Hydrodynamic transport in holography and in clean graphene. Andrew Lucas

Transcription

1 Hydrodynamic transport in holography and in clean graphene Andrew Lucas Harvard Physics Special Seminar, King s College London March 8, 2016

2 Collaborators 2 Subir Sachdev Harvard Physics & Perimeter Institute Philip Kim Harvard Physics/SEAS Koenraad Schalm Leiden Lorentz Institute Jesse Crossno Harvard Physics/SEAS Sašo Grozdanov Leiden Lorentz Institute Kin Chung Fong Raytheon BBN

3 Transport 3 Ohm s law the simplest experiment: I V = IR R 1 σ V

4 Transport 3 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 )

5 Fermi Liquids 4...are Really a Quantum Gas FL theory describes electrons in iron, copper, etc. Fermi sea k B T µ

6 Fermi Liquids 4...are Really a Quantum Gas Fermi sea k B T FL theory describes electrons in iron, copper, etc. long-lived quasiparticles; (quantum) kinetic theory µ

7 Fermi Liquids 4...are Really a Quantum Gas Fermi sea k B T µ FL theory describes electrons in iron, copper, etc. long-lived quasiparticles; (quantum) kinetic theory interaction time: t ee µ (k B T ) s

8 Fermi Liquids 4...are Really a Quantum Gas Fermi sea k B T µ FL theory describes electrons in iron, copper, etc. long-lived quasiparticles; (quantum) kinetic theory interaction time: t ee µ (k B T ) s QP interacts with impurity or phonon every s (exceptions are rare, e.g. GaAs, doped graphene)

9 Fermi Liquids 5 Wiedemann-Franz Law experimentally measured thermal conductivity: Q J=0 κ T, κ = κ T α2 σ.

10 Fermi Liquids 5 Wiedemann-Franz Law experimentally measured thermal conductivity: Q J=0 κ T, κ = κ T α2 σ. Wiedemann-Franz law in a Fermi liquid: L κ σt π2 kb 2 3e W Ω K 2. [Kumar, Prasad, Pohl, Journal of Materials Science (1993)]

11 Quantum Criticality 6 What is Quantum Criticality? quantum critical tee ~ 10 kb T 13 s (T = 100 K) T 0 g gc

12 Quantum Criticality 6 What is Quantum Criticality? quantum critical tee I ~ 10 kb T 13 s (T = 100 K) T quantum criticality in experiments: strange metal superfluid-insulator (cold atoms) 0 graphene g gc

13 Quantum Criticality 7 Emergent Lorentz Invariance l & l ee l a l. a hydrodynamics finite T QFT lattice model many quantum critical models have emergent Lorentz invariance

14 Quantum Criticality 7 Emergent Lorentz Invariance l & l ee l a l. a hydrodynamics finite T QFT lattice model many quantum critical models have emergent Lorentz invariance we re going to focus on l ξ hydrodynamics

15 Hydrodynamics 8 What is Hydrodynamics? l ee T (0) µ(0) u µ (0) T ( ) µ( ) u µ ( ) T (2 ) µ(2 ) u µ (2 ) T (3 ) µ(3 ) u µ (3 ) T (4 ) µ(4 ) u µ (4 ) effective description of relaxing to thermal equilibrium on long length scales

16 Hydrodynamics 8 What is Hydrodynamics? l ee T (0) µ(0) u µ (0) T ( ) µ( ) u µ ( ) T (2 ) µ(2 ) u µ (2 ) T (3 ) µ(3 ) u µ (3 ) T (4 ) µ(4 ) u µ (4 ) effective description of relaxing to thermal equilibrium on long length scales perturbative expansion in l ee /ξ opposite of usual QFT

17 Hydrodynamics 8 What is Hydrodynamics? l ee T (0) µ(0) u µ (0) T ( ) µ( ) u µ ( ) T (2 ) µ(2 ) u µ (2 ) T (3 ) µ(3 ) u µ (3 ) T (4 ) µ(4 ) u µ (4 ) effective description of relaxing to thermal equilibrium on long length scales perturbative expansion in l ee /ξ opposite of usual QFT classical equations of motion: conservation laws ν T µν = F µν J µ, µ J µ = 0

