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

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

AdS COLLAPSE AND RELAXATION IN CLOSED QUANTUM SYSTEMS

MOTIVATION

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

• stochasticity threshold

MOTIVATION

Fundamental questions: for

• 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
• no dynamical chaos since time evolution is linear
• discrete spectrum
• how conserved quantities constraint relaxation

¿How do closed quantum systems thermalise?

h ˆ Oi(t)i

MOTIVATION

A

Sd−1

Poincaré patch Global AdS Heavy Ion Collisions time dependent AdS/CFT

PLAN

• Relaxation in Closed Quantum Systems
• Dynamics of Entanglement Entropy
• Holographic Revivals

no erasure of initial condition

When addition conserved charges prethermalization, described by Generalized Gibbs Ensemble

f(px)

Relaxation in Quantum Closed Systems

2006

Relaxation in Quantum Closed Systems

ˆ H = U 2 ˆ n(ˆ n − 1)

ψ(t) = √ ¯ n e–¯

n(1−cos Ut)ei¯ n sin Ut

trev = 2π U

trev

tcol = 1 ¯ nU

tcol

2002

Relaxation in Quantum Closed Systems

vL

R

what is the speed ?

2012

2014

Quantum systems described by a sum

• f local Hamiltonians

emergent maximum speed

vL

R

1972

trev = p L vL

R

XY model

R.W. Robinett, “Quantum wave packet revivals”, Physics Reports 392 (2004) 1-119

Relaxation in Quantum Closed Systems

2010 2009

• are there revivals?
• what observables are nice to monitor

ii) non-local:

• Wilson line
• entanglement entropy

i) local:

h ˆ Ai(t) = hψ(t)| ˆ A|ψ(t)i

WC = Tr ✓ P exp i ~ Z

C

A ◆

• at strong coupling ? at large c?

Quantum Field Theory

T = 1 β 6= 0

Sβ

A = c

3 log ✓ ⇡✏ sinh ⇡l

• ◆

− → 8 > > < > > :

l⌧β

− → c 3 log l ✏

lβ

− → c 3 log ✓ 2⇡✏ ◆ + ⇡c 3 l + ...

SA = c 3 log ✓ l ✏ ◆ T = 0

B B

Exact results for CFT in 1+1 dimensions

A

l

|ψi = |0i

• C. Holzey, F. Larsen, F. Wilczek 1994
• P. Calabrese & J. Cardy 2003

ˆ ρA = TrB ˆ ρ

SA = −Tr ˆ ρA log ˆ ρA

Quantum Field Theory

µ

• P. Calabrese & J. Cardy 2005

reaches an extensive EE with a

Sl(t) ∼        πc 6τ0 t t < l/2, πc 12τ0 l l/2 < t

l, t τ0 ⇠ 1/m

for

T ∼ 1/4τ0

H(0)CF T , |ψ(t)i

At t = 0 the Hamiltonian changes abruptly, leaving an excited state. Watch it evolve.

H(µ0) , |ψ(0)i

t

ENTANGLEMENT ENTROPY AFTER A QUENCH

B A B B A B

kinematical argument: quench leaves an excited sea of quasiparticle pairs entangled pairs fly apart at the speed of light t = 0

0 ≤ t ≤ l/2 Sl(t) ∼ l Sl(t) ∼ t

ENTANGLEMENT ENTROPY AFTER A QUENCH

A

B

B

entanglement entropy minimal surface homologous to A

s

quench in CFTd shell collapse in AdSd+1

r

HOLOGRAPHIC ENTANGLEMENT ENTROPY

• S. Ryu & T. Takayanagi 2006

analytic case: AdSd+2 -Vaidya: collapse of radiation shell

ds2 = − ✓ r2 − m(v) rd−1 ◆ dv2 + 2drdv + r2

d

X

i=1

dx2

i

Tvv = dm0(v) 2rd

m(v)

r

v

t

γA

S(θ) = Area(γA) 4Gd+1

A

B

B

γA

• J. Abajo-Arrastia, J. Aparicio & E. López 2010
• T. Albash & C. V. Johnson 2011

Sl(t)

Sl(t) ∼ t Sl(t) ∼ l

is the radial position of the shell dual to the entangled pair separation? are we seeing a light-cone-like effect for entangled quasiparticles?

