Non-Equilibrium Steady States Beyond Integrability

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1 Non-Equilibrium Steady States Beyond Integrability Joe Bhaseen TSCM Group King s College London Beyond Integrability: The Mathematics and Physics of Integrability and its Breaking in Low-Dimensional Strongly Correlated Systems Centre De Recherches Mathématiques University of Montréal July 2015

2 Strings, Cosmology & Condensed Matter Benjamin Doyon Koenraad Schalm Andy Lucas Ben Simons Julian Sonner Jerome Gauntlett Toby Wiseman King s, Leiden, Harvard, Cambridge, Imperial

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4 Outline AdS/CFT and far from equilibrium dynamics Quenches and thermalization Energy flow between CFTs AdS/CFT offers new insights Higher dimensions Non-equilibrium fluctuations Current status and future developments MJB, Benjamin Doyon, Andrew Lucas, Koenraad Schalm Energy flow in quantum critical systems far from equilibrium Nature Physics (2015)

5 Progress in AdS/CMT Transport Coefficients Viscosity, Conductivity, Hydrodynamics, Bose Hubbard, Graphene Strange Metals Non-Fermi liquids, instabilities, cuprates Holographic Duals Superfluids, Fermi liquid, O(N), Luttinger liquid Equilibrium or close to equilibrium

6 Gauge-Gravity Duality AdS/CFT correspondence (Anti-de Sitter/Conformal Field Theory) Maldacena (1997); Gubser, Klebanov, Polyakov (1998); Witten (1998) S. Hartnoll, Science 322, 1639 (2008) For an overview see for example John McGreevy, Holographic duality with a view toward many body physics, arxiv:

7 AdS/CFT Correspondence AdS d+1 d 1,1 R minkowski... IR z UV Gubser Klebanov Polyakov Witten Z[φ 0 ] CFT e S AdS[φ] φ φ0 atz=0 φ(z) z d φ 0 (1+...)+z φ 1 (1+...) Fields in AdS operators in dual CFT φ O

8 Utility of Gauge-Gravity Duality Quantum dynamics Classical Einstein equations Finite temperature Black holes Real time approach to finite temperature dynamics in interacting quantum systems Possibility of anchoring to integrable models Beyond Integrability Higher Dimensions Non-Equilibrium Beyond linear response Organising principles out of equilibrium

9 Non-Equilibrium AdS/CMT Quenches in Holographic Superfluids MJB, Gauntlett, Simons, Sonner & Wiseman, Holographic Superfluids and the Dynamics of Symmetry Breaking PRL (2013) Quasi-Normal-Modes Current Noise Sonner and Green, Hawking Radiation and Nonequilibrium Quantum Critical Current Noise, PRL 109, (2013) Hawking Radiation Superfluid Turbulence Chesler, Liu and Adams, Holographic Vortex Liquids and Superfluid Turbulence, Science 341, 368 (2013); also arxiv: Fractal Horizons Metric quenches, entanglement entropy, thermalization...

10 Thermalization Condensed matter and high energy physics T L T R Why not connect two strongly correlated systems together and see what happens?

11 AdS/CFT Energy flow may be studied within pure Einstein gravity S = 1 16πG N d d+2 x g(r 2Λ) z g µν T µν

12 Einstein Centenary The Field Equations of General Relativity (1915) R µν 1 2 Rg µν +Λg µν = 8πG c 4 T µν field equations g µν metric R µν Ricci curvature R scalar curvature Λ cosmological constant T µν energy-momentum tensor Coupled Nonlinear PDEs Schwarzchild Solution (1916) R S = 2MG c 2 Black Holes

13 Non-Equilibrium CFT Bernard & Doyon, Energy flow in non-equilibrium conformal field theory, J. Phys. A: Math. Theor. 45, (2012) Two critical 1D systems (central charge c) at temperatures T L & T R TL TR Join the two systems together T L T R Alternatively, take one critical system and impose a step profile Local Quench

