entanglement tsunami hong liu. josephine suh mark mezei
TRANSCRIPT
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Entanglement Tsunami
Hong Liu
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Josephine Suh Mark Mezei
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Based on
HL and Josephine Suh, 1305.7244, PRL 112, 011601 (2014)HL and Josephine Suh, 1311.1200, PRD 89, 066012 (2014)HL, Mezei, Suh, unpublished, Casini, unpublishedCasini, Hubeny, HL, Maxfield, Mezei, Suh, to appear
See also
Hyungwon Kim and Huse arXiv:1306.4306
Shenker and Stanford arXiv: 1306.0622, 1312.3296
Hubeny and Maxfield arXiv: 1312.6887
Hartman and Maldacena arXiv:1303.1080
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Entanglement generation
A B
How fast can entanglement be generated?
In most physical systems: Local Hamiltonian
C D
: UV cutoff
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Small incremental entangling conjecture/theorem
A B
C D SA : entanglement entropy of A
For spin systems:
dC : dimension of Hilbert space of C
Dur, Vidal et al, Bravyi, KitaevBennett et al, Van Acoleyen, Marien, Verstraete
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A B
C D
For more general quantum systems, e.g. a QFT
In this talks we will describe some hints.
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A simple setup: global quenches
1. 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
A
Also a question of interest for thermalization.
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VA : volume of region A
In equilibrium, system behaves macroscopically as a thermal state, with entanglement entropy disguised as thermal entropy:
Essentially all d.o.f. inside A becomes long ranged entangled with those outside A.
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A A
essentially no long range entanglement
almost all d.o.f. long range entangled
t=0 equilibrium
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Previous results in (1+1)-d CFTs
ΔS/Seq
t/RtS
1. Linear growth with time
Calabrese and Cardy
(not too early and too late)
2R
2. Slope = 1
3. Can be reproduced by free particles
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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.
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String theory to the rescue!
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Earlier work:
Hubeny, Rangamani, Takayanagi: arXiv:0705.0016
Abajo-Arrastia, Aparicio and Lopez, arXiv:1006.4090
Albash and Johnson, arXiv:1008.3027
Balasubramanian, Bernamonti, de Boer, Copland, Craps, Keski-Vakkuri, Muller, Schafer, Shigemori, Staessens arXiv:1012.4753, arXiv:1103.2683
Aparicio and Lopez, arXiv:1109.3571
Caceres and A. Kundu, arXiv:1205.2354
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Holographic description of quench
quench: thin shell collapse to form a black hole.
Black hole
AdS
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Holographic Entanglement entropyRyu, TakayanagiHubeny, Rangamani, Takayanagi
A
Extremal surface: M
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A
R: characteristic size of the region
Interested in long-distance physics:
Area:
Volume:
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Gravity description
Each extremal surface can also be specified bythe location of (and boundary conditions at) the tip.
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Large size and critical extremal surfaces
Critical extremal surfaces determinelarge R, large time behavior
In general a rather complicated problemto determine timeevolution of extremal surfaces
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Four scaling regimes in general dimensions
Late timememory loss
Linear growth
In the large size R limit:
Saturation
Early quadratic growth
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Linear growth
For
independent of shape, holographic theories under consideration, the nature of equilibrium state, also likelythermalization processes
vE: dimensionless number characterizing final eq state.
See also Hartman, Maldacena
: Equilibrium entropy density
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Critical extremal surface for linear growth
: horizon size
The critical extremal surface runs along a constant radial slice inside the horizon
D: # of spatial dimensions
Determined by equilibrium state
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Entanglement Tsunami
suggests a picture of tsunami wave of entanglement, with a sharp wave front.
natural with evolution from a local Hamiltonian
vE
A - vEt
d.o.f. in the region covered by the wave is now entangled with those outside A
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Tsunami velocity
Neutral system (AdS Schwarzschild ):
Turning on chemical potential reduces vE.
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Upper bound on vE ?
Null energy condition important
vE should be constrained by causality.
In all gravity examples:
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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,
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In strongly coupled systems, entanglement tsunami propagates faster than those from free particles traveling at speed of light !
For D=1:
HL, Mezei, SuhCasini
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Bound on entanglement growth?
For any non-equilibrium processes:
Indications from gravity: after local equilibration (t >>1/T)
dimensionless, can be compared among region A of different shapes, sizes, and systems of different number of d.o.f.
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Comparing with small incremental entangling conjecture/theorem:
A B
C D
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Future directions
• More examples:
Both holographic and field theoretical
• Continuum limit of small incremental theorem
• Implications for black hole physics
• More physical intuition on
• Direct probe of entanglement tsunami
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Thank You
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Entanglement in the vacuum
A short-range entanglement near Σ, cutoff dependent
long range entanglement, insensitive to UV physics near Σ
Vacuum: R: characteristic size of A
Long ranged entangled d.o.f. are measure zero.
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Entanglement in equilibrium stateThe system behaves macroscopically as a thermal state, with entanglement entropy disguised as thermal entropy:
Essentially all d.o.f. inside A becomes long ranged entangled with those outside A.
VA : volume of region A