18 Hydrodynamics 9 The Gradient Expansion for a Lorentz-Invariant Fluid expand T µν, J µ in perturbative parameter l ee µ : ( T µν = P η µν + (ɛ + P )u µ u ν 2P µρ P νσ η (ρ u σ) P µν ζ 2η ) ρu ρ +, d J µ = nu µ σ qp µρ ( ρµ µ T ρt uν F ρν ) +, P µν η µν + u µ u ν, Q i = J i µt ti

19 Hydrodynamics 9 The Gradient Expansion for a Lorentz-Invariant Fluid expand T µν, J µ in perturbative parameter l ee µ : ( T µν = P η µν + (ɛ + P )u µ u ν 2P µρ P νσ η (ρ u σ) P µν ζ 2η ) ρu ρ +, d J µ = nu µ σ qp µρ ( ρµ µ T ρt uν F ρν ) +, P µν η µν + u µ u ν, Q i = J i µt ti fluid has both electrons/holes; not separately conserved. distinct from usual plasma physics

20 Hydrodynamics 9 The Gradient Expansion for a Lorentz-Invariant Fluid expand T µν, J µ in perturbative parameter l ee µ : ( T µν = P η µν + (ɛ + P )u µ u ν 2P µρ P νσ η (ρ u σ) P µν ζ 2η ) ρu ρ +, d J µ = nu µ σ qp µρ ( ρµ µ T ρt uν F ρν ) +, P µν η µν + u µ u ν, Q i = J i µt ti fluid has both electrons/holes; not separately conserved. distinct from usual plasma physics quantum physics values of P, σ q, etc...

21 Hydrodynamics 9 The Gradient Expansion for a Lorentz-Invariant Fluid expand T µν, J µ in perturbative parameter l ee µ : ( T µν = P η µν + (ɛ + P )u µ u ν 2P µρ P νσ η (ρ u σ) P µν ζ 2η ) ρu ρ +, d J µ = nu µ σ qp µρ ( ρµ µ T ρt uν F ρν ) +, P µν η µν + u µ u ν, Q i = J i µt ti fluid has both electrons/holes; not separately conserved. distinct from usual plasma physics quantum physics values of P, σ q, etc... transport equations well-posed at first order (dissipation allowed) [Hartnoll, Kovtun, Müller, Sachdev, Physical Review B (2007)]

22 Hydrodynamic Transport 10 Conductivity of a Clean Metal J =0 J = Qv and E =0

23 Hydrodynamic Transport J =0 10 Conductivity of a Clean Metal J =0 J = nv and E =0 J = Qv and E =0

24 Hydrodynamic Transport J =0 10 Conductivity of a Clean Metal J =0 J = nv and E =0 J = Qv and E =0 σ sensitive to how translational symmetry broken

25 Hydrodynamic Transport 11 Mean Field: Drude Model mean field treatment of translational symmetry breaking: [Hartnoll, Kovtun, Müller, Sachdev, Physical Review B (2007)] µ T µi = T it τ + ne i.

26 Hydrodynamic Transport 11 Mean Field: Drude Model mean field treatment of translational symmetry breaking: [Hartnoll, Kovtun, Müller, Sachdev, Physical Review B (2007)] constitutive relations : µ T µi = T it τ + ne i. T it = (ɛ + P )v i J i = σ q E i + nv i :

27 Hydrodynamic Transport 11 Mean Field: Drude Model mean field treatment of translational symmetry breaking: [Hartnoll, Kovtun, Müller, Sachdev, Physical Review B (2007)] constitutive relations : average over space: µ T µi = T it σ(ω) = σ q + τ + ne i. T it = (ɛ + P )v i J i = σ q E i + nv i : n 2 τ (ɛ + P )(1 iωτ).

