AdS COLLAPSE AND RELAXATION IN CLOSED QUANTUM SYSTEMS Javier Mas - PowerPoint PPT Presentation

AdS COLLAPSE AND RELAXATION IN CLOSED QUANTUM SYSTEMS Javier Mas Universidad de Santiago de Compostela Javier Abajo-Arrastia, Emilia da Silva, Esperanza López, J.M. & Alexandre Serantes. arXiv:1403.2632 and 1412.6002

MOTIVATION Classical Dynamics : one expects non-linear dynamics should lead on its own to thermalisation: - stochasticity threshold

MOTIVATION ¿How do closed quantum systems thermalise? - no dynamical chaos since time evolution is linear - discrete spectrum - how conserved quantities constraint relaxation h ˆ Fundamental questions: for O i ( t ) i - is there a stationary state being reached? - can it be described by a Gibbs ensemble? - are initial conditions erased? Addressable with : - recent advances in ultracold atom systems - exact results in integrable chains and CFT - AdS/CFT

MOTIVATION time dependent AdS/CFT Poincaré patch Heavy Ion Collisions Global AdS A S d − 1

PLAN - Relaxation in Closed Quantum Systems - Dynamics of Entanglement Entropy - Holographic Revivals

Relaxation in Quantum Closed Systems When addition conserved charges prethermalization, described by Generalized Gibbs Ensemble 2006 no erasure of initial condition f ( p x )

Relaxation in Quantum Closed Systems 2002 t rev t col H = U ˆ 2 ˆ n (ˆ n − 1) n e –¯ n (1 − cos Ut ) e i ¯ n sin Ut √ ψ ( t ) = ¯ t rev = 2 π 1 t col = U nU ¯

Relaxation in Quantum Closed Systems emergent Quantum systems described by a sum maximum speed of local Hamiltonians v L R 1972 2012 what is the speed ? v L R 2014

Relaxation in Quantum Closed Systems XY model 2010 t rev = p L v L R 2009 R.W. Robinett, “ Quantum wave packet revivals”, Physics Reports 392 (2004) 1-119

Quantum Field Theory - are there revivals? - at strong coupling ? at large c? - what observables are nice to monitor h ˆ A i ( t ) = h ψ ( t ) | ˆ i) local: A | ψ ( t ) i ii) non-local: ✓ ◆ P exp i Z - Wilson line W C = Tr A ~ C - entanglement entropy

Quantum Field Theory S A = − Tr ˆ ρ A = Tr B ˆ ˆ ρ A log ˆ ρ ρ A C. Holzey, F. Larsen, F. Wilczek 1994 Exact results for CFT in 1+1 dimensions P. Calabrese & J. Cardy 2003 ✓ l B A B ◆ S A = c 3 log T = 0 l ✏ | ψ i = | 0 i 3 log l c 8 l ⌧ β − → ✓ � > ✏ > ◆ A = c ⇡✏ sinh ⇡ l < T = 1 S β 3 log ✓ � β 6 = 0 − → � ◆ c + ⇡ c l � β > 3 log 3 � l + ... > − → : 2 ⇡✏

ENTANGLEMENT ENTROPY AFTER A QUENCH At t = 0 the Hamiltonian changes abruptly, leaving an excited state. Watch it evolve. µ H ( µ 0 ) , | ψ (0) i H (0) CF T , | ψ ( t ) i t π c P. Calabrese & J. Cardy 2005  t t < l/ 2 ,  6 τ 0   for l, t � τ 0 ⇠ 1 /m S l ( t ) ∼ π c  l l/ 2 < t   12 τ 0 reaches an extensive EE with a T ∼ 1 / 4 τ 0

ENTANGLEMENT ENTROPY AFTER A QUENCH kinematical argument: quench leaves an excited sea of quasiparticle pairs t = 0 entangled pairs fly apart at the speed of light 0 ≤ t ≤ l/ 2 B A B S l ( t ) ∼ t B A B S l ( t ) ∼ l

HOLOGRAPHIC ENTANGLEMENT ENTROPY entanglement entropy minimal surface homologous to A r γ A S. Ryu & T. Takayanagi 2006 S ( θ ) = Area( γ A ) 4 G d +1 A B B quench in CFT d shell collapse in AdS d+1 s analytic case: AdS d+2 -Vaidya: collapse of radiation shell d ✓ r 2 − m ( v ) ◆ ds 2 = − dv 2 + 2 drdv + r 2 X dx 2 i r d − 1 i =1 T vv = dm 0 ( v ) 2 r d m ( v ) t v r

HOLOGRAPHIC ENTANGLEMENT ENTROPY J. Abajo-Arrastia, J. Aparicio & E. López 2010 T. Albash & C. V. Johnson 2011 S l ( t ) S l ( t ) ∼ t S l ( t ) ∼ l t = l/ 2 are we seeing a light-cone-like effect for entangled quasiparticles? γ A A B B is the radial position of the shell dual to the entangled pair separation?

