Building a theory of transport for strange metals. Andrew Lucas

Transcription

1 Building a theory of transport for strange metals Andrew Lucas Stanford Physics 290K Seminar, UC Berkeley February 13, 2017

2 Collaborators 2 Julia Steinberg Harvard Physics Subir Sachdev Harvard Physics & Perimeter Institute Philip Kim Harvard Physics/SEAS Yingfei Gu Stanford Physics Koenraad Schalm Leiden: Lorentz Institute Jesse Crossno Harvard Physics/SEAS Xiao-Liang Qi Stanford Physics 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 Ordinary Metals: a Review 4 The Fermi Liquid describes electrons in ordinary metals Fermi sea k B T µ

6 Ordinary Metals: a Review 4 The Fermi Liquid Fermi sea k B T µ describes electrons in ordinary metals interaction time constrained by near-fermi surface phase space: t ee µ (k B T ) 2

7 Ordinary Metals: a Review 4 The Fermi Liquid Fermi sea k B T µ describes electrons in ordinary metals interaction time constrained by near-fermi surface phase space: t ee µ (k B T ) 2 long-lived quasiparticles; (quantum) kinetic theory

8 Ordinary Metals: a Review 5 Residual Conductivity cartoon of semiclassical transport theory: σ = ne2 τ m, 1 τ 1 Im(Σ)

9 Ordinary Metals: a Review 5 Residual Conductivity cartoon of semiclassical transport theory: σ = ne2 τ m, 1 τ 1 Im(Σ) neglect e-e interactions, invoke Mattheisen s rule : Im(Σ) Im(Σ e ph ) }{{} + Im(Σ e imp ) }{{} electron-phonon, umklapp electron-impurity scattering k Σ + k k k V imp

10 Ordinary Metals: a Review 5 Residual Conductivity cartoon of semiclassical transport theory: σ = ne2 τ m, 1 τ 1 Im(Σ) neglect e-e interactions, invoke Mattheisen s rule : Im(Σ) Im(Σ e ph ) }{{} + Im(Σ e imp ) }{{} electron-phonon, umklapp electron-impurity scattering k Σ + k k k V imp single-particle quantum mechanics, Fermi s golden rule: d d k Im(Σ e imp ) k,k (2π) d k V imp k 2

11 Ordinary Metals: a Review 6 Wiedemann-Franz Law thermal conductivity κ; electrical conductivity σ: Q J=0 κ T, J T =0 = σe.

12 Ordinary Metals: a Review 6 Wiedemann-Franz Law thermal conductivity κ; electrical conductivity σ: Q J=0 κ T, J T =0 = σe. 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)]

13 Ordinary Metals: a Review Failures of the Old Theory this theory is wrong when interactions important 7

14 Ordinary Metals: a Review Failures of the Old Theory this theory is wrong when interactions important I can occur in ultra-pure FLs (e.g. GaAs, graphene), 7

15 Ordinary Metals: a Review 7 Failures of the Old Theory this theory is wrong when interactions important I I can occur in ultra-pure FLs (e.g. GaAs, graphene), or if FL is destabilized, well-defined quasiparticles absent: σ tee 1 T

16 Ordinary Metals: a Review 7 Failures of the Old Theory this theory is wrong when interactions important I I can occur in ultra-pure FLs (e.g. GaAs, graphene), or if FL is destabilized, well-defined quasiparticles absent: σ tee I 1 T Quantum Monte Carlo? sign problem, imaginary time

17 Weakly Disordered Metals 8 A Universal Drude Formula Ward identity for momentum conservation: t g i = iωg i = g i ρe }{{} i }{{} τ Lorentz force momentum relaxation

18 Weakly Disordered Metals 8 A Universal Drude Formula Ward identity for momentum conservation: t g i = iωg i = hydrodynamic constitutive relations: g i }{{} momentum density g i ρe }{{} i }{{} τ Lorentz force momentum relaxation = Mv i, }{{} J i = ρv i, charge current

19 Weakly Disordered Metals 8 A Universal Drude Formula Ward identity for momentum conservation: t g i = iωg i = hydrodynamic constitutive relations: g i }{{} momentum density g i ρe }{{} i }{{} τ Lorentz force momentum relaxation = Mv i, }{{} J i = ρv i, charge current Drude peak: σ(ω) = J i = ρ2 1. E i M 1 τ iω

