Molecular dynamics simulation of entanglement growth in generalized - PowerPoint PPT Presentation

Molecular dynamics simulation of entanglement growth in generalized hydrodynamics M´ arton Mesty´ an SISSA Based on 1905.03206 Joint work with Vincenzo Alba M´ arton Mesty´ an (SISSA) Molecular dynamics of GHD & entanglement dynamics

Collaborators and Funding Vincenzo Alba (Univ. Amsterdam / D-ITP) M´ arton Mesty´ an (SISSA) Molecular dynamics of GHD & entanglement dynamics

Subject Quasiparticle picture of entanglement Soliton gas picture of Generalized evolution Hydrodynamics (GHD) Calabrese, Cardy (JStat 2005) Yoshimura, Doyon, Caux (PRL 2018) Alba, Calabrese (PNAS 2017) t M´ arton Mesty´ an (SISSA) Molecular dynamics of GHD & entanglement dynamics

1. Entanglement evolution Quantum quench ρ ( t ) = e − iHt ρ (0) e iHt ρ ( t = 0) := | Ψ 0 �� Ψ 0 | , Von Neumann entanglement entropy S A ( t ) = − Tr ρ A ( t ) ln ρ A ( t ) , ρ A ( t ) = Tr B ρ ( t ) B A B Typical behaviour of S A ( t ) S A ( t ) ∼ t ( v M t ≪ ℓ ) , S A ( t ) /ℓ ∼ S th ( t → ∞ ) Exact analytical results on the lattice: ◮ XY chain (free fermionic) Fagotti, Calabrese (PRA 2008) ◮ Kicked Ising chain (chaotic) Bertini, Kos, Prosen (PRX 2019) Effective description: ◮ Minimal membrane picture (non-integrable): Nahum, Ruhman, Vija, Haah (PRX 2017) ◮ Quasiparticle picture (integrable): Calabrese, Cardy (JStat 2005) M´ arton Mesty´ an (SISSA) Molecular dynamics of GHD & entanglement dynamics

2. Quenches in integrable quantum systems Homogeneous systems: GGE Inhomogeneous systems: GHD ◮ GHD: hydrodynamics with infinite ◮ Infinite number of (quasi)local number of continuity equations conserved charges: [ ˆ Q i , ˆ Q j ] = 0. q i ( x, t ) + ∂ x ˆ ∂ t ˆ j i ( x, t ) = 0 ◮ Expectation values of local operators in the steady state are described by a Bertini, Collura, De Nardis, Fagotti (2016) Generalized Gibbs Ensemble Castro-Alvaredo, Doyon, Yoshimura (2016) ◮ Recently confirmed in ultracold − � ρ GGE = 1 j β j ˆ Q j ˆ Z e atomic experiment ⟨ ^ O ( t )⟩ GGE t Rigol, Dunjko, Yurovski, Olshanii (2007) Schemmer, Bouchoule, Doyon, Dubail (2019) M´ arton Mesty´ an (SISSA) Molecular dynamics of GHD & entanglement dynamics

3. Thermodynamic limit of Bethe ansatz solvable systems Energy eigenstates .. . .. . k − 2 k k + 2 k − 1 k + 1 Energy eigenstates are enumerated by sets of (half)integer quantum numbers, which correspond to a set of rapidities λ j − 2 λ j − 1 λ j λ j + 1 λ j + 2 |{ I j } N j =1 � → |{ λ j } N j =1 � λ Densities Expectation values of conserved charges In the thermodynamic limit, eigenstates are characterized by the density of states, � particles and holes in rapidity space: � � ˆ q j � = dλρ n,λ q j,n ( λ ) n ρ t ,n,λ = ρ n,λ + ρ h ,n,λ Yang–Yang entropy ( ∼ ln # of eigenstates) Bethe–Gaudin–Takahashi equations � � � s YY = dλρ t ,λ ln ρ t ,λ � ρ t ,n,λ = a n ( λ ) − dµT nm ( λ − µ ) ρ m,µ n m − ρ λ ln ρ λ − ρ h ,λ ln ρ h ,λ Review: M. Takahashi (Cambridge University Press, 1999) M´ arton Mesty´ an (SISSA) Molecular dynamics of GHD & entanglement dynamics

