Holographic transport with random-field disorder. Andrew Lucas

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1 Holographic transport with random-field disorder Andrew Lucas Harvard Physics Quantum Field Theory, String Theory and Condensed Matter Physics: Orthodox Academy of Crete September 1, 2014

2 Collaborators 2 Subir Sachdev Harvard Physics/Perimeter Institute Koenraad Schalm Leiden Lorentz Institute/Harvard Physics Lucas, Sachdev, Schalm, Physical Review D (2014)

3 Reviews 3 Random Fields in QFT random-field disorder in a low-energy EFT (d spatial dimensions): H = H 0 + d d x O(x)g(x), }{{} linear coupling of disorder E[g(x)] = 0, disorder averages: E[ ]. E[g(x)g(y)] = ε 2 δ(x y).

4 Reviews 3 Random Fields in QFT random-field disorder in a low-energy EFT (d spatial dimensions): H = H 0 + d d x O(x)g(x), }{{} linear coupling of disorder E[g(x)] = 0, disorder averages: E[ ]. E[g(x)g(y)] = ε 2 δ(x y). suppose H 0 invariant under Z 2 symmetry, O O. Z 2 is locally, not globally, broken

5 Reviews 3 Random Fields in QFT random-field disorder in a low-energy EFT (d spatial dimensions): H = H 0 + d d x O(x)g(x), }{{} linear coupling of disorder E[g(x)] = 0, disorder averages: E[ ]. E[g(x)g(y)] = ε 2 δ(x y). suppose H 0 invariant under Z 2 symmetry, O O. Z 2 is locally, not globally, broken defects act as random-fields in the solid-state lab: missing atom wrong atom

6 Reviews 4 The Harris Criterion at Any z, θ most general scale invariant isotropic theory: characterized by z, θ.

7 Reviews 4 The Harris Criterion at Any z, θ most general scale invariant isotropic theory: characterized by z, θ. dynamical exponent z: t x z.

8 Reviews 4 The Harris Criterion at Any z, θ most general scale invariant isotropic theory: characterized by z, θ. dynamical exponent z: t x z. hyperscaling violation exponent θ: degrees of freedom live in d eff = d θ spatial dimensions.

9 Reviews 4 The Harris Criterion at Any z, θ most general scale invariant isotropic theory: characterized by z, θ. dynamical exponent z: t x z. hyperscaling violation exponent θ: degrees of freedom live in d eff = d θ spatial dimensions. Fermi surface: θ = d 1.

10 Reviews 4 The Harris Criterion at Any z, θ most general scale invariant isotropic theory: characterized by z, θ. dynamical exponent z: t x z. hyperscaling violation exponent θ: degrees of freedom live in d eff = d θ spatial dimensions. Fermi surface: θ = d 1. example: entropy density s T (d θ)/z.

11 Reviews 4 The Harris Criterion at Any z, θ most general scale invariant isotropic theory: characterized by z, θ. dynamical exponent z: t x z. hyperscaling violation exponent θ: degrees of freedom live in d eff = d θ spatial dimensions. Fermi surface: θ = d 1. example: entropy density s T (d θ)/z. Harris criterion: O(0)O(x) x 2 ; disorder relevant if < d θ 2 + z

12 Ising-Nematic Quantum Critical Point 5 Pomeranchuk Instability Pomeranchuk instability: Fermi surface distortion (d = 2) h i =0 h i6=0 X k (cos(ak x ) cos(ak y )) c k c k Z 2

13 Ising-Nematic Quantum Critical Point 5 Pomeranchuk Instability Pomeranchuk instability: Fermi surface distortion (d = 2) h i =0 h i6=0 X k (cos(ak x ) cos(ak y )) c k c k Z 2 experiments: strange metal (critical?) and Ising-nematic order...

Ising-Nematic Quantum Critical Point 5 Pomeranchuk Instability Pomeranchuk instability: Fermi surface distortion (d = 2) h i =0 h i6=0 X k (cos(ak x ) cos(ak y )) c k c k Z 2 experiments: strange

14 Ising-Nematic Quantum Critical Point 5 Pomeranchuk Instability Pomeranchuk instability: Fermi surface distortion (d = 2) h i =0 h i6=0 X k (cos(ak x ) cos(ak y )) c k c k Z 2 experiments: strange metal (critical?) and Ising-nematic order... Ising-nematic quantum critical point: L = 1 2 ( φ)2 rc 2 φ2 u 24 φ4 + c (i t ɛ(i ))c λc c( 2 x 2 y)φ Metlitski, Sachdev, Physical Review B (2010)

15 Ising-Nematic Quantum Critical Point 6 Random Field Disorder what is electrical resistivity ρ dc associated to (charged) electron c?