20 Weakly Disordered Metals 8 A Universal Drude Formula Ward identity for momentum conservation: t g i = iωg i = hydrodynamic constitutive relations: g i }{{} momentum density g i ρe }{{} i }{{} τ Lorentz force momentum relaxation = Mv i, }{{} J i = ρv i, charge current Drude peak: σ(ω) = J i = ρ2 1. E i M 1 τ iω transport sensitive to momentum relaxation

21 Weakly Disordered Metals 9 Transport in Weakly Disordered Metals Drude formula can be derived rigorously at weak disorder

22 Weakly Disordered Metals 9 Transport in Weakly Disordered Metals Drude formula can be derived rigorously at weak disorder consider the Hamiltonian H = H 0 d d x h(x)o(x). with H 0 admitting continuous translation symmetry

23 Weakly Disordered Metals 9 Transport in Weakly Disordered Metals Drude formula can be derived rigorously at weak disorder consider the Hamiltonian H = H 0 d d x h(x)o(x). with H 0 admitting continuous translation symmetry to leading order in small h: M d d τ k k 2 Im(G R (2π) d d h(k) 2 OO (k, ω)) lim. ω 0 ω

24 Weakly Disordered Metals 9 Transport in Weakly Disordered Metals Drude formula can be derived rigorously at weak disorder consider the Hamiltonian H = H 0 d d x h(x)o(x). with H 0 admitting continuous translation symmetry to leading order in small h: M d d τ k k 2 Im(G R (2π) d d h(k) 2 OO (k, ω)) lim. ω 0 ω derived from holography: [Lucas; JHEP, ] memory functions: [Hartnoll, Hofman; PRL, ]; [Lucas, Sachdev; PRB, ] hydrodynamics : [Lucas; NJP, ]

25 Weakly Disordered Metals 10 Strong Disorder? magnetoresistance: Drude: σ xy σ xx T 1 experiment: σ xy σ xx T 2

26 Weakly Disordered Metals 10 Strong Disorder? magnetoresistance: Drude: σ xy σ xx T 1 experiment: σ xy σ xx T 2 hydrodynamics? solvable at strong disorder

27 Weakly Disordered Metals 10 Strong Disorder? magnetoresistance: Drude: σ xy σ xx T 1 experiment: σ xy σ xx T 2 hydrodynamics? solvable at strong disorder unclear if related to σ T 1

28 Weakly Disordered Metals 10 Strong Disorder? magnetoresistance: Drude: σ xy σ xx T 1 experiment: σ xy σ xx T 2 hydrodynamics? solvable at strong disorder unclear if related to σ T 1 relevant for other experiments: GaAs, graphene, etc.

29 Weakly Disordered Metals 10 Strong Disorder? magnetoresistance: Drude: σ xy σ xx T 1 experiment: σ xy σ xx T 2 hydrodynamics? transport bounds? solvable at strong disorder unclear if related to σ T 1 relevant for other experiments: GaAs, graphene, etc. universality natural?

30 Weakly Disordered Metals 10 Strong Disorder? magnetoresistance: Drude: σ xy σ xx T 1 experiment: σ xy σ xx T 2 hydrodynamics? transport bounds? solvable at strong disorder unclear if related to σ T 1 relevant for other experiments: GaAs, graphene, etc. universality natural? speculative

31 Weakly Disordered Metals 10 Strong Disorder? magnetoresistance: Drude: σ xy σ xx T 1 experiment: σ xy σ xx T 2 hydrodynamics? transport bounds? solvable at strong disorder unclear if related to σ T 1 relevant for other experiments: GaAs, graphene, etc. universality natural? speculative recent interest: connections with chaos, AdS/CFT

32 Hydrodynamic Electron Flow: Fermi Liquids 11 Metals are Disordered t ee t imp e?) ordinary metal (iron etc.) t ee s t imp =mess

33 Hydrodynamic Electron Flow: Fermi Liquids 11 Metals are Disordered t ee t imp t ee t imp e?) ordinary metal (iron etc.) ultraclean metal (GaAs, graphene?) t ee t imp r J =0 t ee

34 Hydrodynamic Electron Flow: Fermi Liquids 12 Hydrodynamic Equations classical effective theory of thermal system

35 Hydrodynamic Electron Flow: Fermi Liquids 12 Hydrodynamic Equations classical effective theory of thermal system long wavelength gapless modes = conserved currents: t ρ + J = 0.

