Universal Thermal Transport from Holography and Hydrodynamics Joe - - PowerPoint PPT Presentation

Universal Thermal Transport from Holography and Hydrodynamics

Joe Bhaseen

TSCM Group King’s College London Benjamin Doyon Koenraad Schalm Andy Lucas Oxford 21st January 2014

Outline

• AdS/CMT and far from equilibrium dynamics
• Quenches and thermalization
• Recent work on heat flow between CFTs
• Exact results for average current and fluctuations in 1 + 1D
• Numerical simulations in lattice models
• Beyond integrability
• Potential for AdS/CFT to offer new insights
• Higher dimensions and non-equilibrium fluctuations
• Current status and future developments
• M. J. Bhaseen, Benjamin Doyon, Andrew Lucas, Koenraad Schalm

“Far from equilibrium energy flow in quantum critical systems” arXiv:1311.3655

Progress in AdS/CMT

Transport Coefficients Viscosity, Conductivity, Hydrodynamics, Bose–Hubbard, Graphene Strange Metals Non-Fermi liquid theory, instabilities, cuprates Holographic Duals Superfluids, Fermi Liquid, O(N), Luttinger Liquid Equilibrium or close to equilibrium

AdS/CFT Correspondence

For an overview see for example John McGreevy, Holographic duality with a view toward many body physics, arXiv:0909.0518

d−1,1

z R AdS

d+1

minkowski UV IR

...

Generating function for correlation functions Z[φ0]CFT ≡ e−

• dxdt φ0(x,t)O(x,t)CFT

Gubser–Klebanov–Polyakov–Witten Z[φ0]CFT ≃ e−SAdS[φ]|φ∼φ0 at z=0 φ(z) ∼ zd−∆φ0(1 + . . . ) + z∆φ1(1 + . . . ) Fields in AdS ↔ operators in dual CFT φ ↔ O

Utility of Gauge-Gravity Duality

Quantum dynamics Classical Einstein equations Finite temperature Black holes Real time approach to finite temperature quantum dynamics in interacting systems, with the possibility of anchoring to 1 + 1 and generalizing to higher dimensions Non-Equilibrium Beyond linear response Temporal dynamics in strongly correlated systems Combine analytics with numerics Dynamical phase diagrams Organizing principles out of equilibrium

Progress

Simple protocals and integrability Methods of integrability and CFT have been invaluable in classifying equilibrium phases and phase transitions in 1+1 Do do these methods extend to non-equilibrium problems? Quantum quench Parameter in H abruptly changed H(g) → H(g′) System prepared in state |Ψg but time evolves under H(g′) Quantum quench to a CFT

Calabrese & Cardy, PRL 96, 136801 (2006)

Spin chains, BCS, AdS/CFT ... Thermalization

Experiment

Weiss et al “A quantum Newton’s cradle”, Nature 440, 900 (2006)

Non-Equilibrium 1D Bose Gas Integrability and Conservation Laws

AdS/CFT

Heat flow may be studied within pure Einstein gravity S =

1 16πGN

• dd+2x √−g(R − 2Λ)

z

gµν ↔ Tµν

Possible Setups

Local Quench Driven Steady State Spontaneous

Thermalization

R TL T

Why not connect two strongly correlated systems together and see what happens?

Non-Equilibrium CFT

Bernard & Doyon, Energy flow in non-equilibrium conformal field theory, J. Phys. A: Math. Theor. 45, 362001 (2012) Two critical 1D systems (central charge c) at temperatures TL & TR

T T L R

Join the two systems together

TL TR

Alternatively, take one critical system and impose a step profile Local Quench

Steady State Heat Flow

Bernard & Doyon, Energy flow in non-equilibrium conformal field theory, J. Phys. A: Math. Theor. 45 362001 (2012) If systems are very large (L ≫ vt) they act like heat baths For times t ≪ L/v a steady heat current flows

TL TR

Non-equilibrium steady state J = cπ2k2

B

6h

(T 2

L − T 2 R)

Universal result out of equilibrium Direct way to measure central charge; velocity doesn’t enter Sotiriadis and Cardy. J. Stat. Mech. (2008) P11003.

Linear Response

Bernard & Doyon, Energy flow in non-equilibrium conformal field theory, J. Phys. A: Math. Theor. 45, 362001 (2012) J = cπ2k2

B

6h

(T 2

L − T 2 R)

TL = T + ∆T/2 TR = T − ∆T/2 ∆T ≡ TL − TR J = cπ2k2

B

3h

T∆T ≡ g∆T g = cg0 g0 = π2k2

BT

3h

Quantum of Thermal Conductance g0 = π2k2

BT

3h

≈ (9.456 × 10−13 WK−2) T Free Fermions Fazio, Hekking and Khmelnitskii, PRL 80, 5611 (1998) Wiedemann–Franz

κ σT = π2 3e2

σ0 = e2

h

κ0 = π2k2

BT

3h

Conformal Anomaly Cappelli, Huerta and Zemba, Nucl. Phys. B 636, 568 (2002)

Experiment

Schwab, Henriksen, Worlock and Roukes, Measurement of the quantum of thermal conductance, Nature 404, 974 (2000) Quantum of Thermal Conductance