HOLOGRAPHIC ENTANGLEMENT ENTROPY

t = l/2

t = π θ ∈ (0, 2π)

A

CFT = 1+1 Free Fermion

• T. Takayanagi & T. Ugajin 2010

Confirmed for:

Sl(t)

RCFT =minimal models

John Cardy 2014

ENTANGLEMENT EVOLUTION IN COMPACT SPACE

A

γA

Sd−1

x

π 2

AdS-Vaydia leads to a direct black hole formation CFT = 1+1 Free Fermion

• T. Takayanagi & T. Ugajin 2010

Confirmed for:

Sl(t)

ENTANGLEMENT EVOLUTION IN GLOBAL AdS

ds2 = 1 cos2 x

• −dt2 + dx2 + sin2 x dΩ2

d−1

A

γA

Sd−1

x

π 2

S = Z dd+1x√g ✓ 1 2κ2 R + d(d − 1) l2 − 1 2∂µφ∂µφ ◆

Collapse of a massless scalar field in AdSd+1

ds2 = 1 cos2 x ✓ −A(x, t)e−2δ(x,t)dt2 + dx2 A(x, t) + sin2 x dΩ2

d

◆

homogeneous ansatz (Bizon & Rostworowski 2011)

0 ≤ x ≤ π/2

apparent horizon forms whenever

A(xh, th) = 0

Equations of motion + boundary conditions φ(x, t) ∼ ( π

2 − x)dφ∞ + ...

A(x, t) ∼ 1 − ( π

2 − x)dM + ...

δ(x, t) ∼ ( π

2 − x)2dφ∞ + ...

ENTANGLEMENT EVOLUTION IN GLOBAL AdS

A

γA

Sd−1

x

π 2

x

π/2

select a class of initial conditions

ENTANGLEMENT EVOLUTION IN GLOBAL AdS

(0, x) = ✏ 12 ⇡ exp ✓ −4 tan2( π

2 − x)

2 ◆ cosd x

φ(0, x)

✏

is related to the quench energy

✏

δt

σ

σ

is related to the quenching time δt

σ

M(✏, ) = ✏2f()

AdS4 : COLLAPSES

σ = 1/16

✏ = 100

x

π/2

|φ|

ds2 = 1 cos2 x ✓ −A(x, t)e−2δ(x,t)dt2 + dx2 A(x, t) + sin2 x dΩ2

d

◆

0 ≤ x ≤ π/2

1 2 3 4 5 6 t = proper time at boundary 0.2 0.4 0.6 0.8 1.0 Amin

the periodicity is !

≥ π

weak turbulence in action

AdS4 : COLLAPSES

✏ = 42.4

x

π/2

|φ|

σ = 1/16

✏

σ = 1 16

• P. Bizón & A. Rostworowski

2011

xH

42.6 33.4 27.5

Small BH

300

Large BH

M = 0.73 M = 0.014

AdS4 : COLLAPSES

• M. Choptuik 1992

∼ M

x

π/2

φ(0, x)

σ

✏

1 bounce

2 bounces 0 bounces 3 bounces

• J. Abajo-Arrastia, E da Silva, E. López, J.M. & A. Serantes.

Pre-collapse

AdS4 : ENTANGLEMENT ENTROPY

θ = 0.3, 0.4, ..., 1.4

θ

• J. Abajo-Arrastia, E da Silva, E. López, J.M. & A. Serantes.

Pre-collapse

0.3

1.4

1.2

. . .

AdS4 : ENTANGLEMENT ENTROPY

θ = 0.3, 0.4, ..., 1.4

θ

• J. Abajo-Arrastia, E da Silva, E. López, J.M. & A. Serantes.

Pre-collapse

1.4

1.2

. . .

AdS4 : ENTANGLEMENT ENTROPY

θ = 0.3, 0.4, ..., 1.4

θ

• J. Abajo-Arrastia, E da Silva, E. López, J.M. & A. Serantes.

Pre-collapse

1.4

1.2

. . .

AdS4 : ENTANGLEMENT ENTROPY

θ = 0.3, 0.4, ..., 1.4

θ

• J. Abajo-Arrastia, E da Silva, E. López, J.M. & A. Serantes.

Pre-collapse

1.4

1.2

. . .

AdS4 : ENTANGLEMENT ENTROPY

θ = 0.3, 0.4, ..., 1.4

θ

• J. Abajo-Arrastia, E da Silva, E. López, J.M. & A. Serantes.

Pre-collapse

0.3

1.4

1.2

. . .

AdS4 : ENTANGLEMENT ENTROPY

θ = 0.3, 0.4, ..., 1.4

θ

Pre-collapse

0.3

1.4

1.2

. . .

