Page 1 Non-equilibrium steady states in many-body quantum systems Benjamin Doyon Department of Mathematics, King’s College London, UK Collaborators: Denis Bernard , Olalla A. Castro-Alvaredo, Andrea De Luca, Jacopo Viti; Joe Bhassen, Andrew Lucas, Koenraad Schalm Students: Y. Chen, M. Hoogeveen Amsterdam, 1 July 2015
Page 2 Works in preparation: with D. Bernard; and with M.J. Bhaseen, A. Lucas, K. Schalm Published: M.J. Bhaseen, B.D., A. Lucas, K. Schalm: Energy flow in quantum critical systems far from equilibrium, Nature Physics 11 (2015) 509–514 B.D., A. Lucas, K. Schalm, M.J. Bhaseen: Non-equilibrium steady states in the Klein-Gordon theory, J. Phys. A: Math. Theor. 48 (2015) 095002 B.D.: Lower bounds for ballistic current and noise in non-equilibrium quantum steady states, Nucl. Phys. B 892 (2015), 190–210 Y. Chen, B.D., Form factors in equilibrium and non-equilibrium mixed states of the Ising model, J. Stat. Mech. (2014) P09021 O. A. Castro-Alvaredo, Y. Chen, B.D., M. Hoogeveen: Thermodynamic Bethe ansatz for non-equilibrium steady states: exact energy current and fluctuations in integrable QFT, J. Stat. Mech. (2014) P03011
Page 3 D. Bernard, B.D.: Non-equilibrium steady states in conformal field theory, Ann. Henri Poincar´ e 16 (2015) 113–161 B.D., M. Hoogeveen, D. Bernard: Energy flow and fluctuations in non-equilibrium conformal field theory on star graphs, J. Stat. Mech. (2013) P03002 A. De Luca, J. Viti, D. Bernard, B.D., Non-equilibrium thermal transport in the quantum Ising chain, Phys. Rev. B 88, 134301 (2013) D. Bernard, B.D.: Time-reversal symmetry and fluctuation relations in non-equilibrium quantum steady states, J. Phys. A : Math. Theor. 46 (2013) 372001 D. Bernard, B.D.: Energy flow in non-equilibrium conformal field theory, J. Phys. A: Math. Theor. 45 (2012) 362001
Page 4 Partitioning approach [Caroli et. al. 1971; Rubin et. al. 1971; Spohn et. al. 1977] Consider some extended, local many-body quantum system separated into two halves, independently thermalized. Then suddenly connect them (local quench) and wait for a long time (unitary evolution).
Page 5 Generically, expect steady state to be trivial: thermalization, no flows. In what situation can there be a nontrivial current? Asymptotic baths very far; steady state translation invariant ⇒ No gradients ⇒ no diffusive transport (cf Fourier’s law). Current emerges in steady-state region iff there is ballistic transport
Page 6 Ballistic steady state • By stationarity and Eigenstate Thermalization Hypothesis [Deutsch 1991, Srednicki 1994, Rigol, Dunjko, Olshanii 2008] , steady state described by (semi-)local conserved charges . • By cluster property, steady states is exponential of local conserved charges (cf GGE). Need a parity-odd conserved charge P : e − β H + ν P + ... O � � ⟨ O ⟩ stat = Tr e − β H + ν P + ... , Tr ( e − β H + ν P + ... ) Steady-state limit: only in central region, for local observables, vL � t →∞ ⟨ e i Ht O e − i Ht ⟩ 0 , ρ 0 = e − β l H l − β r H r , ⟨ O ⟩ stat = lim H = H l + δ H l r + H r
Page 7 Near quantum criticality Near zero-temperature quantum criticality: continuous translation invariance emerges Momentum P Universal steady state near criticality, with “diffusion time” t di ff ( T l , T r ) set by temperatures, t diff ( T l ,T r ) ,vL � t →∞ ⟨ e i Ht O e − i Ht ⟩ 0 . ⟨ O ⟩ stat = lim
Page 8 If the total current is a conserved quantity Let j be a current observable for transport of quantity q , i.e. ∂ t q + ∇ · j = 0 . Let j := j 1 be longitudinal component, and assume that there is some k such that ∂ t j + ∇ · k = 0 . d d x j is conserved ⇒ nonzero Drude peak, linear-response conductivity � Example: Lorentz invariant energy transport ( z = 1 near-critical systems), ∂ µ T µ ν = 0 and T µ ν = T ν µ Set q = h := T 00 , j = p := T 0 i , k = T 1 i , and we have P = d d x j . �
Page 9 Linear response: sound velocity Take small variations about local Gibbs equilibrium ⟨ q ( x, t ) ⟩ 0 ≈ ⟨ q ⟩ + δ q ( x, t ) , ⟨ j ( x, t ) ⟩ 0 ≈ δ j ( x, t ) , ⟨ k ( x, t ) ⟩ 0 ≈ ⟨ k ⟩ + δ k ( x, t ) Assume local thermalization: Equation of state ⟨ k ⟩ = F ( ⟨ q ⟩ ) valid at every point: δ k ( x, t ) = F ′ ( ⟨ q ⟩ ) δ q ( x, t ) � Conservation equations imply wave equation with sound velocity v s = F ′ ( ⟨ q ⟩ ) : δ q ( x, t ) = f ( x − v s t ) + g ( x + v s t ) , δ j ( x, t ) = v s ( f ( x − v s t ) − g ( x + v s t )) Solving with initial zero-current step profile: δ j stat = δ k l − δ k r . 2 v s
Page 10 An inequality that quantifies non-equilibrium ballistic transport [BD 2014] If “pressure” k is monotonic on large scales in transient regions, then j stat ≥ k l − k r 2 v where v is Lieb-Robinson velocity and k l , r are thermal averages in left and right reservoir. Can define “transient velocities”: v l , r := ± k l , r − k stat , v l , r ≤ v. j stat From the linear response calculation: equilibrium v l , r = v s = sound velocity . lim
Page 11 Shocks Suppose two-shock picture of linear response remains “mostly true”: o ( t ) transient regions . Take integral form of conservation equations through shocks ∂ t h + ∂ x j = 0 , ∂ t p + ∂ x k = 0 Four connection equations (Rankine-Hugoniot): v l ( h l − h stat ) = j stat v l p stat = k l − k stat v r ( h stat − h r ) = j stat v r p stat = k stat − k r If we know h ( β , ν ) , j ( β , ν ) , p ( β , ν ) and k ( β , ν ) : Four equations, four unknowns β stat , ν stat , v l , v r ⇒ unique solution (?)
