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 R minkowski AdS d+1 ... UV IR z Generating function for correlation functions � Z [ φ 0 ] CFT ≡ � e − d x dt φ 0 ( x ,t ) O ( x ,t ) � CFT Gubser–Klebanov–Polyakov–Witten Z [ φ 0 ] CFT ≃ e − S AdS [ φ ] | φ ∼ φ 0 at z =0 φ ( z ) ∼ z d − ∆ φ 0 (1 + . . . ) + z ∆ φ 1 (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 | Ψ g � 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 d d +2 x √− g ( R − 2Λ) 1 � S = 16 πG N z g µν ↔ T µν

Possible Setups Local Quench Driven Steady State Spontaneous

Thermalization TL T R 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 T L & T R 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 = cπ 2 k 2 ( T 2 L − T 2 R ) B 6 h 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 = cπ 2 k 2 ( T 2 L − T 2 R ) B 6 h T R = T − ∆ T/ 2 ∆ T ≡ T L − T R T L = T + ∆ T/ 2 J = cπ 2 k 2 g 0 = π 2 k 2 B T T ∆ T ≡ g ∆ T g = cg 0 B 3 h 3 h Quantum of Thermal Conductance g 0 = π 2 k 2 ≈ (9 . 456 × 10 − 13 WK − 2 ) T B T 3 h Free Fermions Fazio, Hekking and Khmelnitskii, PRL 80 , 5611 (1998) κ 0 = π 2 k 2 σT = π 2 σ 0 = e 2 B T κ Wiedemann–Franz 3 e 2 h 3 h 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 � dk � J = 2 π � ω m ( k ) v m ( k )[ n m ( T L ) − n m ( T R )] T m ( k ) m 1 v m ( k ) = ∂ω m /∂k n m ( T ) = e β � ωm − 1 J = f ( T L ) − f ( T R ) Consider just a single mode with ω = vk and T = 1 � ∞ � ∞ e β � vk − 1 = k 2 B T 2 e x − 1 = k 2 B T 2 � v 2 k π 2 dk x x ≡ � vk f ( T ) = dx 0 2 π h 0 h 6 k B T Velocity cancels out J = π 2 k 2 6 h ( T 2 L − T 2 R ) B For a 1+1 critical theory with central charge c J = cπ 2 k 2 ( T 2 L − T 2 R ) B 6 h

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 � ∂S � ∂P � � � T − P = T V − P T = T ∂V ∂V ∂T � ∂P � − P u = T ∂T V For black body radiation P = u/ 3 � ∂u 4 u 3 = T du 4 u = dT 1 � 4 ln u = ln T + const . 3 ∂T T V 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   u   P   T µν = Traceless P = u/d     P     . . . Thermodynamics � ∂P � u ∝ T d +1 u = T V − P ∂T For 1 + 1 Dimensional CFT u = πck 2 B T 2 J = A v ≡ A T 2 2 ( T 2 L − T 2 R ) 6 � v

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 S BH of geometry S BH = π 2 2 N 2 V 3 T 3 Entropy at Weak Coupling = 8 N 2 free massless bosons & fermions S 0 = 2 π 2 3 N 2 V 3 T 3 Relationship between strong and weak coupling S BH = 3 4 S 0 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 ) ≡ lim t →∞ t − 1 ln � e iλ ∆ t Q � Exact Result � � F ( λ ) = cπ 2 iλ iλ β l ( β l − iλ ) − 6 h β r ( β r + iλ ) Denote z ≡ iλ � � � + z 2 � � � F ( z ) = cπ 2 1 1 1 1 z l − l + + . . . β 2 β 3 6 h β 2 β 3 r r � J � = cπ 2 � δJ 2 � ∝ cπ 2 6 h k 2 B ( T 2 L − T 2 6 h k 3 B ( T 3 L + T 3 R ) R ) � ∞ e − βǫ ( e iλǫ − 1) dǫ = iλ Poisson Process 0 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) t →∞ t − 1 ln � e iλ ∆ t Q � = cπ 2 � � iλ iλ F ( λ ) ≡ lim β l ( β l − iλ ) − 6 h β r ( β r + iλ ) F ( i ( β r − β l ) − λ ) = F ( λ ) Irreversible work fluctuations in isolated driven systems P ( W ) P ( − W ) = e β ( W − ∆ F ) Crooks relation ˜ � e − βW � = e − β ∆ F Jarzynski relation Entropy production in non-equilibrium steady states P ( S ) P ( − S ) = e S Esposito et al , “Nonequilibrium fluctuations, fluctuation theorems, and counting statistics in quantum systems” , RMP 81 , 1665 (2009)

Lattice Models T T L R Quantum Ising Model � ij � S z i S z i S x H = J � j + Γ � i Γ = J/ 2 Critical c = 1 / 2 Anisotropic Heisenberg Model (XXZ) j + S y i S y S x i S x j + ∆ S z i S z � � H = J � j � ij � − 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 0.16 (a) 0 h(n,t) ∆ =0.8, λ =1, b=0 t=4 T L =0.5 0.08 -0.2 T L =0.5 t=10 T R =0.2 T R =5 -20 0 n 20 T R =0.67 0 n=2 T R =5 n=0 -0.08 (b) ∆ =0.5, b=0 0.08 < J E (n,t) > λ =1 λ =0.8 0.04 T L = ∞ λ =0.6 T R =0.48 0 (c) ∆ = − 0.85, λ =1 0.06 b=0 b=0.1 b=0.2 0.03 T L = ∞ b=0.4 T R =0.48 0 0 5 10 15 t  1 n odd  Staggered b n = ( − 1) n b Dimerization J n = ∆ n = ∆ 2 λ n even 

Time-Dependent DMRG Karrasch, Ilan and Moore, Non-equilibrium thermal transport and its relation to linear response , arXiv:1211.2236 (a) (b) vertical shift of ∆ =2 ~T 2 T L = ∞ T L =2 T L =1 curves in (a) 0.2 T L =0.5 T L =0.25 curves collapse if 10 -1 ∆ =0.5 shifted vertically! 0.1 0 < J E (t →∞ ) > < J E (t →∞ ) > ∆ =0, exact ∆ =2 -0.2 T L = ∞ ∆ =0, DMRG ~T − 1 0 1 2 3 4 5 10 -2 0 T L =1 0.64 0.115 c=1.01 c=0.48 T L =0.5 0.11 0.6 fit to quantum a+c π /12 T 2 T L =0.2 Ising model 0.105 10 -3 0.56 -0.1 ∆ =0.5 T L =0.125 0 0.1 0.2 0 0.4 0.8 0 1 2 3 4 5 0.1 1 T T R R  T 2 T ≪ 1  lim t →∞ � J E ( n, t ) � = f ( T L ) − f ( T R ) f ( T ) ∼ T − 1 T ≫ 1  Beyond CFT to massive integrable models (Doyon)