Heuristic Interpretation of CFT Result

J =

• m

dk 2π ωm(k)vm(k)[nm(TL) − nm(TR)]Tm(k) vm(k) = ∂ωm/∂k nm(T) =

1 eβωm−1

J = f(TL) − f(TR) Consider just a single mode with ω = vk and T = 1 f(T) = ∞

dk 2π v2k eβvk−1 = k2

BT 2

h

∞ dx

x ex−1 = k2

BT 2

h π2 6

x ≡ vk

kBT

Velocity cancels out J = π2k2

B

6h (T 2 L − T 2 R)

For a 1+1 critical theory with central charge c J = cπ2k2

B

6h

(T 2

L − T 2 R)

Stefan–Boltzmann

Cardy, The Ubiquitous ‘c’: from the Stefan-Boltzmann Law to Quantum Information, arXiv:1008.2331 Black Body Radiation in 3 + 1 dimensions dU = TdS − PdV ∂U

∂V

• T = T

∂S

∂V

• T − P = T

∂P

∂T

• V − P

u = T ∂P ∂T

• V

− P For black body radiation P = u/3

4u 3 = T 3

∂u

∂T

• V

du 4u = dT T 1 4 ln u = ln T + const.

u ∝ T 4

Stefan–Boltzmann and CFT

Cardy, The Ubiquitous ‘c’: from the Stefan-Boltzmann Law to Quantum Information, arXiv:1008.2331 Energy-Momentum Tensor in d + 1 Dimensions Tµν =        u P P . . .        Traceless P = u/d Thermodynamics u = T ∂P

∂T

• V − P

u ∝ T d+1 For 1 + 1 Dimensional CFT u = πck2

BT 2

6v ≡ AT 2 J = Av 2 (T 2

L − T 2 R)

Stefan–Boltzmann and AdS/CFT

Gubser, Klebanov and Peet, Entropy and temperature of black 3-branes, Phys. Rev. D 54, 3915 (1996). Entropy of SU(N) SYM = Bekenstein–Hawking SBH of geometry SBH = π2 2 N 2V3T 3 Entropy at Weak Coupling = 8N 2 free massless bosons & fermions S0 = 2π2 3 N 2V3T 3 Relationship between strong and weak coupling SBH = 3

4S0

Gubser, Klebanov, Tseytlin, Coupling constant dependence in the thermodynamics of N = 4 supersymmetric Yang-Mills Theory,

• Nucl. Phys. B 534 202 (1998)

Energy Current Fluctuations

Bernard & Doyon, Energy flow in non-equilibrium conformal field theory, J. Phys. A: Math. Theor. 45, 362001 (2012) Generating function for all moments F(λ) ≡ limt→∞ t−1 lneiλ∆tQ Exact Result F(λ) = cπ2

6h

• iλ

βl(βl−iλ) − iλ βr(βr+iλ)

• Denote z ≡ iλ

F(z) = cπ2

6h

• z
• 1

β2

l −

1 β2

r

• + z2

1 β3

l +

1 β3

r

• + . . .
• J = cπ2

6h k2 B(T 2 L − T 2 R)

δJ2 ∝ cπ2

6h k3 B(T 3 L + T 3 R)

Poisson Process ∞ e−βǫ(eiλǫ − 1)dǫ =

iλ β(β−iλ)

Non-Equilibrium Fluctuation Relation

Bernard & Doyon, Energy flow in non-equilibrium conformal field theory,

• J. Phys. A: Math. Theor. 45, 362001 (2012)

F(λ) ≡ lim

t→∞ t−1 lneiλ∆tQ = cπ2

6h

• iλ

βl(βl − iλ) − iλ βr(βr + iλ)

• F(i(βr − βl) − λ) = F(λ)

Irreversible work fluctuations in isolated driven systems Crooks relation

P (W ) ˜ P (−W ) = eβ(W −∆F )

Jarzynski relation e−βW = e−β∆F Entropy production in non-equilibrium steady states

P (S) P (−S) = eS

Esposito et al, “Nonequilibrium fluctuations, fluctuation theorems, and counting statistics in quantum systems”, RMP 81, 1665 (2009)

Lattice Models

R L T T

Quantum Ising Model H = J

ij Sz i Sz j + Γ i Sx i

Γ = J/2 Critical c = 1/2 Anisotropic Heisenberg Model (XXZ) H = J

ij

• Sx

i Sx j + Sy i Sy j + ∆Sz i Sz j

• −1 < ∆ < 1

Critical c = 1

Time-Dependent DMRG

Karrasch, Ilan and Moore, Non-equilibrium thermal transport and its relation to linear response, arXiv:1211.2236

0.04 0.08 5 10 15

t

0.03 0.06

<JE(n,t)>

• 0.08

0.08 0.16

• 20

20

n

• 0.2

h(n,t)

λ=1 ∆=−0.85, λ=1 T

R=5

T

R=0.67

T

R=0.2

T

L=0.5

b=0 ∆=0.8, λ=1, b=0 b=0.1 b=0.2 b=0.4 TL=∞ T

R=0.48 t=4 t=10

TL=∞ T

R=0.48

∆=0.5, b=0 λ=0.8 λ=0.6

TL=0.5 T

R=5

n=2 n=0

(a) (b) (c)