28 Hydrodynamic Transport 11 Mean Field: Drude Model mean field treatment of translational symmetry breaking: [Hartnoll, Kovtun, Müller, Sachdev, Physical Review B (2007)] constitutive relations : average over space: µ T µi = T it σ(ω) = σ q + τ + ne i. T it = (ɛ + P )v i J i = σ q E i + nv i : n 2 τ (ɛ + P )(1 iωτ). but missing sharp predictions for τ...

29 Hydrodynamic Transport 12 Wiedemann-Franz Revisited L 0 = L = κ σt L 0 (1 + (n/n 0 ) 2 ) 2 (ɛ + P )τ σ q T 2, n 2 0 = (ɛ + P )σ q τ

30 Hydrodynamic Transport 12 Wiedemann-Franz Revisited L 0 = L = κ σt L 0 (1 + (n/n 0 ) 2 ) 2 (ɛ + P )τ σ q T 2, n 2 0 = (ɛ + P )σ q τ L/LWF heat/charge currents nearly same hydro FL decoupled heat/charge n

31 Hydrodynamic Transport 13 Non-Perturbative Approach s(x) > 0 l ee non-perturbative hydrodynamic transport: disorder on scale ξ l ee n(x) > 0 n(x) < 0 n x

32 Hydrodynamic Transport 13 Non-Perturbative Approach non-perturbative hydrodynamic transport: disorder on scale ξ l ee l ee n(x) > 0 s(x) > 0 n(x) < 0 n x static fluid in an inhomogeneous chemical potential µ 0 (x)

33 Hydrodynamic Transport 13 Non-Perturbative Approach non-perturbative hydrodynamic transport: disorder on scale ξ l ee l ee n(x) > 0 s(x) > 0 n(x) < 0 n x static fluid in an inhomogeneous chemical potential µ 0 (x) transport from linearized hydrodynamic equations

34 Hydrodynamic Transport 13 Non-Perturbative Approach l ee n(x) > 0 s(x) > 0 n(x) < 0 n x non-perturbative hydrodynamic transport: disorder on scale ξ l ee static fluid in an inhomogeneous chemical potential µ 0 (x) transport from linearized hydrodynamic equations mean-field formalism with specific τ as µ 0 0: 1 τ ( µ 0) 2 ( ) n 2 [ µ 1 σ q (ɛ + P ) + 4ηµ 2 ξ 2 (ɛ + P ) 3 [Lucas, NJP (2015)]; [Lucas et al, PRB (2016)] similar ideas in FL: [Andreev, Kivelson, Spivak, PRL (2011)] ]

35 Hydrodynamic Transport 14 Exact Statements Onsager reciprocity

36 Hydrodynamic Transport 14 Exact Statements Onsager reciprocity variational (Thomson s) principle for DC transport: resistances power dissipated on arbitrary unit flows analogous to resistor network theory [Lucas, New Journal of Physics (2015)]

37 g MN T µ Holography 15 AdS/CMT Correspondence horizon boundary compute σ, etc. in mystery strongly coupled QFT via gravity in one higher dimension z =(d + 1)/(4 T ) z z =0 x

38 g MN T µ Holography 15 AdS/CMT Correspondence horizon boundary compute σ, etc. in mystery strongly coupled QFT via gravity in one higher dimension controlled calculations with no reference to quasiparticles z =(d + 1)/(4 T ) z z =0 x

39 Holography 15 AdS/CMT Correspondence horizon boundary g MN T µ compute σ, etc. in mystery strongly coupled QFT via gravity in one higher dimension controlled calculations with no reference to quasiparticles finite T and n = (asymptotically AdS?) charged black hole z =(d + 1)/(4 T ) z z =0 x

40 Holography 16 Holographic Lattices, Disorder and Massive Gravity how to break translational invariance in holography: explicitly (numerical GR): [Horowitz, Santos, Tong, Journal of High Energy Physics (2012)]

41 Holography 16 Holographic Lattices, Disorder and Massive Gravity how to break translational invariance in holography: explicitly (numerical GR): [Horowitz, Santos, Tong, Journal of High Energy Physics (2012)] mean field methods e.g. [Andrade, Withers, JHEP (2014)] L = R 2Λ 1 2 ( φ i) 2, φ i = mx i