AdS4 : ENTANGLEMENT ENTROPY

Javier Abajo-Arrastia, Emilia da Silva, Esperanza López, J.M. & Alexandre Serantes. arXiv:1403.2632

θ = 0.3, 0.4, ..., 1.4

θ

0.3

θ

t = θ = l/2

AdS4 : SPECTRAL ANALYSIS

AdS4 : ABSORTIVE PHASE

horizon formation 1 bounce 2 bounces Schwarzchild BH

0.02 0.04 0.06

x

0.5 1

A

AdS4 : ABSORTIVE PHASE

Post-collapse: two regimes, elastic and absorptive

0.3

1.4 1.2

. . .

T ∼ 1/✏2

AdS4 : ABSORTIVE PHASE

Bouncing phase: period changes with mass: two time scales

AdS4 : CHANGE INITIAL CONDITIONS

trev

tcol

Bose

trev

tcol trev tcol = ¯ n collapse time: tcol = θ revival time: trev = f(M)

✏

0.5 1.0 1.5

x

0.5 1.0 1.5 2.0 2.5

Ρ0,x

Σ20 Σ0.6 Σ0.05

• no collapse occurs for 0.3 ≤ σ ≤ 16

Buchel, Liebling & Lehner 1304.4166

• new period appears

π/3

AdS4 : CHANGE INITIAL CONDITIONS σ

50 100 150 200 t 0.6 0.7 0.8 0.9 1.0 min(A)

✏

xH

M = 1

Bounce σ = 1 4

• J. Jalmuzna & P. Bizón 2013

Stable (Integrable)

Unstable (Caotic)

Collapse

AdS3 ~ 1+1 CFT

chaotic

tr

2 4 6 8 10

t

0.1 0.2 0.3 0.4 0.5

LΘ

M >1

1 1.0025 1.0046 30 70 110 0.0 0.2 0.4 0.6 0.8 1.0

M

10 20 30 40

tr

10 20 30 40

t

0.2 0.4 0.6 0.8

LΘ

θ = π/2

AdS3 ~ 1+1 CFT

Autocorrelation superperiod

0.5 1.0 1.5 x 0.1 0.2 0.3 0.4 0.5 0.6 0.7 ρ(t,x)

it=3 , it=657 200 400 600 800 1000 1200 it 0.2 0.4 0.6 0.8 1.0 1 M

• π/2

tan(x)|ρ(t,x)ρ(0,x)

1/2ⅆx

M=0.5

C(t) = 1 M Z π/2 tan(x)|ρ(t, x)ρ(0, x)|1/2dx

AdS3 ~ 1+1 CFT

PLANAR AdS5 with a Hard Wall

rh < rHW

r = ∞

Craps, Lindgren, Taliotis, Vanhoof & Zhang 1406.1454

PLANAR AdS5 with a Hard Wall

r = ∞

rHW < rh Craps, Lindgren, Taliotis, Vanhoof & Zhang 1406.1454

PLANAR AdS5 with a Hard Wall

r = ∞

rHW < rh da SIlva, J.M., A. Serantes, E. López 1508.xxxx

1.05 1.10 1.15 1.20 1.25 M 10 20 30 40 t collapseHML

σ = 0.1

σ = 0.2 σ = 0.3 σ = 0.4

σ = 0.5 σ = 0.5

da SIlva, J.M., A. Serantes, E. López 1508.xxxx

Thin Shell Oscillations in AdS

Veff = 1 + r2

s − 1

2mr1−d

s

− m2 4M 2 r2α

s

− 1 4M 2r−2(d−1+α)

s

r = ru r = rl

0.5 1.0 1.5 2.0

r

• 0.4
• 0.2

0.2 0.4

Veff

0.0 0.2 0.4 0.6 0.8 1.0 m 0.1 0.2 0.3 0.4

M

p = ⇣α d ⌘ σ Equation of state Effective potential

Keranen, Nishimura, Stricker, Taanila, & Vuorinen, 1405.7015 Serantes & J.M. 1507.xxxx

• landscape of evolutions in global AdS for different initial conditions

and dimensions ¿ ? routes to thermalization in closed quantum systems at strong coupling

• revivals allow for precission measurements
• search for generic features: two regimes, two timescales, collapse

blocking, additional frequency, etc.

• need some exact results from the QFT side at strong coupling and

large c in different dimensions and geometries

• collapse in AdS3 may shed light on the role of integrability breaking
• the massive case needs better understanding/formulation

Conclusions & outlook

π/d

CONCLUSION

A

Sd−1

Poincaré patch Global AdS Heavy Ion Collisions AdS/CFT Revivals

thank you