Transport bounds for condensed matter physics. Andrew Lucas

Transport bounds for condensed matter physics Andrew Lucas Stanford Physics High Energy Physics Seminar, University of Washington May 2, 2017

Collaborators 2 Julia Steinberg Harvard Physics Subir Sachdev Harvard Physics & Perimeter Institute Yingfei Gu Stanford Physics Koenraad Schalm Leiden: Lorentz Institute Sean Hartnoll Stanford Physics Xiao-Liang Qi Stanford Physics Sašo Grozdanov Leiden: Lorentz Institute

Introduction to Transport 3 Transport in Metals Ohm s law the simplest experiment... E = ρj (V = IR)

Introduction to Transport 3 Transport in Metals Ohm s law the simplest experiment... E = ρj (V = IR)...yet ρ hard to compute in interesting systems: electron-electron interactions translation symmetry breaking gases transport QFT fluids stat.mech. black holes chaos

Introduction to Transport 3 Transport in Metals Ohm s law the simplest experiment... E = ρj (V = IR)...yet ρ hard to compute in interesting systems: electron-electron interactions translation symmetry breaking gases transport QFT fluids stat.mech. black holes chaos can we at least bound ρ?

Introduction to Transport 4 The Drude Model ρ governed by scattering? ρ = m 1 ne 2 τ impurities phonons electron interactions (umklapp) ρ T 0 ρ T d+2 (low T ) ρ T 2 ρ T (high T )

Introduction to Transport 4 The Drude Model ρ governed by scattering? ρ = m 1 ne 2 τ impurities phonons electron interactions (umklapp) ρ T 0 ρ T d+2 (low T ) ρ T 2 ρ T (high T ) scattering rates add (Mattheisen s rule )? ρ = ρ e,imp + ρ e,ph + ρ ee?

Introduction to Transport 5 Momentum Conservation: A Theorem J =0 J = Qv and E =0

Introduction to Transport 5 Momentum Conservation: A Theorem J =0 J = nv and E =0 J = Qv and E =0

Introduction to Transport 5 Momentum Conservation: A Theorem J =0 J = nv and E =0 J = Qv and E =0 if collisions cannot relax momentum, ρ = 0

Introduction to Transport 5 Momentum Conservation: A Theorem J =0 J = nv and E =0 J = Qv and E =0 if collisions cannot relax momentum, ρ = 0 Mattheisen s rule is not generally true

Introduction to Transport 5 Momentum Conservation: A Theorem J =0 J = nv and E =0 J = Qv and E =0 if collisions cannot relax momentum, ρ = 0 Mattheisen s rule is not generally true Noether s theorem: momentum relaxation = spatially inhomogeneous

Strange Metals in Experiment 6 Electron-Electron Interaction Limited Resistivity in Fermi Liquids in a Fermi liquid (ordinary metal): τ ee µ (k B T ) 2, ρ = BT 2 1 τ ee...

Strange Metals in Experiment 6 Electron-Electron Interaction Limited Resistivity in Fermi Liquids in a Fermi liquid (ordinary metal): τ ee µ (k B T ) 2, ρ = BT 2 1... τ ee B depends on thermodynamics (not disorder?): [Jacko, Fjaerestad, Powell; Nature Physics, 0805.4275]

Strange Metals in Experiment 7 Linear Resistivity: A Challenge in a theory without quasiparticles: τ ee k B T.

Strange Metals in Experiment 7 Linear Resistivity: A Challenge in a theory without quasiparticles: τ ee k B T. Drude ρ = m 1 ne 2 m k B T τ ee ne 2 : [Bruin, Sakai, Perry, Mackenzie; Science (2013)]

Diffusion Bounds 8 From Viscosity to Transport ρ bounded due to bounds on diffusion constant D? [Hartnoll; Nature Physics, 1405.3651] ρ = 1 χd, D = v 2 microτ ee, τ ee k B T, D v2 micro k B T

Diffusion Bounds 8 From Viscosity to Transport ρ bounded due to bounds on diffusion constant D? [Hartnoll; Nature Physics, 1405.3651] ρ = 1 χd, D = vmicroτ 2 ee, τ ee k B T, D v2 micro k B T inspiration: viscosity [Kovtun, Son, Starinets; PRL, hep-th/0405231] η s 4πk B? η ɛτ ee, τ ee k B T, η ɛ k B T s k B.

