symmetry without breaking homogeneity: Bianchi spacetimes: [Iizuka et al, ] ds 2 = A(z)dz 2 B(z)dt 2 + C(z)dx D(z) ( ω2 2 + ω3) 2, ω 2 + iω 3 = e ikx 1 (dx 2 + dx 3) Q-lattices: [Donos, Gauntlett, ] S dis = 1 d d+2 x g χ 2, χ e ikx. 2 linear axion model: [Andrade, Withers, ] S dis = 1 d d+2 x g ( χ I) 2, χ I mx i δ I i 2 I massive gravity [Vegh, ]; [Baggioli, Pujolas, ]

59 Transport 18 Mean Field Disorder: Ground States simplest axion model: simple homogeneous black hole forms with area/entropy density s max(t, m) d

60 Transport 18 Mean Field Disorder: Ground States simplest axion model: simple homogeneous black hole forms with area/entropy density s max(t, m) d at large m, commonly studied phases may have negative energy density: ɛ < 0 (m > m c ), and these phases appear unstable [Caldarelli, Christodoulou, Papadimitriou, Skenderis: ]

61 Transport 18 Mean Field Disorder: Ground States simplest axion model: simple homogeneous black hole forms with area/entropy density s max(t, m) d at large m, commonly studied phases may have negative energy density: ɛ < 0 (m > m c ), and these phases appear unstable [Caldarelli, Christodoulou, Papadimitriou, Skenderis: ] Bianchi models: k of lattice vector k may be tuned through Harris criterion: metal-insulator transition [Donos, Hartnoll: ]

62 Transport 19 Mean Field Disorder: Drude Conductivity simple analytic formula for σ: for example, σ = ( s ) (d 2)/d Z } 4π{{} σ q + 4πn2 sm 2 }{{} Drude (though this interpretation is not quite right...) [Davison, Goutéraux: ], [Blake: ]

63 Transport 19 Mean Field Disorder: Drude Conductivity simple analytic formula for σ: for example, σ = ( s ) (d 2)/d Z } 4π{{} σ q + 4πn2 sm 2 }{{} Drude (though this interpretation is not quite right...) [Davison, Goutéraux: ], [Blake: ] as m 0, long lived momentum: coherent (Drude form)

64 Transport 19 Mean Field Disorder: Drude Conductivity simple analytic formula for σ: for example, σ = ( s ) (d 2)/d Z } 4π{{} σ q + 4πn2 sm 2 }{{} Drude (though this interpretation is not quite right...) [Davison, Goutéraux: ], [Blake: ] as m 0, long lived momentum: coherent (Drude form) as m ; σ σ 0 > 0: incoherent

65 Transport 20 Conductivity Bounds and Incoherent Transport? diffusion bounds control conductivity? [Hartnoll, ] σ D v 2 bt ee v2 b T?

66 Transport 20 Conductivity Bounds and Incoherent Transport? diffusion bounds control conductivity? [Hartnoll, ] σ D v 2 bt ee v2 b T? t ee 1/T can be understood from quantum chaos [Maldacena, Shenker, Stanford, ]

67 Transport 20 Conductivity Bounds and Incoherent Transport? diffusion bounds control conductivity? [Hartnoll, ] σ D v 2 bt ee v2 b T? t ee 1/T can be understood from quantum chaos [Maldacena, Shenker, Stanford, ] experiments in many metals [Bruin, Sakai, Perry, Mackenzie (2013)] ρ = 1 σ = τ m ne 2 τ k B T

68 Transport 21 Butterfly Velocity Conjecture conjecture: v b in diffusion bound is butterfly velocity: [Blake, , ] A(t, x)b(0)a(t, x)b(0) 1 1 [ ( N 2 exp λ t x )] v b

69 Transport 21 Butterfly Velocity Conjecture conjecture: v b in diffusion bound is butterfly velocity: [Blake, , ] A(t, x)b(0)a(t, x)b(0) 1 1 [ ( N 2 exp λ t x )] v b energy diffusion bound holds in many AdS 2 R d geometries [Blake, Donos: ], [Davison et al: ]