Page 12 Relativistic thermodynamics If j = p then: • Stress-energy tensor in state e − β H + ν P � � cosh θ (with β = β rest cosh θ , ν = β rest sinh θ , u = ): sinh θ T µ ν = k rest η µ ν + ( h rest + k rest ) u µ u ν where k rest = k ( T rest ) , h rest = h ( T rest ) (thermal averages) • Temperature dependence in thermal state e − β H : T d d T k ( T ) = h ( T ) + k ( T ) (thermal averages) ⇒ Thermal equation of state k ( T ) = F ( h ( T )) fixes everything. � h ( T ) d ℓ F ′ ( ℓ ) � k ( T ) d ℓ log T = ℓ + F − 1 ( ℓ ) = ℓ + F ( ℓ ) . Example: conformal relativistic fluid in d dimensions, k ( T ) = d h ( T ) .
Page 13 Refinement: pure hydrodynamics • Assume local generalized thermalization: β = β ( x, t ) and ν = ν ( x, t ) . • Hydrodynamic equations are ∂ t h ( β , ν ) + ∂ x j ( β , ν ) = 0 , ∂ t p ( β , ν ) + ∂ x k ( β , ν ) = 0 • Solve using step-profile initial condition • Shocks are weak self-similar solutions Further refinement: viscous hydrodynamics, entropy considerations • Viscosity terms (higher-derivatives)... • 2 nd law of thermodynamics (entropy production)... • Rarefaction waves (other self-similar solutions)...
Page 14 Example 1: 1+1-dimensional conformal field theory [...; Sotiriadis, Cardy 2008; Bernard, BD 2012; ...] Here j = p and k = h . Right- and left-moving combinations : h + = h + p h − = h − p = h + ( x − t ) , = h − ( x + t ) . 2 2 Same as linear-response calculation! t →∞ ⟨ h + ( − t ) − h − ( t ) ⟩ 0 = ⟨ h + ⟩ l − ⟨ h − ⟩ r = k l − k r j stat = lim . 2 Using CFT results, j stat = π ck 2 T 2 l − T 2 � � B . Verified numerically [Karrasch, Ilan, Moore r 12 � 2012] and experimentally [Jezouin, Parmentier, Anthore, Gennser, Cavanna, Jin, Pierre 2013] . Remarks: Inequalities saturated , v l = v r = v s = v . Sharp shock waves (up to non-universal scales) Steady state reached “immediately” (idem)
Page 15 Density matrix for steady state [Bernard, BD 2012; Bhaseen, BD, Lucas, Schalm 2015] : � β l + β r H + β l − β r � e − β l H + − β r H − = exp − P 2 2 H ± = total energy of right- / left- moving modes; boost of a thermal state with β rest = √ β l β r , tanh θ = β r − β l β l + β r .
Page 16 Example 2: T ¯ T -perturbation of CFT [Bernard, BD in preparation] � � dx ( T ( x ) + ¯ dx T ( x ) ¯ H = T ( x )) + g T ( x ) . Irrelevant perturbation: low-energy correction to universal behaviour. Currents at O ( g ) : h ( x ) = T ( x ) + ¯ T ( x ) + gT ( x ) ¯ p ( x ) = T ( x ) − ¯ T ( x ) , T ( x ) k ( x ) = h ( x ) + 2 gT ( x ) ¯ j ( x ) = p ( x ) + ∂ x ( · · · ) , T ( x ) + ∂ x ( · · · ) 2 ⟨ h ⟩ 2 ⇒ Thermodynamics is relativistic: ⟨ j ⟩ = ⟨ p ⟩ , eqn of state ⟨ k ⟩ = ⟨ h ⟩ + g Can determine exact thermal averages, e.g. h ( T ) = c π 1 − gc π 6 T 2 � 8 T 2 � .
Page 17 Speed of sound is v s ( T ) = 1 + gc π 12 T 2 , and we find • Shocks with velocities v l = v s ( T l ) and v r = v s ( T r ) • Current j stat = c π � T 2 l /v l − T 2 � r /v r : still left-right separation in agreement with 12 numerics [Karrasch, Ilan, Moore 2012] • Steady state density matrix with T rest = √ T l T r 1 − gc π � 48 ( T l − T r ) 2 � and tanh θ = T l − T r 1 − gc π ; we still have β = β l + β r � � 12 T l T r T l + T r 2 • Shocks of sublinear extent O ( t 1 / 3 ) (conjecture) √ • Generic approach O (1 / t ) (conjecture)
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