Dimerization Jn =    1 n odd λ n even ∆n = ∆ Staggered bn = (−1)nb

2

Time-Dependent DMRG

Karrasch, Ilan and Moore, Non-equilibrium thermal transport and its relation to linear response, arXiv:1211.2236

1 2 3 4 5

T

R

• 0.1

0.1

<JE(t→∞)>

1 2 3 4 5

• 0.2

0.2

(a)

T

L=∞

T

L=1

T

L=0.5

T

L=0.2

∆=0.5

∆=2

T

L=∞ T L=2 T L=1

T

L=0.5 T L=0.25

curves collapse if shifted vertically!

T

L=0.125

0.1 1

T

R

10-3 10-2 10-1

<JE(t→∞)>

∆=0, exact ∆=0, DMRG

0.1 0.2 0.105 0.11 0.115 0.4 0.8 0.56 0.6 0.64

(b)

vertical shift of curves in (a) c=1.01 fit to a+cπ/12 T2

~T2 ∆=0.5 ∆=2 ~T−1

quantum Ising model c=0.48

limt→∞JE(n, t) = f(TL) − f(TR) f(T) ∼    T 2 T ≪ 1 T −1 T ≫ 1 Beyond CFT to massive integrable models (Doyon)

Energy Current Correlation Function

Karrasch, Ilan and Moore, Non-equilibrium thermal transport and its relation to linear response, arXiv:1211.2236

0.03 0.06 10 20

t

0.008 0.016

<JE(t)JE(0)> / N

0.008 0.016 λ=0.8 ∆=−0.85, λ=1, T=0.5 λ=0.6 λ=0.4 b=0 ∆=0.5, b=0, T=1 λ=1 ∆=0.5, b=0, T=0.25 λ=1 λ=0.8 λ=0.6 b=0.1 λ=0.4 b=0.2 b=0.4

Bethe ansatz

(a) (b) (c)

Beyond Integrability Importance of CFT for pushing numerics and analytics

AdS/CFT

Steady State Region

General Considerations

∂µT µν = 0 ∂0T 00 = −∂xT x0 ∂0T 0x = −∂xT xx Stationary heat flow = ⇒ Constant pressure ∂0T 0x = 0 = ⇒ ∂xT xx = 0 In a CFT P = u/d = ⇒ ∂xu = 0 No energy/temperature gradient Stationary homogeneous solutions

Solutions of Einstein Equations

S =

1 16πGN

• dd+2x√−g(R − 2Λ)

Λ = −d(d + 1)/2L2 Unique homogeneous solution = boosted black hole ds2 = L2 z2 dz2 f(z) − f(z)(dt cosh θ − dx sinh θ)2+ (dx cosh θ − dt sinh θ)2 + dy2

⊥

• f(z) = 1 −
• z

z0

d+1 z0 = d+1

4πT

Fefferman–Graham Coordinates Tµνs =

Ld 16πGN limZ→0

d

dZ

d+1 Z2

L2 gµν(z(Z))

z(Z) = Z/R − (Z/R)d+2/[2(d + 1)zd+1 ] R = (d!)1/(d−1)

Boost Solution

Lorentz boosted stress tensor of a finite temperature CFT T µνs = ad T d+1 (ηµν + (d + 1)uµuν) ηµν = diag(−1, 1, · · · , 1) uµ = (cosh θ, sinh θ, 0, . . . , 0) T txs = 1

2ad T d+1(d + 1) sinh 2θ

ad = (4π/(d + 1))d+1Ld/16πGN One spatial dimension a1 =

Lπ 4GN

c =

3L 2GN

TL = Teθ TR = Te−θ T tx = cπ2k2

B

6h

(T 2

L − T 2 R)

Can also obtain complete steady state density matrix

Shock Solutions

Rankine–Hugoniot Energy-Momentum conservation across shock T txs = ad

• T d+1

L

− T d+1

R

uL + uR

• Invoking boosted steady state gives uL,R in terms of TL,R:

uL = 1

d

• χ+d

χ+d−1

uR =

• χ+d−1

χ+d

χ ≡ (TL/TR)(d+1)/2 Steady state region is a boosted thermal state with T = √TLTR Boost velocity (χ − 1)/

• (χ + d)(χ + d−1) Agrees with d = 1

Shock waves are non-linear generalizations of sound waves EM conservation: uLuR = c2

s, where cs = v/

√ d is speed of sound cs < uR < v cs < uL < c2

s/v

reinstated microscopic velocity v

Numerics I

Excellent agreement with predictions

Numerics II

Excellent agreement far from equilibrium Asymmetry in propagation speeds

Conclusions

Average energy flow in arbitrary dimension Lorentz boosted thermal state Energy current fluctuations Exact generating function of fluctuations Acknowledgements

• B. Benenowski, D. Bernard, P. Chesler, A. Green
• D. Haldane C. Herzog, D. Marolf, B. Najian, C.-A. Pillet
• S. Sachdev, A. Starinets