4. The quasiparticle picture of entanglement evolution Calabrese, Cardy (JStat 2005) Generic integrable systems: Alba, Calabrese (PNAS 2017) ◮ Valid at large space-time scales ◮ Each segment [ x, x + ∆ x ] is a source of quasiparticles ◮ In the quenches considered here, quasiparticles are emitted in pairs with rapidity ± λ ◮ Different configurations are possible Bertini, Tartaglia, Calabrese (JStat 2018) ◮ Quasiparticles move linearly with the effective velocity v n,λ ◮ A pair contributes to the entanglement iff one of them is in A and the other is outside ◮ Each shared pair contributes to the entanglement s n,λ , the Yang–Yang entropy density of the GGE ◮ S A ( t ) is obtained by counting shared pairs and integrating over all modes     � �   � � S A ( t ) ∼ 2 t dλ | v n,λ | s n,λ + ℓ dλs n,λ   n n   2 | v n,λ | t<ℓ 2 | v n,λ | t>ℓ M´ arton Mesty´ an (SISSA) Molecular dynamics of GHD & entanglement dynamics

5. The quasiparticle velocities Bonnes, Essler, Lauchli (PRL 2014) .. . k − 2 k + 2 .. . k k − 1 k + 1 λ j − 2 λ j − 1 λ j μ λ j + 1 λ j + 2 λ ◮ When a quasiparticle is added, the rapidities of other quasiparticles are shifted ◮ This results in a dressing of charges N � q dr � q j,n (˜ λ k ) − q k,n ( λ k ) � j,n ( µ ) = q j,n ( µ ) + k =1 ◮ The effective velocities of quasiparticles are v n,λ = e dr ′ n ( λ ) p dr ′ n ( λ ) ◮ In the TDL, � dµT nm ( λ − µ ) � v n,λ = v bare n,λ + ρ m ( µ )( v n,λ − v m,µ ) a n ( λ ) m M´ arton Mesty´ an (SISSA) Molecular dynamics of GHD & entanglement dynamics

6. Generalized hydrodynamics (at ballistic scale) Infinite number of conservation laws + local quasi-stationarity q i ( x, t ) + ∂ x ˆ ∂ t ˆ j i ( x, t ) = 0 Quasi-stationary GGE Continuity equations for modes ∂ t ρ n,λ ( x, t ) + ∂ x ( v n,λ ( x, t ) ρ n,λ ( x, t )) = 0 Castro-Alvaredo, Doyon, Yoshimura (PRX 2016) Bertini, Collura, De Nardis, Fagotti (PRL 2016) M´ arton Mesty´ an (SISSA) Molecular dynamics of GHD & entanglement dynamics

7. An inhomogeneous setting XXZ Heisenberg spin chain L � � j +1 + S y j S y � S x j S x j +1 + ∆ S z j S z H = j +1 j =1 Bipartite quantum quench - extension of quasiparticle picture | Ψ( t ) � = e − iHt | Ψ 0 � , | Ψ 0 � = | Ψ 0 , L � ⊗ | Ψ 0 , R � Example of an initial state � 1 + T � ( | ↑↓� ) ⊗ L/ 2 | Ψ L � = | N´ eel � ≡ √ 2 � 1 + T �� | ↑↓� − | ↓↑� � ⊗ L/ 2 | Ψ R � = | dimer � ≡ √ √ 2 2 M´ arton Mesty´ an (SISSA) Molecular dynamics of GHD & entanglement dynamics