36 Hydrodynamic Electron Flow: Fermi Liquids 12 Hydrodynamic Equations classical effective theory of thermal system long wavelength gapless modes = conserved currents: t ρ + J = 0. perturbative expansion in derivatives: J = D(ρ) ρ and all microscopic (quantum) details contained in D

37 Hydrodynamic Electron Flow: Fermi Liquids 12 Hydrodynamic Equations classical effective theory of thermal system long wavelength gapless modes = conserved currents: t ρ + J = 0. perturbative expansion in derivatives: J = D(ρ) ρ and all microscopic (quantum) details contained in D diffusion (for lone conserved charge) t ρ = (D ρ)

38 Hydrodynamic Electron Flow: Fermi Liquids 12 Hydrodynamic Equations classical effective theory of thermal system long wavelength gapless modes = conserved currents: t ρ + J = 0. perturbative expansion in derivatives: J = D(ρ) ρ and all microscopic (quantum) details contained in D diffusion (for lone conserved charge) t ρ = (D ρ) local second law of thermodynamics: D 0

39 Hydrodynamic Electron Flow: Fermi Liquids 13 Linearized Hydrodynamics of a Disordered Fermi Liquid neglect thermal effects; on length scales l l ee :

40 Hydrodynamic Electron Flow: Fermi Liquids 13 Linearized Hydrodynamics of a Disordered Fermi Liquid neglect thermal effects; on length scales l l ee : conservation of charge/ conservation of momentum: (nv) n v = 0. P η 2 v Γ v c µ l ee 2 v v l imp

41 Hydrodynamic Electron Flow: Fermi Liquids 13 Linearized Hydrodynamics of a Disordered Fermi Liquid neglect thermal effects; on length scales l l ee : conservation of charge/ conservation of momentum: (nv) n v = 0. P η 2 v Γ v c µ l ee 2 v v l imp length scales: Ohmic diffusion: creep flow : l l ee l imp l l ee l imp

42 Hydrodynamic Electron Flow: Fermi Liquids 14 Flow Through Thin Opening [Levitov, Falkovich; Nature Physics, ] [Torre, Tomadin, Geim, Polini; PRB, ]

43 Hydrodynamic Electron Flow: Fermi Liquids 15 Experimental Evidence experimental geometry: [Bandurin et al; Science, ]

44 Hydrodynamic Electron Flow: Fermi Liquids 15 Experimental Evidence experimental geometry: [Bandurin et al; Science, ]

45 Hydrodynamic Electron Flow: Fermi Liquids 15 Experimental Evidence experimental geometry: [Bandurin et al; Science, ] but no signal when n = 0! use relativistic hydro here...

46 Hydrodynamic Electron Flow: Fermi Liquids 16 Stokes Paradox L R I [Lucas; ]

47 Hydrodynamic Electron Flow: Fermi Liquids 16 Stokes Paradox L (differential) resistance: Ohmic: R l ee l imp : R I R R2 l imp hydrodynamic: l ee R l ee l imp : R l ee log(r 1 l ee l imp ) ballistic: l ee R: [Lucas; ] R R 2

48 Hydrodynamic Electron Flow: Fermi Liquids 16 Stokes Paradox L (differential) resistance: Ohmic: R l ee l imp : R I R R2 l imp R (a.u.) ξ/r = 1 ξ/r = 3 ξ/r = 10 ξ/r = 30 hydrodynamic: l ee R l ee l imp : R l ee log(r 1 l ee l imp ) T/T ballistic: l ee R: R R 2 [Lucas; ]

49 Relativistic Hydrodynamics in Graphene 17 The Dirac Fluid µ Fermi liquid

50 Relativistic Hydrodynamics in Graphene 17 The Dirac Fluid µ µ Fermi liquid Dirac fluid

51 Relativistic Hydrodynamics in Graphene 18 Thermal and Electrical Conductivity at Charge Neutrality rt E

52 Relativistic Hydrodynamics in Graphene 18 Thermal and Electrical Conductivity at Charge Neutrality rt E in a clean charge neutral metal: κ =, σ = finite.