16 Ising-Nematic Quantum Critical Point 6 Random Field Disorder what is electrical resistivity ρ dc associated to (charged) electron c? Boltzmann transport (near-fs c scatter off φ) ρ dc T 4/3.

17 Ising-Nematic Quantum Critical Point 6 Random Field Disorder what is electrical resistivity ρ dc associated to (charged) electron c? Boltzmann transport (near-fs c scatter off φ) ρ dc T 4/3. two types of random-field disorder to add: L dis = V (x)c c h(x)φ

18 Ising-Nematic Quantum Critical Point 6 Random Field Disorder what is electrical resistivity ρ dc associated to (charged) electron c? Boltzmann transport (near-fs c scatter off φ) ρ dc T 4/3. two types of random-field disorder to add: L dis = V (x)c c h(x)φ random-fields dominate transport at low T : ρ dc E[V 2 ] + E[h 2 ] T log(t /T ) calculation breaks down as T 0! Hartnoll, Mahajan, Punk, Sachdev, Physical Review B (2014)

19 Momentum Loss 7 (Almost) Hydrodynamics and Memory Matrices how to understand? strongly-coupled theory relaxes quickly to a hydrodynamic regime: µ T µt = 0, µ T µi = 1 τ T ti.

20 Momentum Loss 7 (Almost) Hydrodynamics and Memory Matrices how to understand? strongly-coupled theory relaxes quickly to a hydrodynamic regime: µ T µt = 0, µ T µi = 1 τ T ti. τ dominated by fastest momentum-dissipating process

21 Momentum Loss 7 (Almost) Hydrodynamics and Memory Matrices how to understand? strongly-coupled theory relaxes quickly to a hydrodynamic regime: µ T µt = 0, µ T µi = 1 τ T ti. τ dominated by fastest momentum-dissipating process τ also controls ρ dc [P x tot, J x tot] 0, µ J µ = 0 = same process controlling momentum loss controls ρ dc

22 Momentum Loss 7 (Almost) Hydrodynamics and Memory Matrices how to understand? strongly-coupled theory relaxes quickly to a hydrodynamic regime: µ T µt = 0, µ T µi = 1 τ T ti. τ dominated by fastest momentum-dissipating process τ also controls ρ dc [P x tot, J x tot] 0, µ J µ = 0 = same process controlling momentum loss controls ρ dc memory matrix formalism (L dis go, g ε): ρ dc ε 2 d 2 k k 2 Im GR OO (ω, k) ω

23 Momentum Loss 8 A Massive Gravity Analogy holographic systems are fluid-like for low-ω transport computations

24 Momentum Loss 8 A Massive Gravity Analogy holographic systems are fluid-like for low-ω transport computations break translation symmetry by breaking some of the diffeomorphism symmetry in bulk (graviton mass)

25 Momentum Loss 8 A Massive Gravity Analogy holographic systems are fluid-like for low-ω transport computations break translation symmetry by breaking some of the diffeomorphism symmetry in bulk (graviton mass) in holography with graviton mass m, one finds that τ m 2.

26 Momentum Loss 8 A Massive Gravity Analogy holographic systems are fluid-like for low-ω transport computations break translation symmetry by breaking some of the diffeomorphism symmetry in bulk (graviton mass) in holography with graviton mass m, one finds that τ m 2. computation of ρ dc : ρ dc τ m 2 m 2 perturbatively computable for holographic lattice (holographic Higgs mechanism) Vegh, arxiv: Davison, Physical Review D (2013) Blake, Tong, Vegh, Physical Review Letters (2014)

27 Disordered Holography 9 Beyond Replicas we start with finite charge density EMD geometry (g MN, A M, Φ) for given z, θ [ ds 2 = L2 dr 2 r 2 r 2θ/(d θ) f fdt 2 + dx2 r2d(z 1)/(d θ) ] ( ) d+dz/(d θ) r, f = 1 r h

28 Disordered Holography 9 Beyond Replicas we start with finite charge density EMD geometry (g MN, A M, Φ) for given z, θ [ ds 2 = L2 dr 2 r 2 r 2θ/(d θ) f fdt 2 + dx2 r2d(z 1)/(d θ) ] ( ) d+dz/(d θ) r, f = 1 r h sanity check: r h T (1 θ/d)/z ; s r d h T (d θ)/z.

