Exact results for quantum quenches in the Ising chain Maurizio - PowerPoint PPT Presentation

Florence, 22 May 2012 Exact results for quantum quenches in the Ising chain Maurizio Fagotti ( Oxford ) Pasquale Calabrese ( Pisa ) Fabian Essler ( Oxford )

OUTLINE Introduction Correlation functions after a sudden quench in the TFIC Thermalization vs. GGE ... towards a stationary state Dynamics? MF and P. Calabrese, Phys. Rev. A 78 , 010306(R) (2008) P. Calabrese, F. H. L. Essler, and MF, Phys. Rev. Lett. 106 , 227203 (2011) arXiv:1204.3911 (2012) arXiv:1205.2211 (2012) F. H. L. Essler, S. Evangelisti, and MF, in preparation F. H. L. Essler and MF, in preparation

Quantum systems out of equilibrium main theoretical questions: • Correlation functions • Are there emergent phenomena? “macroscopic | Ψ ( t ) � = e − i H t | Ψ 0 � non-equilibrium steady states? description” • How to evaluate averages? • many-body physics with ultracold atomic gases interaction and external potentials can be changed dynamically • weakly coupled to the environment coherence for long times (also mesoscopic heterostructures, quantum dots, ... ) not only academic

Quantum Newton’s Cradle Kinoshita, Wenger, and Weiss ( 2006 ) 1D thousands of oscillations non-thermal momentum distribution (no thermalization?) 3D a few oscillations and then thermalization 1D close to integrability (without trap) N ∂ 2 � � H N = − + 2 c δ ( x j − x k ) ∂ x 2 j j = 1 N ≥ j > k ≥ 1 basic features (dimensionality, integrability, ...) play a central role in non-equilibrium physics

Sudden quench ∣ ϕ 0 ⟩ consider a system in the ground state of a local Hamiltonian depending on certain parameters (magnetic field, interaction) ∣ ⟩ H ( h 0 ,... ) ∣ ϕ 0 ⟩ = E ( h 0 ,... ) ∣ ϕ 0 ⟩ G.S. at a given time the parameters are changed ∣ ϕ t ⟩ = e − i H ( h,... ) t ∣ ϕ 0 ⟩ extensive excess of energy global quench [ H ( h 0 ,... ) ,H ( h,... )] ≠ 0 � ϕ t | ˆ O ( r 1 , . . . , r i , . . . ) | ϕ t � time evolution of (local) correlation functions ℓ S time evolution of subsystems S ( ) r 1 r 2 r 3 ( ) //////////////////////// ///////////////////// ρ ( t ) ≡ e − i H ( h,... ) t ∣ ϕ 0 ⟩⟨ ϕ 0 ∣ e i H ( h,... ) t ρ S ( t ) ≡ Tr ¯ S ρ ( t ) � → late-time regime and emergence of stationary behavior t → ∞ Tr [ ρ S ( t ) ˆ O ] ∃ lim t →∞ ρ S ( t ) ? ∃ lim ? v m ax t ≫ ℓ S , r

Transverse Field Ising chain simplest paradigm of a quantum phase transition central charge c= 1 /2 L � [ σ x j σ x j + 1 + h σ z H = − J j ] j = 1 � σ x � = 0 � σ x � � = 0 0 1 h spontaneously order parameter broken Z 2 symmetry: rotation of around z-axis π Mapping to free fermions �� � σ z l , H R ( N S ) = 0 two fermionic sectors l  j σ y � � σ z j σ z σ z a 2 l = a 2 l − 1 =  l σ z l σ z l l  H = 1 − � H R + 1 + �  l l H N S j < l j < l 2 2  { a l , a n } = 2 δ ln  a l H R ( N S ) a n � ln H R ( N S ) = 4 Jordan-Wigner transformation l,n Wick theorem imaginary antisymmetric

Quench dynamics in the TFIC ordered phase disordered phase H ( h 0 ) → H ( h ) 0 1 h Approach I: Block-Toeplitz determinants: (in the thermodynamic limit) expectation values of even operators Pfaffians of structured matrices multi-dimensional � ϕ 0 | σ x 1 ( t ) σ x ℓ + 1 ( t ) | ϕ 0 � ∼ Block-Toeplitz matrix stationary phase approximation Approach II: “Form-Factor” Sums: 1. large finite volume L √ | ¯ 0 � R ± | ¯ � � 0 � N S / 2 (ordered phase) 2. initial state | ¯ 0 � N S (disordered phase) 3. express this in terms of the final Bogoliobov fermions � θ h ( p ) − θ h 0 ( p ) � � � � p α † | ¯ K ( p ) α † 0 � R ( N S ) = exp i | 0 � R ( N S ) K ( p ) = tan − p 2 0 < p ∈ R ( N S ) 4. Lehmann representation in terms of the final Bogolioubov fermions n l 1 K ( k j ) e 2 i t ε k j N S � ¯ 0 | σ x 1 ( t ) | ¯ � � � � K ( p i ) e − 2 i t ε p i N S � { − k, k} n | σ x 0 � R = 1 |{ p, − p} l � R n ! l ! j = 1 i = 1 l,n ≥ 0 k 1 , . . . k n known exactly for p 1 , . . . , p l K ( p ) 5. as expansion parameter: low density of excitations the lattice model