53 Relativistic Hydrodynamics in Graphene 19 Wiedemann-Franz Law Violations in Experiment Tbath (K) phonon-limited disorder-limited n (10 9 cm -2 ) L / L0 [Crossno et al; Science, ]

54 Relativistic Hydrodynamics in Graphene 20 Relativistic Hydrodynamics hydrodynamic transport: disorder on scale ξ l ee l ee s(x) > 0 n(x) > 0 n(x) < 0 n x

55 Relativistic Hydrodynamics in Graphene 20 Relativistic Hydrodynamics hydrodynamic transport: disorder on scale ξ l ee l ee s(x) > 0 n(x) > 0 n(x) < 0 n x not perturbative in disorder amplitude: µ 0 /µ 0!

56 Relativistic Hydrodynamics in Graphene 21 Linearized Hydrodynamics of the Disordered Dirac Fluid T = T 0 = constant, µ 0 = µ 0 (x), n = n(µ 0 ), etc. conservation of charge: ( (nv + σ q E µ + µ )) 0 T = 0. T 0

57 Relativistic Hydrodynamics in Graphene 21 Linearized Hydrodynamics of the Disordered Dirac Fluid T = T 0 = constant, µ 0 = µ 0 (x), n = n(µ 0 ), etc. conservation of charge: ( (nv + σ q E µ + µ )) 0 T = 0. T 0 conservation of heat: ( (T 0 sv µ 0 σ q E µ + µ )) 0 T = 0. T 0

58 Relativistic Hydrodynamics in Graphene 21 Linearized Hydrodynamics of the Disordered Dirac Fluid T = T 0 = constant, µ 0 = µ 0 (x), n = n(µ 0 ), etc. conservation of charge: ( (nv + σ q E µ + µ )) 0 T = 0. T 0 conservation of heat: ( (T 0 sv µ 0 σ q E µ + µ )) 0 T = 0. T 0 Navier-Stokes equation n( µ E) + s T = (η( v + v T )) (η v)

59 Relativistic Hydrodynamics in Graphene 21 Linearized Hydrodynamics of the Disordered Dirac Fluid T = T 0 = constant, µ 0 = µ 0 (x), n = n(µ 0 ), etc. conservation of charge: ( (nv + σ q E µ + µ )) 0 T = 0. T 0 conservation of heat: ( (T 0 sv µ 0 σ q E µ + µ )) 0 T = 0. T 0 Navier-Stokes equation n( µ E) + s T = (η( v + v T )) (η v) solve equations; spatially average charge/heat current [Lucas, Crossno, Fong, Kim, Sachdev; PRB, ]

60 Relativistic Hydrodynamics in Graphene 22 Comparing Theory to Experiment (k 1 ) hole FL Dirac fluid elec. FL apple (nw/k) hole FL Dirac fluid elec. FL (k 1 ) n (µm 2 ) puddle FLs Dirac fluid 10 Figure 1: testing phonons 8 apple (nw/k) n (µm 2 ) puddle FLs Dirac fluid phonons T (K) T (K) Figure 1: testing [Lucas, Crossno, Fong, Kim, Sachdev; PRB, ]

61 Transport Bounds 23 Revisiting the Strange Metal in many metals: [Bruin, Sakai, Perry, Mackenzie; Science (2013)] ρ = 1 σ = τ m ne 2 τ k B T

62 Transport Bounds 23 Revisiting the Strange Metal in many metals: [Bruin, Sakai, Perry, Mackenzie; Science (2013)] ρ = 1 σ = τ m ne 2 τ k B T diffusion bound? [Hartnoll; Nature Physics, ] D v 2 t ee 1 T?

63 Transport Bounds 24 Connections to Quantum Chaos? quantum coherence: schematically A(x, t)b(0, 0)A(x, t)b(0, 0) 1 1 N e(t x /vb)/τl. e t/ l t = x v

64 Transport Bounds 24 Connections to Quantum Chaos? quantum coherence: schematically A(x, t)b(0, 0)A(x, t)b(0, 0) 1 1 N e(t x /vb)/τl. e t/ l [Blake; PRL, ; PRD, ] proposed D v 2 bτ l. t = x v

65 Transport Bounds 24 Connections to Quantum Chaos? quantum coherence: schematically A(x, t)b(0, 0)A(x, t)b(0, 0) 1 1 N e(t x /vb)/τl. e t/ l [Blake; PRL, ; PRD, ] proposed D v 2 bτ l. [Maldacena, t = x Shenker, Stanford; JHEP, ] proved: v τ l 2πk B T.