42 Holography 17 The Mean Field Zoo mean-field models capture Drude-like DC physics: [e.g. Goutéraux, Journal of High Energy Physics (2014)] σ = ( s ) (d 2)/d Z } 4π{{} σ q + 4πn2 sm 2 }{{} Drude

43 Holography 17 The Mean Field Zoo mean-field models capture Drude-like DC physics: [e.g. Goutéraux, Journal of High Energy Physics (2014)] σ = ( s ) (d 2)/d Z } 4π{{} σ q + 4πn2 sm 2 }{{} Drude finite ω beyond Drude model [Davison, Goutéraux, Journal of High Energy Physics (2015)] [Blake, Journal of High Energy Physics (2015)]

44 Holography 17 The Mean Field Zoo mean-field models capture Drude-like DC physics: [e.g. Goutéraux, Journal of High Energy Physics (2014)] σ = ( s ) (d 2)/d Z } 4π{{} σ q + 4πn2 sm 2 }{{} Drude finite ω beyond Drude model [Davison, Goutéraux, Journal of High Energy Physics (2015)] [Blake, Journal of High Energy Physics (2015)] stranger: m well-defined; σ > 0 at all m

45 Holography 17 The Mean Field Zoo mean-field models capture Drude-like DC physics: [e.g. Goutéraux, Journal of High Energy Physics (2014)] σ = ( s ) (d 2)/d Z } 4π{{} σ q + 4πn2 sm 2 }{{} Drude finite ω beyond Drude model [Davison, Goutéraux, Journal of High Energy Physics (2015)] [Blake, Journal of High Energy Physics (2015)] stranger: m well-defined; σ > 0 at all m can we explicitly confirm these mean field ideas?

46 Holography 18 Weak Disorder lattice /disorder perturbatively = mean field [Blake, Tong, Vegh, Physical Review Letters (2014)]

47 Holography 18 Weak Disorder lattice /disorder perturbatively = mean field [Blake, Tong, Vegh, Physical Review Letters (2014)] holography perturbatively reproduces Drude formula, with τ given by memory matrix methods: [Lucas, Journal of High Energy Physics (2015)] 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!

48 Holography 19 The Horizon Fluid non-perturbative results? (focus on d = 2) membrane paradigm = exact transport from horizon PDEs [Donos, Gauntlett, Physical Review D (2015)]

49 Holography 19 The Horizon Fluid non-perturbative results? (focus on d = 2) membrane paradigm = exact transport from horizon PDEs [Donos, Gauntlett, Physical Review D (2015)] natural interpretation of strange relativistic fluid: [Lucas, New Journal of Physics (2015)]; [Grozdanov, Lucas, Sachdev, Schalm, Physical Review Letters (2015)] i J i = i Q i = 0 Q i = T 0 sv i, s = 4π J i = nv i + σ qγ ij (E j j µ), σ q = 1 2η j (i v j) = n( i µ E i ) + s( i T T 0 ζ i ), η = 1 with covariant derivatives taken w.r.t. horizon metric γ ij

50 Holography 19 The Horizon Fluid non-perturbative results? (focus on d = 2) membrane paradigm = exact transport from horizon PDEs [Donos, Gauntlett, Physical Review D (2015)] natural interpretation of strange relativistic fluid: [Lucas, New Journal of Physics (2015)]; [Grozdanov, Lucas, Sachdev, Schalm, Physical Review Letters (2015)] i J i = i Q i = 0 Q i = T 0 sv i, s = 4π J i = nv i + σ qγ ij (E j j µ), σ q = 1 2η j (i v j) = n( i µ E i ) + s( i T T 0 ζ i ), η = 1 with covariant derivatives taken w.r.t. horizon metric γ ij employ general hydro tricks to analyze strong disorder solutions!

51 Holography 20 Simple Holographic Models are Conductors simple holographic model, dual to uncharged theory: σ = 1 e 2

52 Holography 20 Simple Holographic Models are Conductors simple holographic model, dual to uncharged theory: σ = 1 e 2 variational result for finite charge density: σ 1 e 2.

