HOLOGRAPHIC QUENCHES WITH A GAPentangle2016/PLopez.pdfAdS boundary AdS3 with an IR cutoff + massless...

HOLOGRAPHIC

QUENCHES WITH A GAP

Esperanza Lopez

together with E. da Silva, J. Mas and A. Serantes, arXiv:1604. 08765

quantum quench

Quenches & revivals in holography

H0 , → H , |Ψ ⟩gs H0 |Ψ(t)⟩H

quantum quench


H0 , → H , |Ψ ⟩gs H0 |Ψ(t)⟩H

revival |Ψ(t+τ)⟩ ≃ |Ψ(t)⟩

holographic thermalization

quantum quench


H0 , → H , |Ψ ⟩gs H0 |Ψ(t)⟩H



reflecting boundary

bouncing geometries in AdS(Bizon, Rostworowski 2011; Buchel, Lehner, Liebling, 2012;......)

quantum quench


H0 , → H , |Ψ ⟩gs H0 |Ψ(t)⟩H



reflecting boundary

bouncing geometries in AdS(Bizon, Rostworowski 2011; Buchel, Lehner, Liebling, 2012;......)

quantum quench


H0 , → H , |Ψ ⟩gs H0 |Ψ(t)⟩H


↕ (Abajo, da Silva, Lopez, Mas, Serantes, 2014)

Outlook

holographic quenches: H ≠ H0

Outlook


compare the parameter space for revivals and oscillating geometries

Outlook



typical versus fine-tuned initial state and its holographic representation

Outlook



typical versus fine-tuned initial state and its holographic representation

compare with results from other approaches: tensor networks

The model

quench dynamics of a 1+1 massive QFT

The model AdS boundary

IR hard wallAdS3 with an IR cutoff + massless scalar




D,N boundary conditions at the wall




D,N boundary conditions at the wall energy inflow from AdS boundary (Craps et al, 2013; Craps et al, 2014)




energy is conserved at the IR wall





energy is conserved at the IR wall time dependent profiles at the wall

redistribution of energy between metric and scalar





energy is conserved at the IR wall time dependent profiles at the wall

redistribution of energy between metric and scalar

Δt < IR scale out of equilibrium



Static solutions with Φ=0

ds

2 =1z

2

✓�(1�M0z

2) dt

2 +dz

2

1�M0z2

+ dx

2

◆


AdS boundary: Minkowskids

2 =1z

2

✓�(1�M0z

2) dt

2 +dz

2

1�M0z2

+ dx

2

◆



2 =1z

2

✓�(1�M0z

2) dt

2 +dz

2

1�M0z2

+ dx

2

◆

M0=0 : AdS3

M0>0 : black hole

M0<0 : naked singularity

AdS-hard wall



2 =1z

2

✓�(1�M0z

2) dt

2 +dz

2

1�M0z2

+ dx

2

◆

M0=0 : AdS3

M0>0 : black hole


z0

AdS-hard wall



2 =1z

2

✓�(1�M0z

2) dt

2 +dz

2

1�M0z2

+ dx

2

◆

M0=0 : AdS3

M0>0 : black hole


z0

M0 z0 >1 : black hole2

AdS-hard wall



2 =1z

2

✓�(1�M0z

2) dt

2 +dz

2

1�M0z2

+ dx

2

◆

M0=0 : AdS3

M0>0 : black hole


z0


M0 z0 <1 : AdS, AdS-BH, naked singularities2

AdS-hard wall



2 =1z

2

✓�(1�M0z

2) dt

2 +dz

2

1�M0z2

+ dx

2

◆

M0=0 : AdS3

M0>0 : black hole


z0


M0 z0 <1 : AdS, AdS-BH, naked singularities2 M0: wall mass

AdS-hard wall



2 =1z

2

✓�(1�M0z

2) dt

2 +dz

2

1�M0z2

+ dx

2

◆

M0=0 : AdS3

M0>0 : black hole


z0



no reason to exclude M0<0

M0: wall mass

AdS-hard wall

= 1



2 =1z

2

✓�(1�M0z

2) dt

2 +dz

2

1�M0z2

+ dx

2

◆

M0=0 : AdS3

M0>0 : black hole


z0



no reason to exclude M0<0

M0: wall mass

Solitonic solutionsDirichlet bc at the wall :

static solutions with a non-trivial scalar profile



solitons up to a threshold Φ0 (M0)