Diffusion Bounds 8 From Viscosity to Transport ρ bounded due to bounds on diffusion constant D? [Hartnoll; Nature Physics, 1405.3651] ρ = 1 χd, D = vmicroτ 2 ee, τ ee k B T, D v2 micro k B T inspiration: viscosity [Kovtun, Son, Starinets; PRL, hep-th/0405231] η s 4πk B? η ɛτ ee, τ ee k B T, η ɛ k B T s. k B counter-examples to both bounds. in particular: ρ T 0 (static impurities)

Diffusion Bounds 9 Connections to Quantum Chaos? if D v 2 τ, what v? what τ?

Diffusion Bounds 9 Connections to Quantum Chaos? if D v 2 τ, what v? what τ? a rigorous bound from quantum chaos: [Maldacena, Shenker, Stanford; JHEP, 1503.01409] where (schematically) τ l [A(x, t), B(0, 0)] 2 2πk B T. ( ) 1 2 N e(t x /vb)/τl.

Diffusion Bounds 9 Connections to Quantum Chaos? if D v 2 τ, what v? what τ? a rigorous bound from quantum chaos: [Maldacena, Shenker, Stanford; JHEP, 1503.01409] where (schematically) τ l [A(x, t), B(0, 0)] 2 2πk B T. ( ) 1 2 N e(t x /vb)/τl. [Blake; PRL, 1603.08510; PRD, 1604.01754] proposed D v 2 bτ l.

Diffusion Bounds 9 Connections to Quantum Chaos? if D v 2 τ, what v? what τ? a rigorous bound from quantum chaos: [Maldacena, Shenker, Stanford; JHEP, 1503.01409] where (schematically) τ l [A(x, t), B(0, 0)] 2 2πk B T. ( ) 1 2 N e(t x /vb)/τl. [Blake; PRL, 1603.08510; PRD, 1604.01754] proposed D v 2 bτ l. many examples confirm conjecture but examples are homogeneous

Diffusion Bounds 10 Diffusion and Chaos in Inhomogeneous Media diffusion bound fails in inhomogeneous systems: diffusion: 1 D 1 D x e t/ l t = v x

Diffusion Bounds 10 Diffusion and Chaos in Inhomogeneous Media diffusion bound fails in inhomogeneous systems: diffusion: 1 D 1 D x butterfly velocity: e t/ l 1 v b τl D x t = x v t = v x

Diffusion Bounds 10 Diffusion and Chaos in Inhomogeneous Media diffusion bound fails in inhomogeneous systems: diffusion: 1 D 1 D x butterfly velocity: e t/ l 1 τl v b D x x t = v t = v x Cauchy-Schwarz inequality: D v 2 bτ l.

Diffusion Bounds 10 Diffusion and Chaos in Inhomogeneous Media diffusion bound fails in inhomogeneous systems: diffusion: 1 D 1 D x butterfly velocity: e t/ l 1 τl v b D x x t = v t = v x Cauchy-Schwarz inequality: D v 2 bτ l. we ve explicitly confirmed this: Dcharge : holography [Lucas, Steinberg; JHEP, 1608.03286] Denergy : SYK chains [Gu, Lucas, Qi; 1702.08462]

Hydrodynamic Resistivity Bounds 11 Resistor Network Bounds Thomson s principle: R<1 resistance of R resistor = 1 network obeys I0R 2 eff Ie 2 R e edges e for arbitrary conserved currents I e : I true I loop

Hydrodynamic Resistivity Bounds 11 Resistor Network Bounds Thomson s principle: R<1 resistance of R resistor = 1 network obeys I0R 2 eff Ie 2 R e edges e for arbitrary conserved currents I e : I true I loop Thomson s principle in the continuum limit: ρ 1 d d x J 2 Jx,avg 2 V σ loc (x), J = 0.

Hydrodynamic Resistivity Bounds 12 Generalized Hydrodynamics hydrodynamic limit of transport: lee s(x) > 0 n(x) > 0 n(x) < 0 n x

Hydrodynamic Resistivity Bounds 12 Generalized Hydrodynamics hydrodynamic limit of transport: lee s(x) > 0 n(x) > 0 n(x) < 0 n x the resistor network is simplest (Ohmic) hydrodynamics : J = 0, J = σ loc (x) µ +

Hydrodynamic Resistivity Bounds 12 Generalized Hydrodynamics hydrodynamic limit of transport: lee s(x) > 0 n(x) > 0 n(x) < 0 n x the resistor network is simplest (Ohmic) hydrodynamics : J = 0, J = σ loc (x) µ + more conserved quantities? (including momentum): J A = 0, J A = n A v Σ AB µ B, 0 n A µ A (η v). }{{} P

Hydrodynamic Resistivity Bounds 13 Hydrodynamic Bounds power dissipated = entropy production T Ṡ