53 Holography 20 Simple Holographic Models are Conductors simple holographic model, dual to uncharged theory: σ = 1 e 2 variational result for finite charge density: σ 1 e 2. compare to mean field approach: σ = 1 e 2 + 4πn2 sm 2 1 e 2.

54 Holography 20 Simple Holographic Models are Conductors simple holographic model, dual to uncharged theory: σ = 1 e 2 variational result for finite charge density: σ 1 e 2. compare to mean field approach: σ = 1 e 2 + 4πn2 sm 2 1 e 2. no disorder-driven metal-insulator transition [Grozdanov, Lucas, Sachdev, Schalm, PRL (2015)]

55 Holography 21 Incoherent Metal? similar to incoherent metal [Hartnoll, Nature Physics (2015)]

56 Holography 21 Incoherent Metal? similar to incoherent metal [Hartnoll, Nature Physics (2015)] microscopic σ q (charge diffusion) = σ bounded from below (use resistor network analogy) coherent incoherent Q 0 u

57 Holography 21 Incoherent Metal? similar to incoherent metal [Hartnoll, Nature Physics (2015)] microscopic σ q (charge diffusion) = σ bounded from below (use resistor network analogy) coherent incoherent Q 0 u relevant for experiments: σ A/T...

58 Dirac Fluid in Graphene 22 Crash Course in Graphene 1nm

59 Dirac Fluid in Graphene 23 Crash Course in Graphene a = ~v F k + V int = e r T Dirac fluid hole FL electron FL 0 n

60 Dirac Fluid in Graphene 23 Crash Course in Graphene a = ~v F k + V int = e r T Dirac fluid hole FL electron FL 0 n marginally irrelevant 1/r Coulomb interactions: α 0 α eff = 1 + (α 0 /4) log((10 5 K)/T ), α 0 1 c v F ɛ r

61 Dirac Fluid in Graphene 23 Crash Course in Graphene a = ~v F k + V int = e r T Dirac fluid hole FL electron FL 0 n marginally irrelevant 1/r Coulomb interactions: α eff = α (α 0 /4) log((10 5 K)/T ), α effective quantum critical description at room T! c 0.5. v F ɛ r e.g. [Sheehy, Schmalian, Physical Review Letters (2007)] [Müller, Fritz, Sachdev, Physical Review B (2008)]

62 Dirac Fluid in Graphene 24 Graphene: an Ideal Experimental Platform fabricating ultra pure monolayer graphene: [Dean et al, Nature Nanotechnology (2010)] hbn monolayer graphene hbn

63 Dirac Fluid in Graphene 24 Graphene: an Ideal Experimental Platform fabricating ultra pure monolayer graphene: [Dean et al, Nature Nanotechnology (2010)] b 100 nm hbn µ 30 K hbn monolayer graphene hbn c E d (mev) weak disorder: charge puddles [Xue et al, Nature Materials (2011)] Figure 4 Spatial maps of the density of states of graphene on hbn and SiO2. a, Topography of graphene on hbn. b, Tip voltage at the Dirac point as a function of position for graphene on hbn. c, Tip voltage at the Dirac SiO 2 µ 300 K

64 Dirac Fluid in Graphene 25 Wiedemann-Franz Law Violations in Experiment Tbath (K) phonon-limited disorder-limited n (10 9 cm -2 ) L / L0 [Crossno et al, Science (2016)]

65 n min (cm -2 ) Dirac Fluid in Graphene K 150 K 250 K Wiedemann-Franz Law Violations in Experiment T bath (K) Thermal Conductivity (nw/k) 8 C 75 K 6 4 D T dis κe T el-ph σtl0 2 0 V 0 E K 10 mm K V Vg (V) Tbath (K) L/L 0 S3 Disorder Thermally Limited Limited Temperature (K) C H +V e h h e C -V L / L Temperature (K) H (ev/µm 2 ) T (K) n (10 10 cm 2 ) compare to perturbative prediction L = L DF (1 + (n/n 0 ) 2 ) 2 [Crossno et al, Science (2016)]

66 Dirac Fluid in Graphene 27 3 Experimental Puzzles experimental puzzles: σ(n) is nearly constant

67 Dirac Fluid in Graphene 27 3 Experimental Puzzles experimental puzzles: σ(n) is nearly constant what is controlling the T -dependence of κ near n = 0?