ENTANGLEMENT EVOLUTION IN COMPACT SPACE Confirmed for: CFT = 1+1 Free Fermion T. Takayanagi & T. Ugajin 2010 S l ( t ) t = π R CFT =minimal models John Cardy 2014 A θ ∈ (0 , 2 π )

ENTANGLEMENT EVOLUTION IN GLOBAL AdS Confirmed for: AdS- Vaydia leads to a direct black hole formation CFT = 1+1 Free Fermion T. Takayanagi & T. Ugajin 2010 S l ( t ) π γ A x 2 1 − dt 2 + dx 2 + sin 2 x d Ω 2 ds 2 = � � cos 2 x d − 1 A S d − 1

ENTANGLEMENT EVOLUTION IN GLOBAL AdS Collapse of a massless scalar field in AdS d+1 ✓ 1 2 κ 2 R + d ( d − 1) − 1 ◆ Z d d +1 x √ g 2 ∂ µ φ∂ µ φ S = π l 2 γ A x 2 homogeneous ansatz (Bizon & Rostworowski 2011) dx 2 1 ✓ ◆ ds 2 = − A ( x, t ) e − 2 δ ( x,t ) dt 2 + A ( x, t ) + sin 2 x d Ω 2 cos 2 x d 0 ≤ x ≤ π / 2 Equations of motion + boundary conditions 2 − x ) d φ ∞ + ... φ ( x, t ) ( π ∼ 2 − x ) d M + ... A ( x, t ) 1 − ( π ∼ 2 − x ) 2 d φ ∞ + ... δ ( x, t ) ( π ∼ A S d − 1 apparent horizon forms whenever A ( x h , t h ) = 0

ENTANGLEMENT EVOLUTION IN GLOBAL AdS φ (0 , x ) select a class of initial conditions π ✏ γ A x 2 σ π / 2 x 0 − 4 tan 2 ( π 2 − x ) � (0 , x ) = ✏ 12 ✓ ◆ cos d x ⇡ exp � 2 is related to the quench energy ✏ σ is related to the quenching time δ t σ δ t A M ( ✏ , � ) = ✏ 2 f ( � ) S d − 1

AdS 4 : COLLAPSES | φ | ✏ = 100 σ = 1 / 16 π / 2 x dx 2 1 ✓ ◆ ds 2 = − A ( x, t ) e − 2 δ ( x,t ) dt 2 + A ( x, t ) + sin 2 x d Ω 2 cos 2 x d 0 ≤ x ≤ π / 2

AdS 4 : COLLAPSES | φ | σ = 1 / 16 ✏ = 42 . 4 π / 2 x Amin 1.0 weak turbulence in action 0.8 0.6 the periodicity is ! ≥ π 0.4 0.2 t = proper time at boundary 1 2 3 4 6 5

φ (0 , x ) AdS 4 : COLLAPSES σ = 1 ✏ ✏ ∼ M 16 σ π / 2 Large BH x 0 M = 0 . 73 300 Small BH 0 bounces M = 0 . 014 M. Choptuik 1992 42 . 6 1 bounce P. Bizón & A. Rostworowski 2011 33 . 4 2 bounces 27 . 5 3 bounces x H

AdS 4 : ENTANGLEMENT ENTROPY Pre-collapse θ = 0 . 3 , 0 . 4 , ..., 1 . 4 θ J. Abajo-Arrastia, E da Silva, E. López, J.M. & A. Serantes.

AdS 4 : ENTANGLEMENT ENTROPY Pre-collapse θ = 0 . 3 , 0 . 4 , ..., 1 . 4 1 . 4 1 . 2 θ . . . 0 . 3 J. Abajo-Arrastia, E da Silva, E. López, J.M. & A. Serantes.

AdS 4 : ENTANGLEMENT ENTROPY Pre-collapse θ = 0 . 3 , 0 . 4 , ..., 1 . 4 1 . 4 1 . 2 θ . . . J. Abajo-Arrastia, E da Silva, E. López, J.M. & A. Serantes.