Hydrodynamic Resistivity Bounds 13 Hydrodynamic Bounds power dissipated = entropy production T Ṡ in the hydrodynamic limit: T ṡ Σ AB µ A µ B + η( v) 2 (Σ 1 ) AB (J A n A v) (J B n B v) + η( v) 2

Hydrodynamic Resistivity Bounds 13 Hydrodynamic Bounds power dissipated = entropy production T Ṡ in the hydrodynamic limit: T ṡ Σ AB µ A µ B + η( v) 2 (Σ 1 ) AB (J A n A v) (J B n B v) + η( v) 2 bound: [Lucas; NJP, 1506.02662], [Lucas, Hartnoll; 1704.07384] ρ xx T ṡ[ja, v] Jx,avg 2, if J A = 0.

Hydrodynamic Resistivity Bounds 13 Hydrodynamic Bounds power dissipated = entropy production T Ṡ in the hydrodynamic limit: T ṡ Σ AB µ A µ B + η( v) 2 (Σ 1 ) AB (J A n A v) (J B n B v) + η( v) 2 bound: [Lucas; NJP, 1506.02662], [Lucas, Hartnoll; 1704.07384] J = constant, v = 0: ρ xx T ṡ[ja, v] Jx,avg 2, if J A = 0. ρ xx Σ 1, Σ τ ee (more later...)

Holographic Transport Bounds 14 A Holographic Aside application #1 of Thomson s principle: AdS/CFT known equations of state extend beyond strict hydrodynamic limit

Holographic Transport Bounds 14 A Holographic Aside application #1 of Thomson s principle: AdS/CFT known equations of state extend beyond strict hydrodynamic limit holographic workhorse, dual to 2 + 1 metal: S = d 4 x g (R + 6 F 2 ). 4

Holographic Transport Bounds 14 A Holographic Aside application #1 of Thomson s principle: AdS/CFT known equations of state extend beyond strict hydrodynamic limit holographic workhorse, dual to 2 + 1 metal: S = d 4 x g (R + 6 F 2 ). 4 consider static black hole backgrounds: ds 2 = dr 2 F (r, x)dt 2 + G ij (r, x)dx i dx j, A = p(r, x)dt, connected horizon of T 2 (or R 2...) topology

Holographic Transport Bounds 15 Holographic Thomson s Principle membrane paradigm for uncharged black hole: ( M gf MN ) = 0 (bulk Maxwell equation) d 2 x J x,avg = gf ir, at any r. }{{} V 2 }{{} boundary theory conserved in x

Holographic Transport Bounds 15 Holographic Thomson s Principle membrane paradigm for uncharged black hole: ( M gf MN ) = 0 (bulk Maxwell equation) d 2 x J x,avg = gf ir, at any r. }{{} V 2 }{{} boundary theory conserved in x thermoelectric transport from emergent horizon charge current I and heat current J [Donos, Gauntlett; PRD, 1506.01360]

Holographic Transport Bounds 15 Holographic Thomson s Principle membrane paradigm for uncharged black hole: ( M gf MN ) = 0 (bulk Maxwell equation) d 2 x J x,avg = gf ir, at any r. }{{} V 2 }{{} boundary theory conserved in x thermoelectric transport from emergent horizon charge current I and heat current J [Donos, Gauntlett; PRD, 1506.01360] Thomson s principle without a hydrodynamic limit: [Grozdanov, Lucas, Sachdev, Schalm; PRL, 1507.00003] T ṡ[i, J ] = γ ( I QJ 4πT i I i = 0 }{{} charge conservation ) 2 + 2 γ (4πT ) 2 (i J j) (i J j) i J i = 0 }{{} heat conservation

Holographic Transport Bounds 16 Sharp Transport Bounds fix induced horizon metric: ds 2 hor = ( eω(x,y) dx 2 + dy 2).

Holographic Transport Bounds 16 Sharp Transport Bounds fix induced horizon metric: ds 2 hor = ( eω(x,y) dx 2 + dy 2). plug in J i = 0, I i = e ω δ i x: σ 1.

Holographic Transport Bounds 16 Sharp Transport Bounds fix induced horizon metric: ds 2 hor = ( eω(x,y) dx 2 + dy 2). plug in J i = 0, I i = e ω δ i x: σ 1. plug in I i = 0, J i = e ω δ i x: κ 4π2 T 3.

Holographic Transport Bounds 16 Sharp Transport Bounds fix induced horizon metric: ds 2 hor = ( eω(x,y) dx 2 + dy 2). plug in J i = 0, I i = e ω δ i x: σ 1. plug in I i = 0, J i = e ω δ i x: κ 4π2 T 3. no disorder-driven metal-insulator transition

New Boltzmann Transport 17 Kinetic Theory application #2 of Thomson s principle: kinetic theory

New Boltzmann Transport 17 Kinetic Theory application #2 of Thomson s principle: kinetic theory assume quasiparticles: ψ (k, ω)ψ(k, ω) 1 ω ɛ(k) + icω 2 +.