68 Dirac Fluid in Graphene 27 3 Experimental Puzzles experimental puzzles: σ(n) is nearly constant what is controlling the T -dependence of κ near n = 0? non-monotonic growth in κ(t )?

69 Dirac Fluid in Graphene 27 3 Experimental Puzzles experimental puzzles: σ(n) is nearly constant what is controlling the T -dependence of κ near n = 0? non-monotonic growth in κ(t )? proposed resolution: density/viscosity dependence neglected (can be addressed in perturbation theory!)

70 Dirac Fluid in Graphene 27 3 Experimental Puzzles experimental puzzles: σ(n) is nearly constant what is controlling the T -dependence of κ near n = 0? non-monotonic growth in κ(t )? proposed resolution: density/viscosity dependence neglected (can be addressed in perturbation theory!) perturbative: universal κ T 3 from charge puddles

71 Dirac Fluid in Graphene 27 3 Experimental Puzzles experimental puzzles: σ(n) is nearly constant what is controlling the T -dependence of κ near n = 0? non-monotonic growth in κ(t )? proposed resolution: density/viscosity dependence neglected (can be addressed in perturbation theory!) perturbative: universal κ T 3 from charge puddles momentum relaxation from disorder vs. acoustic phonons? [Lucas et al, Physical Review B (2016)]

72 Dirac Fluid in Graphene 28 A Hydrodynamic Model for Graphene s = C 0 T0 2 + C 2 2 µ2 0 C 4µ 4 0 4T0 2 +, l ee n(x) > 0 s(x) > 0 n(x) < 0 n x n = C 2 µ 0 T 0 + C 4 T 0 µ 3 0 +, η = η 0 T 2 0, ζ = 0, σ q = σ 0 numerically solve linearized hydrodynamic equations: ( ( 0 = i nv i + σ q E i i µ µ 0 ζ i + µ )) 0 i T, T 0 0 = i ( T 0 sv i µ 0 σ q ( E i i µ µ 0 ζ i + µ 0 T 0 i T )), 0 = n( i µ E i ) + s ( i T T 0 ζ i ) j (η ( i v j + j v i δ ij k v k )). [Lucas et al, Physical Review B (2016)]

73 Dirac Fluid in Graphene 29 Comparing Theory to Experiment comparison to experiment at T = 75 K fit parameters: (k 1 ) hole FL Dirac fluid elec. FL apple (nw/k) hole FL Dirac fluid elec. FL n (µm 2 ) n (µm 2 ) Figure 1: testing

74 Dirac Fluid in Graphene 29 Comparing Theory to Experiment comparison to experiment at T = 75 K fit parameters: (k 1 ) hole FL Dirac fluid elec. FL apple (nw/k) hole FL Dirac fluid elec. FL n (µm 2 ) n (µm 2 ) η s 10 k B Figure 1: testing (though possibly substantial overestimate) [Lucas et al, Physical Review B (2016)]

75 Dirac Fluid in Graphene 30 Comparing Theory to Experiment comparison to experiment at n = 0 (no new fit parameters): (k 1 ) puddle FLs Dirac fluid phonons apple (nw/k) puddle FLs Dirac fluid phonons T (K) T (K) Figure 1: testing satisfactory quantitative agreement in Dirac fluid! [Lucas et al, Physical Review B (2016)]

76 Summary 31 new transport physics from quantum critical hydrodynamics

77 Summary 31 new transport physics from quantum critical hydrodynamics simple holographic models are incoherent metals

78 Summary 31 new transport physics from quantum critical hydrodynamics simple holographic models are incoherent metals first quantitative test of relativistic transport models in experiment: Dirac fluid in graphene