70 Transport 21 Butterfly Velocity Conjecture conjecture: v b in diffusion bound is butterfly velocity: [Blake, , ] A(t, x)b(0)a(t, x)b(0) 1 1 [ ( N 2 exp λ t x )] v b energy diffusion bound holds in many AdS 2 R d geometries [Blake, Donos: ], [Davison et al: ] charge diffusion bound fails in hydrodynamically (striped) systems: [Lucas, Steinberg: ]: D 1 E [σ 1 (x)] E [χ(x)], v b 1 E [ χ(x) 1/2 σ(x) 1/2],

71 Transport 22 Conductivity Bounds: Einstein-Maxwell System alternative: (rigorous) conductivity bounds [Lucas: ] ( J Q T ) T ( σ T α T α T κ ) 1 ( J Q T ) T Ṡ[(some) trial functions]

72 Transport 22 Conductivity Bounds: Einstein-Maxwell System alternative: (rigorous) conductivity bounds [Lucas: ] ( J Q T ) T ( σ T α T α T κ ) 1 ( J Q T ) T Ṡ[(some) trial functions] d = 2 isotropic AdS-Einstein-Maxwell (w/ connected horizon, T > 0): [Grozdanov, Lucas, Sachdev, Schalm: ] [Grozdanov, Lucas, Schalm: ] σ 1, κ κ 2 T α2 σ 4π2 T 3

73 Transport 22 Conductivity Bounds: Einstein-Maxwell System alternative: (rigorous) conductivity bounds [Lucas: ] ( J Q T ) T ( σ T α T α T κ ) 1 ( J Q T ) T Ṡ[(some) trial functions] d = 2 isotropic AdS-Einstein-Maxwell (w/ connected horizon, T > 0): [Grozdanov, Lucas, Sachdev, Schalm: ] [Grozdanov, Lucas, Schalm: ] σ 1, κ κ 2 T α2 σ 4π2 T 3 absence of disorder-driven metal-insulator transition

74 Transport 22 Conductivity Bounds: Einstein-Maxwell System alternative: (rigorous) conductivity bounds [Lucas: ] ( J Q T ) T ( σ T α T α T κ ) 1 ( J Q T ) T Ṡ[(some) trial functions] d = 2 isotropic AdS-Einstein-Maxwell (w/ connected horizon, T > 0): [Grozdanov, Lucas, Sachdev, Schalm: ] [Grozdanov, Lucas, Schalm: ] σ 1, κ κ 2 T α2 σ 4π2 T 3 absence of disorder-driven metal-insulator transition these proofs rely on reduction of the dc transport problem to a strange hydrodynamics [Donos, Gauntlett: ]

75 Transport 23 Conductivity Bounds: More General Systems generalize to a more complicated model? S = d 4 x g (R ZF 2 1 ) 4 2 ( Φ)2 V (Φ)

76 Transport 23 Conductivity Bounds: More General Systems generalize to a more complicated model? S = d 4 x g (R ZF 2 1 ) 4 2 ( Φ)2 V (Φ) known conductivity bounds: σ 1 E [Z 1 ], κ 8π2 T E[ V ], (averages on horizon) which are generally difficult to bound sharply

77 Transport 23 Conductivity Bounds: More General Systems generalize to a more complicated model? S = d 4 x g (R ZF 2 1 ) 4 2 ( Φ)2 V (Φ) known conductivity bounds: σ 1 E [Z 1 ], κ 8π2 T E[ V ], (averages on horizon) which are generally difficult to bound sharply no electrical insulator without Z 0 in the IR: all simple holographic theories want to be conductors: constructions where Z depends on linear axions will likely lead to exotic T, m dependence [Baggioli, Pujolas: ] [Goutéraux, Kiritsis, Li: ]

78 Outlook 24 Major Open Questions T 0 limit with relevant disorder?

79 Outlook 24 Major Open Questions T 0 limit with relevant disorder? is the horizon connected? does it break into many pieces? does mean field approach give qualitatively correct physics?

80 Outlook 24 Major Open Questions T 0 limit with relevant disorder? is the horizon connected? does it break into many pieces? does mean field approach give qualitatively correct physics? 1/N corrections? localization as a quantum bulk effect?

81 Outlook 24 Major Open Questions T 0 limit with relevant disorder? is the horizon connected? does it break into many pieces? does mean field approach give qualitatively correct physics? 1/N corrections? localization as a quantum bulk effect? do (useful) transport bounds exist for more general inhomogeneous incoherent metals?