8. Analytical vs. numerical approach The effective velocities � dµT nm ( λ − µ ) � v n,λ ( ζ ) = v bare n,λ ( ζ ) + ρ m,µ ( ζ )( v n,λ ( ζ ) − v m,µ ( ζ )) a n ( λ ) m Possibilities for following quasiparticles & computing S A ( t ) ◮ Analytically, by solving V. Alba, B. Bertini, M. Fagotti (1903.00467) d dtX n,λ ( x, t ) = v n,λ ( X n,λ ( t, x ) , t ) ◮ Numerically, using the flea gas picture of GHD MM, V. Alba (1905.03206) M´ arton Mesty´ an (SISSA) Molecular dynamics of GHD & entanglement dynamics

9. The flea gas picture of GHD The flea gas algorithm for simulating GHD: LL: Yoshimura, Doyon, Caux (PRL 2018) 1. Generate randomly a configuration of quasiparticles according to the initial distributions ρ n,λ ( ±∞ ) 2. Move the particles linearly with their bare velocities v bare n,λ 3. When two particles ( n, λ ) (on the left) and ( m, µ ) (on the right) meet, make them jump with + T nm ( λ − µ ) for ( n, λ ) a n ( λ ) − T mn ( µ − λ ) for ( m, µ ) a m ( µ ) 4. After the simulation time T has elapsed, compute profiles of charges / entropy in the configuration and store it 5. Repeat the above many times ( ∼ 10 2 − 10 5 ) and take average of quantities over realizations M´ arton Mesty´ an (SISSA) Molecular dynamics of GHD & entanglement dynamics

10. The velocities in the flea gas (heuristic argument) t ◮ In a time ∆ t , the number of times a particle ( n, λ ) meets particles ( m, µ ) is (on average) ρ m,µ | v n,λ ( ζ ) − v m,µ ( ζ ) | ∆ t ◮ At each scattering, the particle ( n, λ ) jumps sgn ( v n,λ ( ζ ) − v m,µ ( ζ )) · T nm ( λ − µ ) a n ( λ ) Effective velocities of flea gas particles � dµT nm ( λ − µ ) � v n,λ ( ζ ) = v bare n,λ ( ζ ) + ρ m,µ ( ζ )( v n,λ ( ζ ) − v m,µ ( ζ )) a n ( λ ) m This is the same equation as the effective velocity equation in GHD. M´ arton Mesty´ an (SISSA) Molecular dynamics of GHD & entanglement dynamics

11. Testing effective velocities in the XXZ flea gas Rightmost panels: analytical result from Piroli, De Nardis, Collura, Bertini, Fagotti (2017) M´ arton Mesty´ an (SISSA) Molecular dynamics of GHD & entanglement dynamics

12. Computing entanglement entropy 1. Prepare the initial state with particle pairs with rapidity ± λ i 2. For each pair, compute the Yang-Yang entropy contribution s ( λ i ) 3. Evolve the flea gas in time 4. Find the “shared pairs” and sum their contribution � shared pairs s ( λ i ) 5. Repeat many times and compute the average � � � S A ( t ) = s ( λ i ) shared pairs M´ arton Mesty´ an (SISSA) Molecular dynamics of GHD & entanglement dynamics

13. Test of the flea gas picture against analytical results I. Homogeneous global quench from the N´ eel state M´ arton Mesty´ an (SISSA) Molecular dynamics of GHD & entanglement dynamics

14. Test of the flea gas picture against analytical results II. Bipartite quench from (tilted N´ eel ⊗ dimer) Initial rate: S ′ = � � dλ � sgn( v n,λ (0)) s n,λ (0) + | v n,λ ( σ ∞ ) | s n,λ ( σ ∞ ) � n Analytical: Alba, Bertini, Fagotti (1905.03206) M´ arton Mesty´ an (SISSA) Molecular dynamics of GHD & entanglement dynamics

15. Mutual information I A 1 : A 2 = S A 1 + S A 2 − S A 1 ∪ A 2 M´ arton Mesty´ an (SISSA) Molecular dynamics of GHD & entanglement dynamics