14 Steady State Energy Flow Bernard & Doyon, Energy flow in non-equilibrium conformal field theory, J. Phys. A: Math. Theor (2012) If systems are very large (L vt) they act like heat baths For times t L/v a steady energy current flows T L T R Non-equilibrium steady state J = cπ2 k 2 B 6h (T2 L T2 R ) Universal result out of equilibrium Direct way to measure central charge; velocity doesn t enter Sotiriadis and Cardy. J. Stat. Mech. (2008) P Stefan Boltzmann

15 Energy Current Fluctuations Bernard & Doyon, Energy flow in non-equilibrium conformal field theory, J. Phys. A: Math. Theor. 45, (2012) Generating function for all moments F(λ) lim t t 1 ln e iλ tq F(λ) = cπ2 6h Exact Result ( iλ β l (β l iλ) iλ β r (β r +iλ) ) F(z) = cπ2 6h [ ( z Denote z iλ ( )+z 2 1 β 2 l 1 β 2 r 1 β 3 l ) + 1 βr 3 ] +... J = cπ2 6h k2 B (T2 L T2 R ) δj2 cπ2 6h k3 B (T3 L +T3 R ) Poisson Process 0 e βǫ (e iλǫ 1)dǫ = iλ β(β iλ)

16 AdS/CFT Steady State Region Spatially Homogeneous

17 Solutions of Einstein Equations S = 1 16πG d d+2 x g(r 2Λ) Λ = d(d+1)/2l 2 Unique homogeneous solution = boosted black hole ds 2 = L2 z 2 [ dz 2 f(z) f(z)(dtcoshθ dxsinhθ)2 + (dxcoshθ dtsinhθ) 2 +dy 2 ] f(z) = 1 ( ) d+1 z z 0 z 0 = d+1 4πT Fefferman Graham Coordinates T µν s = Ld 16πG lim ( d ) d+1 Z 2 Z 0 dz L g 2 µν (z(z))

18 Boost Solution Lorentz boosted stress tensor of a finite temperature CFT Perfect fluid T µν s = a d T d+1 (η µν +(d+1)u µ u ν ) η µν = diag( 1,1,,1) u µ = (coshθ,sinhθ,0,...,0) One spatial dimension a 1 = Lπ 4G c = 3L 2G T L = Te θ T R = Te θ T tx = cπ2 k 2 B 6h (T2 L T2 R ) Can also obtain complete steady state density matrix Describes all the cumulants of the energy transfer process The non-equilibrium steady state (NESS) is a Lorentz boosted thermal state Run past a thermal state with temperature T = T L T R

19 Steady State Density Matrix O... = Tr(ρ so...) Tr(ρ s ) ρ s = e βe coshθ+βp x sinhθ β = β L β R e 2θ = β R βl Lorentz boosted thermal density matrix Generalized Gibbs Ensemble (GGE) Describes all the cumulants of the energy transfer process The non-equilibrium steady state (NESS) is a Lorentz boosted thermal state

20 Higher Dimensions Shock Waves Energy-Momentum conservation across the shocks

21 Shock Solutions Rankine Hugoniot Energy-Momentum conservation µ T µν = 0 across shock ( ) T tx T d+1 L T d+1 R s = a d u L +u R Invoking boosted steady state gives u L,R in terms of T L,R : u L = 1 χ+d χ+d d χ+d u 1 R = 1 χ+d χ (T L /T R ) (d+1)/2 Steady state region is a boosted thermal state with T = T L T R Boost velocity (χ 1)/ (χ+d)(χ+d 1 ) Agrees with d = 1 Shock waves are non-linear generalisations of sound waves EM conservation: u L u R = c 2 s, where c s = v/ d is speed of sound c s < u R < v c s < u L < c 2 s/v reinstated microscopic velocity v

22 Numerics I Excellent agreement with predictions

23 Numerics II Excellent agreement far from equilibrium Asymmetry in propagation speeds

24 Conclusions Average energy flow in arbitrary dimension Lorentz boosted thermal state Energy current fluctuations Exact generating function of fluctuations Generalizations Other types of charge noise Non-Lorentz invariant situations Bose Hubbard and Unitary Fermi Gas Different central charges Fluctuation theorems Numerical GR Acknowledgements B. Benenowski, D. Bernard, P. Chesler, A. Green D. Haldane C. Herzog, D. Marolf, B. Najian, C.-A. Pillet S. Sachdev, A. Starinets