One-point function: ∆ k = θ h ( k ) − θ h 0 ( k ) different from zero only for quenches from the ordered phase ... to the ordered phase 0 1 h � π � � d k 1 2 exp � σ x i ( t ) � ≃ ( C x π ε ′ t h ( k ) ln | cos ∆ k | FF ) 0 ... to the disordered phase 0 1 h equation cos ∆ k 0 = 0, � π � � d k 1 1 2 [1 + cos(2 ε h ( k 0 ) t + α ) + . . . ] 2 exp � σ x i ( t ) � = ( C x π ε ′ t h ( k ) ln | cos ∆ k | FP ) 0

less clear behavior in quenches across the critical point Two-point function: ... within the ordered phase 0 1 h � π d k � �� ρ xx F F ( ℓ , t ) ≃ C x � 2 ε ′ h ( k ) t − ℓ FF exp ℓ π ln | cos ∆ k | θ H 0 � π d k � �� � π ε ′ ℓ − 2 ε ′ 2 t h ( k ) ln | cos ∆ k | θ H h ( k ) t × exp 0 ... within the disordered phase 0 1 h � π 4 √ dk K ( k ) 1 ρ xx P P ( ℓ , t ) ≃ ρ xx P P ( ℓ , ∞ ) + ( h 2 − 1) 4 J 2 h sin(2 t ε k − k ℓ ) × π ε k − π � π � dp � � π K 2 ( p ) � ℓ + θ H ( ℓ − 2 t ε ′ p )[2 t ε ′ + . . . exp − 2 p − ℓ ] 0

Stationary state //////////////////////// ( ) ///////////////////// S ρ ( t ) = e − i H ( h,... ) t | ϕ 0 � � ϕ 0 | e i H ( h,... ) t − → ρ S ( t ) = Tr ¯ S ρ ( t ) • at large times after a quench correlations display stationary behavior • entanglement entropies of subsystems become independent of time effective density matrix? t → ∞ Tr [ ρ S ( t ) ˆ O ] t →∞ Tr ¯ lim S ρ ( t ) → ¯ ∃ lim ? ρ S Eigenstate Thermalization? Deutsch ( 1 991 ) , Srednicki ( 1 994 ) intuitive argument: in an infinite system the rest of the system could act as a bath ≫ ∣ ∣ ≫ M. C. Escher ( 1 961 ) integrable systems: e − H / T ef f ? non - integrable systems? ¯ conserved quantities! ≈ ρ Tr e − H / T ef f ( more parameters ) ∃

Quenches to integrable systems operators commuting with the final Hamiltonian are stationary: expectation values are conserved [ H, I ] = 0 ⇒ � ϕ t |I | ϕ t � = � ϕ 0 |I | ϕ 0 � charge with conservation law also local density for the reduced system persistent oscillations at late times stationary behavior: each local charge gives a constraint ρ ∞ = ρ ∞ ( I 1 , I 2 , . . . ) M. C. Escher ( 1 956 ) e − ∑ m β m I m ∣ S ∣ → ∞ ¯ conjecture ( GGE ) : Tr [ e − ∑ m β m I m ] ρ S � � � → Rigol, Dunjko, Yurovsky , and Olshanii ( 2007 )

Infinite time after a quench in the TFIC ordered phase disordered phase H ( h 0 ) → H ( h ) 0 1 h (in the thermodynamic limit) equal-time correlations are stationary (almost all excitations have nonzero � σ z H = velocity (no localized excitations); counterexample: quench to ) l l exponential decay with distance (possibly “dressed” with power-law and oscillatory factors) ρ xx ( ℓ ≫ 1 , t = ∞ ) = C x ( ℓ ) e − ℓ / ξ 0 1 h C x ( ℓ ) ∼ cos( κℓ ) C x ( ℓ ) ∼ ℓ − 3 / 2 c ( ℓ ≫ 1 , t = ∞ ) ≃ C z ℓ − α z e − ℓ / ξ z α z = 1 + θ H ( | log h| − | log h 0 | ) ρ zz 2 local properties described exactly by a GGE (equal-time fermionic two-point functions have thermal structure corresponding to an Hamiltonian that commutes with the final one)

“Pair Ensemble” vs GGE: the role of locality n k = α † k α k � � � p α † | ¯ K ( p ) α † 0 � R ( N S ) = exp i | 0 � R ( N S ) − p � n k n − k O � t = � n k O � t 0 < p ∈ R ( N S ) off-diagonal elements factorization in Pair Ensemble do not contribute pair of quasiparticles different from GGE ! (GGE in the Ising model is Gaussian) Pair Ensemble and GGE are locally indistinguishable my feeling (when GGE works) : we CAN construct the effective density matrix considering only local charges

... towards the stationary state: “practical meaning” of the infinite time limit fundamental question: how long we need to wait to “see” the observables converging to their stationary values finite systems : GGE time must be • smaller (enough) than the system’s size (in units of the maximal velocity) • much larger (how much?) than the typical length (distance in 2-point functions, subsystem’s length, ...) strong limits to the lengths of subsystems that relax! c ( ℓ , ∞ ) + E z ( t ) ℓ e − ℓ / ˜ ξ z + D z ( t ) ℓ 2 ρ zz c ( ℓ , t ) ∼ ρ zz max t 3 + . . . v 2 t 3 / 2 generally J t ≫ ( ℓ / a ) α e κℓ / a at fixed distance, exponentially large times the behavior depends on the observable

Dynamical correlations at infinite times after the quench numerical analysis + general physical arguments t →∞ � O 1 ( t + τ ) O 2 ( t ) � = Tr[ ρ GGE O 1 ( τ ) O 2 ] lim FDT with less symmetries w.r.t. the thermal case 0 1 h x = 30 L = 512 O ( x/ L ) in practice ℓ , τ ≪ t � L →∞ exact results? C ( x, τ ) = � σ x ( L − x ) / 2 ( t + τ ) σ x ( L + x ) / 2 ( t ) �

Conclusions Quantum quenches display a rich phenomenology Many open problems: ... thermalization (non-integrable systems) ... GGE (integrable systems) Importance to have analytic results More general initial states? ... and when subsystems do not relax? Thank you for your attention