New Boltzmann Transport 17 Kinetic Theory application #2 of Thomson s principle: kinetic theory assume quasiparticles: ψ (k, ω)ψ(k, ω) non-equilibrium distribution function: 1 ω ɛ(k) + icω 2 +. f(x, p) particles of momentum p at position x J = d d x V dd p v(p)f(x, p)

New Boltzmann Transport 17 Kinetic Theory application #2 of Thomson s principle: kinetic theory assume quasiparticles: ψ (k, ω)ψ(k, ω) non-equilibrium distribution function: 1 ω ɛ(k) + icω 2 +. f(x, p) particles of momentum p at position x J = d d x V dd p v(p)f(x, p) weak interactions + x p = kinetic theory: t f + v x f + F p f = }{{} C[f] }{{}. free-particle streaming collisions

New Boltzmann Transport 18 Linearized Boltzmann Equation assume: inversion and time reversal symmetry thermodynamic equilibrium feq is not unstable

New Boltzmann Transport 18 Linearized Boltzmann Equation assume: inversion and time reversal symmetry thermodynamic equilibrium feq is not unstable static linear response: f = f eq + δf, v x δf + F p δf + ee v f eq = δc }{{}}{{ ɛ} δf δf eq streaming terms }{{} source term collision operator

New Boltzmann Transport 18 Linearized Boltzmann Equation assume: inversion and time reversal symmetry thermodynamic equilibrium feq is not unstable static linear response: f = f eq + δf, v x δf + F p δf + ee v f eq = δc }{{}}{{ ɛ} δf δf eq streaming terms }{{} source term collision operator BIG linear algebra problem...schematically: L Φ + W Φ = E, δf = f eq ɛ Φ }{{} Φ not singular

New Boltzmann Transport 19 Ziman s Bound historical simplifications: either W = 0 or L = 0

New Boltzmann Transport 19 Ziman s Bound historical simplifications: either W = 0 or L = 0 L = 0: resistivity related to entropy production: ρ = T ṡ Φ W Φ = J 2 Φ E 2, W Φ = E.

New Boltzmann Transport 19 Ziman s Bound historical simplifications: either W = 0 or L = 0 L = 0: resistivity related to entropy production: ρ = T ṡ Φ W Φ = J 2 Φ E 2, W Φ = E. in fact, ρ minimizes entropy production: [Ziman; Canadian J. Phys. (1957)] ρ Φ W Φ Φ E 2, for any Φ.

New Boltzmann Transport 20 Classical Disorder homogeneous Boltzmann transport theory is insufficient to recover hydrodynamics...

New Boltzmann Transport 20 Classical Disorder homogeneous Boltzmann transport theory is insufficient to recover hydrodynamics... charge puddles: H = H clean + d d x n(x)v imp (x): F µ(x) =µ 0 V imp (x)

New Boltzmann Transport 20 Classical Disorder homogeneous Boltzmann transport theory is insufficient to recover hydrodynamics... charge puddles: H = H clean + d d x n(x)v imp (x): F µ(x) =µ 0 V imp (x) for simplicity: neglect umklapp, phonons

New Boltzmann Transport 20 Classical Disorder homogeneous Boltzmann transport theory is insufficient to recover hydrodynamics... charge puddles: H = H clean + d d x n(x)v imp (x): F µ(x) =µ 0 V imp (x) for simplicity: neglect umklapp, phonons W P = 0 ee collisions conserve momentum

New Boltzmann Transport 20 Classical Disorder homogeneous Boltzmann transport theory is insufficient to recover hydrodynamics... charge puddles: H = H clean + d d x n(x)v imp (x): F µ(x) =µ 0 V imp (x) for simplicity: neglect umklapp, phonons W P = 0 ee collisions conserve momentum must include streaming terms: L 0

New Boltzmann Transport 21 Generalized Kinetic Bound we proved a transport bound with L 0: [Lucas, Hartnoll; 1705.0XXXX] ρ T ṡ J 2 = Φ odd W Φ odd Φ odd E 2, subject to J [Φ odd ] = 0 }{{} conservation of charge, energy, imbalance... upon integrating out non-conserved even modes: W = W odd + L T W 1 evenl

New Boltzmann Transport 21 Generalized Kinetic Bound we proved a transport bound with L 0: [Lucas, Hartnoll; 1705.0XXXX] ρ T ṡ J 2 = Φ odd W Φ odd Φ odd E 2, subject to J [Φ odd ] = 0 }{{} conservation of charge, energy, imbalance... upon integrating out non-conserved even modes: W = W odd + L T W 1 evenl hydrodynamic limit: T Ṡ d d x [ (J nv) Σ 1 (J nv) + η( v) 2].