29 Disordered Holography 9 Beyond Replicas we start with finite charge density EMD geometry (g MN, A M, Φ) for given z, θ [ ds 2 = L2 dr 2 r 2 r 2θ/(d θ) f fdt 2 + dx2 r2d(z 1)/(d θ) ] ( ) d+dz/(d θ) r, f = 1 r h sanity check: r h T (1 θ/d)/z ; s r d h T (d θ)/z. AdS/CFT dictionary: random-field disorder random (scalar ψ) field in bulk normalizable mode turned on: L dis = g(k)o(k) ψ(k, r 0) g(k)r ν + }{{} + relevant operator

30 Disordered Holography 9 Beyond Replicas we start with finite charge density EMD geometry (g MN, A M, Φ) for given z, θ [ ds 2 = L2 dr 2 r 2 r 2θ/(d θ) f fdt 2 + dx2 r2d(z 1)/(d θ) ] ( ) d+dz/(d θ) r, f = 1 r h sanity check: r h T (1 θ/d)/z ; s r d h T (d θ)/z. AdS/CFT dictionary: random-field disorder random (scalar ψ) field in bulk normalizable mode turned on: L dis = g(k)o(k) ψ(k, r 0) g(k)r ν + }{{} + relevant operator g(k) ε small enough = background approx ψ-independent. ( memory matrix regime)

31 Disordered Holography 10 Aside: Scalar Fields with θ 0 for this computation to work, must choose S[ψ] = 1 ( 1 d d+2 x 2 2 M ψ M ψ + 1 ) 2 B(Φ)ψ2 with B(Φ) e βφ

32 Disordered Holography 10 Aside: Scalar Fields with θ 0 for this computation to work, must choose S[ψ] = 1 ( 1 d d+2 x 2 2 M ψ M ψ + 1 ) 2 B(Φ)ψ2 with B(Φ) e βφ analogous story with usual AdS/CFT (mass dimension): B(Φ) B 0 e βφ, β fixed by z, θ; ( d(2 θ) 4B 0 = d d θ dz ) 2 ( d + d θ dz ) 2 d θ with dimension of (relevant) operator dual to ψ.

33 Disordered Holography 10 Aside: Scalar Fields with θ 0 for this computation to work, must choose S[ψ] = 1 ( 1 d d+2 x 2 2 M ψ M ψ + 1 ) 2 B(Φ)ψ2 with B(Φ) e βφ analogous story with usual AdS/CFT (mass dimension): B(Φ) B 0 e βφ, β fixed by z, θ; ( d(2 θ) 4B 0 = d d θ dz ) 2 ( d + d θ with dimension of (relevant) operator dual to ψ. dz ) 2 d θ if B(Φ) = m 2, AdS radius L enters the boundary theory as a length scale in ψ correlation functions!

34 Disordered Holography 11 Computing the Resistivity AdS/CFT: electric field: δa x e iωt (ω 0). electric current: d 1 r δa x.

35 Disordered Holography 11 Computing the Resistivity AdS/CFT: electric field: δa x e iωt (ω 0). electric current: d 1 r δa x. linearized fluctuations couple to all spin 1 modes: δa x, δg tx, x δψ, 2 x δψ, 2 2 x δψ,

36 Disordered Holography 11 Computing the Resistivity AdS/CFT: electric field: δa x e iωt (ω 0). electric current: d 1 r δa x. linearized fluctuations couple to all spin 1 modes: δa x, δg tx, x δψ, 2 x δψ, 2 2 x δψ, for computation of ρ (ω = 0 only), only 3 modes couple!

37 Disordered Holography 11 Computing the Resistivity AdS/CFT: electric field: δa x e iωt (ω 0). electric current: d 1 r δa x. linearized fluctuations couple to all spin 1 modes: δa x, δg tx, x δψ, 2 x δψ, 2 2 x δψ, for computation of ρ (ω = 0 only), only 3 modes couple! graviton mass a sum of each mode contribution: ρ dc sm 2, m 2 d d k k 2 ψ(k, r h ) 2

38 Disordered Holography 11 Computing the Resistivity AdS/CFT: electric field: δa x e iωt (ω 0). electric current: d 1 r δa x. linearized fluctuations couple to all spin 1 modes: δa x, δg tx, x δψ, 2 x δψ, 2 2 x δψ, for computation of ρ (ω = 0 only), only 3 modes couple! graviton mass a sum of each mode contribution: ρ dc sm 2, m 2 d d k k 2 ψ(k, r h ) 2 given ρ dc, coefficients κ dc, α dc straightforward to obtain Amoretti, Braggio, Maggiore, Magnoli, Musso, arxiv:

39 Disordered Holography 12 A Hairy Black Hole mode by mode solution: modes are modified Bessel functions: ψ r # (r k 1 θ/d ), ψ exp[ kr d/(d θ) ] (r k 1 θ/d )

