1. Entanglement Tsunami
Hong Liu

2. Josephine Suh
Mark Mezei

3. Based on
HL and Josephine Suh, , PRL 112, (2014)
HL and Josephine Suh, , PRD 89, (2014)
HL, Mezei, Suh, unpublished, Casini, unpublished
Casini, Hubeny, HL, Maxfield, Mezei, Suh, to appear
See also
Hyungwon Kim and Huse arXiv:
Hartman and Maldacena arXiv:
Shenker and Stanford arXiv: ,
Hubeny and Maxfield arXiv:

4. Entanglement generationB C D
How fast can entanglement be generated?
In most physical systems: Local Hamiltonian
: UV cutoff

5. Small incremental entangling conjecture/theoremB C D
SA : entanglement
entropy of A
Dur, Vidal et al, Bravyi, Kitaev
Bennett et al, Van Acoleyen, Marien, Verstraete
For spin systems:
dC : dimension of Hilbert space of C

6. For more general quantum systems, e.g. a QFTB C D
For more general quantum systems, e.g. a QFT
In this talks we will describe some hints.

7. A simple setup: global quenches
Start with a QFT in the
ground state.
2. At t=0 in a very short time
interval inject a uniform energy
density
initial state homogeneous, isotropic,
entanglement properties as vacuum
3. The system evolves to (thermal) equilibrium
Also a question of interest for thermalization.

8. In equilibrium, system behaves macroscopically as a thermal state, with entanglement entropy disguised as thermal entropy:
VA : volume of region A
Essentially all d.o.f. inside A becomes long ranged
entangled with those outside A.

9. A A t=0 equilibrium essentially almost all d.o.f. long range entangled
no long range
entanglement
almost all d.o.f. long range entangled

10. Previous results in (1+1)-d CFTs
Calabrese and Cardy 2R
ΔS/Seq
t/R tS
1. Linear growth with time
(not too early and too late)
2. Slope = 1
3. Can be reproduced by
free particles

11. Special techniques in one spatial dimension do not
apply to higher dimensions:
Equilibration processes: complicated non-equilibrium many-body dynamics, generally out of theoretical control.
Entanglement entropy is notoriously difficult to calculate even for simple regions in the vacuum of a free theory, not to mention for general regions in interacting theories far from equilibrium.

12. String theory to the rescue!

13. …………… Earlier work: Hubeny, Rangamani, Takayanagi: arXiv:0705.0016
Abajo-Arrastia, Aparicio and Lopez, arXiv:
Albash and Johnson, arXiv:
Balasubramanian, Bernamonti, de Boer, Copland, Craps, Keski-Vakkuri, Muller, Schafer, Shigemori, Staessens arXiv: , arXiv:
Aparicio and Lopez, arXiv:
Caceres and A. Kundu, arXiv:
……………

14. Holographic description of quench
Black hole
AdS
quench: thin shell collapse to form a black hole.

15. Holographic Entanglement entropy
Ryu, Takayanagi
Hubeny, Rangamani, Takayanagi A
Extremal surface: M

16. A Area: Volume: R: characteristic size of the region
Interested in long-distance physics:

17. Gravity description Each extremal surface can also be specified by
the location of (and boundary conditions at) the tip.

18. Large size and critical extremal surfaces
In general a rather complicated problem
to determine time
evolution of extremal
surfaces
Critical extremal
surfaces determine
large R, large time
behavior

19. Four scaling regimes in general dimensions
In the large size R limit:
Saturation
Late time
memory loss
Linear growth
Early quadratic growth

20. Linear growth For : Equilibrium entropy density
See also
Hartman, Maldacena
For
: Equilibrium entropy density
independent of shape, holographic theories under
consideration, the nature of equilibrium state, also likely
thermalization processes
vE: dimensionless number characterizing final eq state.

21. Critical extremal surface for linear growth
The critical extremal surface
runs along a constant
radial slice inside the horizon
D: # of spatial dimensions
: horizon size
Determined by equilibrium state

22. Entanglement Tsunami natural with evolution from a local Hamiltonian
suggests a picture of tsunami wave of entanglement, with a sharp wave front. vE
A - vEt
d.o.f. in the region covered
by the wave is now entangled
with those outside A
natural with evolution from a local Hamiltonian

23. Tsunami velocity Neutral system (AdS Schwarzschild ):
Turning on chemical potential reduces vE.

24. Upper bound on vE ? vE should be constrained by causality.
In all gravity examples:
Null energy condition important

25. Comparing with free particle streaming
Assume:
At t=0, there is a uniform density of “photons” with only local entanglement correlations.
Entanglement spreads when photons propagate.
Leading to shape independent linear growth,

26. In strongly coupled systems, entanglement tsunami
For D=1:
HL, Mezei, Suh
Casini
In strongly coupled systems, entanglement tsunami
propagates faster than those from free particles
traveling at speed of light !

27. Bound on entanglement growth?
For any non-equilibrium processes:
dimensionless, can be compared among region A of
different shapes, sizes, and systems of different
number of d.o.f.
Indications from gravity: after local equilibration (t >>1/T)

28. B A Comparing with small incremental entangling conjecture/theorem: CD

29. Future directions …………….. More examples:
Both holographic and field theoretical
More physical intuition on
Direct probe of entanglement tsunami
Continuum limit of small incremental theorem
Implications for black hole physics
……………..

30. Thank You

31. Entanglement in the vacuum
short-range entanglement
near Σ, cutoff dependent
long range entanglement, insensitive to UV physics near Σ
R: characteristic
size of A
Vacuum:
Long ranged entangled d.o.f. are measure zero.

32. Entanglement in equilibrium state
The system behaves macroscopically as a thermal state,
with entanglement entropy disguised as thermal entropy:
VA : volume of region A
Essentially all d.o.f. inside A becomes long ranged
entangled with those outside A.