66 Transport Bounds 25 Two Maximally Chaotic Models theories with holographic dual (Einstein gravity + matter)

67 Transport Bounds 25 Two Maximally Chaotic Models theories with holographic dual (Einstein gravity + matter) SYK chain [Gu, Qi, Stanford; ] H = J ijkl,x χ i,xχ j,xχ k,x χ l,x + J ijkl,xχ i,xχ j,xχ k,x+1 χ l,x+1 ijkl,x ijkl,x

68 Transport Bounds 26 Diffusion and Chaos in Inhomogeneous Media in an inhomogeneous SYK chain: [Gu, Lucas, Qi; XXXX] energy diffusion: 1 D 1 D x e t/ l t = v x

69 Transport Bounds 26 Diffusion and Chaos in Inhomogeneous Media in an inhomogeneous SYK chain: [Gu, Lucas, Qi; XXXX] energy diffusion: 1 D 1 D x butterfly velocity: e t/ l 1 τl v b D x t = x t = v x v

70 Transport Bounds 26 Diffusion and Chaos in Inhomogeneous Media in an inhomogeneous SYK chain: [Gu, Lucas, Qi; XXXX] energy diffusion: 1 D 1 D x butterfly velocity: e t/ l 1 τl v b D x t = x t = v x v Cauchy-Schwarz inequality: D v 2 bτ l.

71 Transport Bounds 26 Diffusion and Chaos in Inhomogeneous Media in an inhomogeneous SYK chain: [Gu, Lucas, Qi; XXXX] energy diffusion: 1 D 1 D x butterfly velocity: e t/ l 1 τl v b D x t = x t = v x v Cauchy-Schwarz inequality: D v 2 bτ l. same physics for charge diffusion in holography [Lucas, Steinberg; JHEP, ]

72 Transport Bounds 27 Transport Bounds from Entropy Production R<1 R = 1 Thomson s principle for resistor networks: Itrue Iloop I 2 R eff I 2 e R e if Kirchoff s current law obeyed

73 Transport Bounds 27 Transport Bounds from Entropy Production R<1 R = 1 Thomson s principle for resistor networks: Itrue Iloop I 2 R eff I 2 e R e if Kirchoff s current law obeyed constrained variational principle for conductivity J 2 σ T Ṡ[trial function] exists in hydrodynamics/holography [Lucas; NJP, ]

74 Transport Bounds 27 Transport Bounds from Entropy Production R<1 R = 1 Thomson s principle for resistor networks: Itrue Iloop I 2 R eff I 2 e R e if Kirchoff s current law obeyed constrained variational principle for conductivity J 2 σ T Ṡ[trial function] exists in hydrodynamics/holography [Lucas; NJP, ] Einstein-Maxwell-AdS 4 holography: σ 1, κ 4π2 T 3 [Grozdanov, Lucas, Sachdev, Schalm; PRL, ]; [Grozdanov, Lucas, Schalm; PRD, ], (natural units)

75 Outlook 28 quantitative confirmation of hydrodynamic behavior in electron fluids

76 Outlook 28 quantitative confirmation of hydrodynamic behavior in electron fluids rigorous theory of transport bounds (not solved even classically )

77 Outlook 28 quantitative confirmation of hydrodynamic behavior in electron fluids rigorous theory of transport bounds (not solved even classically ) Weyl semimetals: observation of (emergent) axial-gravitational anomaly via thermoelectric transport? [Lucas, Davison, Sachdev; PNAS, ]

78 Outlook 28 quantitative confirmation of hydrodynamic behavior in electron fluids rigorous theory of transport bounds (not solved even classically ) Weyl semimetals: observation of (emergent) axial-gravitational anomaly via thermoelectric transport? [Lucas, Davison, Sachdev; PNAS, ] beyond ω = 0 transport universality at ω T : [Lucas et al; PRL, ]