40 Disordered Holography 12 A Hairy Black Hole mode by mode solution: modes are modified Bessel functions: ψ r # (r k 1 θ/d ), ψ exp[ kr d/(d θ) ] (r k 1 θ/d ) horizon imprinted with disorder: l hair T 1/z : (tracks modes that effectively scatter momentum!) k 1 <T 1/z k 2 >T 1/z random Fourier cartoon l hair k (1 2 /d) r T (1 /d)/z

Disordered Holography 12 A Hairy Black Hole mode by mode solution: modes are modified Bessel functions: ψ r # (r k 1 θ/d ), ψ exp[ kr d/(d θ) ] (r k 1 θ/d ) horizon imprinted with disorder: l

41 Disordered Holography 12 A Hairy Black Hole mode by mode solution: modes are modified Bessel functions: ψ r # (r k 1 θ/d ), ψ exp[ kr d/(d θ) ] (r k 1 θ/d ) horizon imprinted with disorder: l hair T 1/z : (tracks modes that effectively scatter momentum!) k 1 <T 1/z k 2 >T 1/z random Fourier cartoon l hair k (1 2 /d) r T (1 /d)/z similar logic valid even when horizon non-perturbative?

42 Disordered Holography 13 The Answer only momenta k < T 1/z contribute to m 2 : ρ dc ε 2 T 2(1+ z)/z ε 2 T (d z+η)/z

43 Disordered Holography 13 The Answer only momenta k < T 1/z contribute to m 2 : ρ dc ε 2 T 2(1+ z)/z ε 2 T (d z+η)/z this scaling also found (easier) with memory matrix

44 Disordered Holography 13 The Answer only momenta k < T 1/z contribute to m 2 : ρ dc ε 2 T 2(1+ z)/z ε 2 T (d z+η)/z this scaling also found (easier) with memory matrix holographic advantage: breakdown of perturbation theory/memory matrix when ψ backreacts on geometry: T T c, T c ε (z +(d θ)/2)/z

45 Disordered Holography 13 The Answer only momenta k < T 1/z contribute to m 2 : ρ dc ε 2 T 2(1+ z)/z ε 2 T (d z+η)/z this scaling also found (easier) with memory matrix holographic advantage: breakdown of perturbation theory/memory matrix when ψ backreacts on geometry: T T c, T c ε (z +(d θ)/2)/z at this breakdown we find ρ dc T (d+2 θ)/z T 2/z s

46 Disordered Holography 13 The Answer only momenta k < T 1/z contribute to m 2 : ρ dc ε 2 T 2(1+ z)/z ε 2 T (d z+η)/z this scaling also found (easier) with memory matrix holographic advantage: breakdown of perturbation theory/memory matrix when ψ backreacts on geometry: T T c, T c ε (z +(d θ)/2)/z at this breakdown we find ρ dc T (d+2 θ)/z T 2/z s ρ dc s for random-field disorder coupled to T tt.

47 Disordered Holography 13 The Answer only momenta k < T 1/z contribute to m 2 : ρ dc ε 2 T 2(1+ z)/z ε 2 T (d z+η)/z this scaling also found (easier) with memory matrix holographic advantage: breakdown of perturbation theory/memory matrix when ψ backreacts on geometry: T T c, T c ε (z +(d θ)/2)/z at this breakdown we find ρ dc T (d+2 θ)/z T 2/z s ρ dc s for random-field disorder coupled to T tt. generically scalar RF disorder has more singular contribution Davison, Schalm, Zaanen, Physical Review B (2014)

48 Disordered Holography 14 Nonperturbative Numerics numerics have recently been performed on (T = 0) disordered horizons with marginal scalars: plot of ψ(x, r h ): Hartnoll, Santos, Physical Review Letters (2014)

49 Summary 15 random-field disorder (almost certainly) present in experiments on strange metal phases

50 Summary 15 random-field disorder (almost certainly) present in experiments on strange metal phases most efficient mechanism (known) for losing momentum

51 Summary 15 random-field disorder (almost certainly) present in experiments on strange metal phases most efficient mechanism (known) for losing momentum elegant holographic implementation: dirty black holes

52 Summary 15 random-field disorder (almost certainly) present in experiments on strange metal phases most efficient mechanism (known) for losing momentum elegant holographic implementation: dirty black holes scale-invariant holography with θ 0: dilaton couplings to new fields for scale-invariant QFT!

53 Summary 15 random-field disorder (almost certainly) present in experiments on strange metal phases most efficient mechanism (known) for losing momentum elegant holographic implementation: dirty black holes scale-invariant holography with θ 0: dilaton couplings to new fields for scale-invariant QFT! in progress: easy interpolating AdS HV geometry at all T for (almost all) z, θ; holographic check of dc HV AdS T

Disordered spacetimes in AdS/CMT. Andrew Lucas

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