New Boltzmann Transport 22 Free Fermions (L 0, W = 0) simple argument: 1 v F τ Drude ξ y V imp y x

New Boltzmann Transport 22 Free Fermions (L 0, W = 0) simple argument: 1 v F τ Drude ξ (not-trivial) recover with bound: y V imp ρ 1 νv F ξ y x

New Boltzmann Transport 22 Free Fermions (L 0, W = 0) simple argument: 1 v F τ Drude ξ (not-trivial) recover with bound: y V imp ρ 1 νv F ξ y entropy production even as W 0 with inhomogeneity x

New Boltzmann Transport 23 Comparison with Viscous Hydrodynamics look for ansatz J A = c AI (x)π I with π I odd conserved quantities

New Boltzmann Transport 23 Comparison with Viscous Hydrodynamics look for ansatz J A = c AI (x)π I with π I odd conserved quantities if we can do this, bound gives ρ l ee ξ 2

New Boltzmann Transport 23 Comparison with Viscous x Hydrodynamics look for ansatz J A = c AI (x)π I with π I odd conserved quantities if we can do this, bound gives ρ l ee ξ 2 y V imp QP random walks sees V imp slower: x ρ 1 τ v Fl ee ξ 2

New Boltzmann Transport 24 Imbalance Limited Transport? if not enough odd conserved quantities: ρ 1 l ee

New Boltzmann Transport 24 Imbalance Limited Transport? if not enough odd conserved quantities: ρ 1 l ee J 1 = J 2 = 0; QPs pushed out of equilibrium: µimbalance `ee J imb J x

New Boltzmann Transport 24 Imbalance Limited Transport? if not enough odd conserved quantities: ρ 1 l ee momentum relaxation: 1 τ l eev F ξ }{{ 2 ξ2 l } 2 v F ee l }{{} ee diffusion imbalance J 1 = J 2 = 0; QPs pushed out of equilibrium: µimbalance `ee mp J imb y J x x x

Experimental Phenomenology 25 Phenomenology: Imbalance Modes in Strange Metals? imbalance modes from pockets/bands?

Experimental Phenomenology 25 Phenomenology: Imbalance Modes in Strange Metals? imbalance modes from pockets/bands? Pomeranchuk criticality?

Experimental Phenomenology 25 Phenomenology: Imbalance Modes in Strange Metals? imbalance empty modes from pockets/bands? Pomeranchuk criticality? filled spin imbalance? empty filled filled

Experimental Phenomenology 26 Phenomenology: Enhanced Resistivity near Criticality I sample phase diagram: T ρ T ρ T2 doping

Experimental Phenomenology 26 Phenomenology: Enhanced Resistivity near Criticality I sample phase diagram: T ρ T ρ T2 doping I our theory: imbalance diffusion causes 1 T strange metal ρ. 2 T Fermi liquid τee

Experimental Phenomenology 26 Phenomenology: Enhanced Resistivity near Criticality I sample phase diagram: T ρ T ρ T2 doping I our theory: imbalance diffusion causes 1 T strange metal ρ. 2 T Fermi liquid τee I ρ increases near quantum critical points?

Experimental Phenomenology 27 Phenomenology: Disorder Independence? out of plane impurities for quasi-2d metal? ed conduction layers (impurity-rich dopant layers make qualitatively large amplitude, large ξ puddles?) filled

Experimental Phenomenology 27 Phenomenology: Disorder Independence? out of plane impurities for quasi-2d metal? ed conduction layers filled (impurity-rich dopant layers make qualitatively large amplitude, large ξ puddles?) failure of Mattheisen s rule at weak disorder? observed in some heavy fermions? [Kadowaki, Woods (1986)]

Outlook 28 it is generally not possible to bound the diffusion of charge or energy (as normally defined)

Outlook 28 it is generally not possible to bound the diffusion of charge or energy (as normally defined) universal (classical) formalism for ρ bounds, new theory for ρ 1 τ ee.

Outlook 28 it is generally not possible to bound the diffusion of charge or energy (as normally defined) universal (classical) formalism for ρ bounds, new theory for ρ 1 τ ee. future directions? extensions and applications of new Boltzmann transport theory? imbalance modes in realistic strange metals? quantum generalization of ρ bound?