AdS 4 : ENTANGLEMENT ENTROPY Pre-collapse θ = 0 . 3 , 0 . 4 , ..., 1 . 4 1 . 4 1 . 2 θ . . . J. Abajo-Arrastia, E da Silva, E. López, J.M. & A. Serantes.

AdS 4 : ENTANGLEMENT ENTROPY Pre-collapse θ = 0 . 3 , 0 . 4 , ..., 1 . 4 1 . 4 1 . 2 θ . . . J. Abajo-Arrastia, E da Silva, E. López, J.M. & A. Serantes.

AdS 4 : ENTANGLEMENT ENTROPY Pre-collapse θ = 0 . 3 , 0 . 4 , ..., 1 . 4 1 . 4 1 . 2 θ . . . 0 . 3 J. Abajo-Arrastia, E da Silva, E. López, J.M. & A. Serantes.

AdS 4 : ENTANGLEMENT ENTROPY Pre-collapse θ = 0 . 3 , 0 . 4 , ..., 1 . 4 θ 1 . 4 1 . 2 θ . . . 0 . 3 0 . 3 t = θ = l/ 2 Javier Abajo-Arrastia, Emilia da Silva, Esperanza López, J.M. & Alexandre Serantes. arXiv:1403.2632

AdS 4 : SPECTRAL ANALYSIS

AdS 4 : ABSORTIVE PHASE

AdS 4 : ABSORTIVE PHASE A 1 horizon formation 1 bounce 2 bounces Schwarzchild BH 0.5 x 0 0 0.02 0.04 0.06

AdS 4 : ABSORTIVE PHASE Post-collapse: two regimes, elastic and absorptive 1 . 4 1 . 2 . . . 0 . 3 T ∼ 1 / ✏ 2

AdS 4 : CHANGE INITIAL CONDITIONS ✏ Bouncing phase: period changes with mass: two time scales Bose t rev t rev n = ¯ t col t col t col t rev collapse time: t col = θ revival time: t rev = f ( M )

AdS 4 : CHANGE INITIAL CONDITIONS σ - no collapse occurs for 0 . 3 ≤ σ ≤ 16 Buchel, Liebling & Lehner 1304.4166 Ρ � 0,x � 2.5 Σ� 0.05 Σ� 0.6 2.0 Σ� 20 1.5 1.0 0.5 x 1.0 0.5 1.5 - new period appears π / 3

AdS 3 ~ 1+1 CFT ✏ σ = 1 4 chaotic t r Collapse min ( A ) 1.0 Unstable (Caotic) M = 1 0.9 0.8 Bounce Stable (Integrable) 0.7 J. Jalmuzna & P. Bizón 2013 0.6 t 50 100 150 200 x H

AdS 3 ~ 1+1 CFT L Θ θ = π / 2 0.5 0.4 0.3 0.2 0.1 t 2 4 6 8 10 L Θ t r 40 0.8 110 M >1 30 0.6 70 20 0.4 30 1 1.0025 1.0046 0.2 10 t 10 20 30 40 M 0.0 0.2 0.4 0.6 0.8 1.0

AdS 3 ~ 1+1 CFT Z π / 2 Autocorrelation superperiod C ( t ) = 1 tan( x ) | ρ ( t, x ) ρ (0 , x ) | 1 / 2 dx M 0 M = 0.5 π / 2 it = 3 , it = 657 1 1 / 2 ⅆ x tan ( x )| ρ ( t,x ) ρ ( 0,x ) ρ ( t,x ) � M 0 0.7 1.0 0.6 0.8 0.5 0.4 0.6 0.3 0.4 0.2 0.1 0.2 x 0.5 1.0 1.5 it 200 400 600 800 1000 1200

PLANAR AdS 5 with a Hard Wall Craps, Lindgren, Taliotis, Vanhoof & Zhang 1406.1454 r h < r HW r = ∞

PLANAR AdS 5 with a Hard Wall Craps, Lindgren, Taliotis, Vanhoof & Zhang 1406.1454 r = ∞ r HW < r h

PLANAR AdS 5 with a Hard Wall da SIlva, J.M., A. Serantes, E. López 1508.xxxx r = ∞ r HW < r h

σ = 0 . 5 da SIlva, J.M., A. Serantes, E. López 1508.xxxx t collapse H M L σ = 0 . 5 40 σ = 0 . 4 30 σ = 0 . 2 20 σ = 0 . 3 σ = 0 . 1 10 M 1.10 1.20 1.05 1.15 1.25