stable and unstable branches

stable solitons with M>1

given M0MXO\

0.2 0.4 0.6 0.8 1.0 1.2f0

0.2

0.4

0.6

0.8

1.0

1.2

•

•

:op dual to Φ

:total mass

Φ0 : scalar field at the wall



-20 -15 -10 -5M0

-2

-1

1

2

f0parameter space




given M0MXO\

0.2 0.4 0.6 0.8 1.0 1.2f0

0.2

0.4

0.6

0.8

1.0

1.2

•

•

:op dual to Φ

:total mass




-20 -15 -10 -5M0

-2

-1

1

2

f0parameter space

boundary conditions can not be changed by bulk dynamics

solitons = vacua { M0 ,Φ0 } = couplings




given M0MXO\

0.2 0.4 0.6 0.8 1.0 1.2f0

0.2

0.4

0.6

0.8

1.0

1.2

•

•

:op dual to Φ

:total mass


Mass gap

lowest pole of 2-point function

Mass gap

⇒ normal modes of Φlowest pole of 2-point function

Mass gap

⇒ normal modes of Φ

M0 ,Φ0 ≠ 0 break Lorentz invariance

field theory in a non-trivial homogeneous medium


Mass gap



field theory in a non-trivial homogeneous medium ≤ 1dωdk

causality


Mass gap




causality


mgap = ω| for the lowest normal modek=0

Mass gap




causality


M0=1/2

M0=-8

the gap increases with negative M0, closes up with growing Φ0

mgap = ω| for the lowest normal modek=0

Entanglement entropy

A

B

S(A)= - Tr ρA log ρA


A

B

Area law

S(A)∝ ∂A



A

B

Area law

S(A)∝ ∂A

ξ : correlation length



A

B

Area law

S(A)∝ ∂A


1+1 gapped theory

• •l



A

B

Area law

S(A)∝ ∂A


1+1 gapped theory

• •l

S(l>ξ ) = const



A

B

Area law

S(A)∝ ∂A


1+1 gapped theory

• •l

S(l>ξ ) = const

l homologous to γS(l )∝ length(γ)

holographic entanglement entropy

γ

IR wall

• •l

(Ryu, Takayanagi, 2006)



A

B

Area law

S(A)∝ ∂A


1+1 gapped theory

• •l

homology relative to IR wall

S(l>ξ ) = const



γ

IR wall

• •l




A

B

Area law

S(A)∝ ∂A


1+1 gapped theory

• •l

homology relative to IR wall

S(l>ξ ) = const

pure state S(A)=S(B) in the absence of horizons: pure states



γ

IR wall

• •l



Correlation length

γγ

ll

Correlation length

S(l )∝log(l )

γγ

ll

Correlation length

S(l )∝log(l ) S(l )=const

γγ

ll

Correlation length


ξ length(con)=length(dis) γγ

ll

Correlation length



ll

M0=1/2

M0=-8

ξ decreases with negative M0, increases with Φ0

Correlation length



ll

expected: mgap ξ = O(1)

M0=1/2

M0=-8


Correlation length



ll

expected: mgap ξ = O(1)

except close to soliton threshold:large N effect

M0=1/2

M0=-8

◉

0.0 0.5 1.0 1.5 2.0f0

1

2

3

4

5

6

7

xw

M0=-∞

M0=1/2

M0=-8


Holographic quench

simple gapped model with 2 couplings, which control IR physics

ds

2 =1z

2

✓�A(t, z)e�2�(t,z)

dt

2 +dz

2

A(t, z)+ dx

2

◆, �(t, z)

⇧0 =

✏

↵

1

cosh

2 t/↵

Holographic quench


ds

2 =1z

2

✓�A(t, z)e�2�(t,z)

dt

2 +dz

2

A(t, z)+ dx

2

◆, �(t, z)

ds

2 =1z

2

✓�A(t, z)e�2�(t,z)

dt

2 +dz

2

A(t, z)+ dx

2

◆, �(t, z)

⇧0 =

✏

↵

1

cosh

2 t/↵

Holographic quench


ds

2 =1z

2

✓�A(t, z)e�2�(t,z)

dt

2 +dz

2

A(t, z)+ dx

2

◆, �(t, z)

ds

2 =1z

2

✓�A(t, z)e�2�(t,z)

dt

2 +dz

2

A(t, z)+ dx

2

◆, �(t, z)

⇧0 =

✏

↵

1

cosh

2 t/↵

dirichlet bc : �̇0 = 0

Holographic quench


ds

2 =1z

2

✓�A(t, z)e�2�(t,z)

dt

2 +dz

2

A(t, z)+ dx

2

◆, �(t, z)

ds

2 =1z

2

✓�A(t, z)e�2�(t,z)

dt

2 +dz

2

A(t, z)+ dx

2

◆, �(t, z)

⇧0 =

✏

↵

1

cosh

2 t/↵

⇧ ⌘ A�1e��̇

⇧0 =

✏

↵

1

cosh

2 t/↵ → time dependent wall profile : dirichlet bc : �̇0 = 0

Holographic quench


ds

2 =1z

2

✓�A(t, z)e�2�(t,z)

dt

2 +dz

2

A(t, z)+ dx

2

◆, �(t, z)

ds

2 =1z

2

✓�A(t, z)e�2�(t,z)

dt

2 +dz

2

A(t, z)+ dx

2

◆, �(t, z)

⇧0 =

✏

↵

1

cosh

2 t/↵

⇧ ⌘ A�1e��̇

⇧0 =

✏

↵

1

cosh

2 t/↵ → time dependent wall profile :

M = M0 + MΦ = const

⤷

⤶

energy exchange


ËË

-12 -10 -8 -6 -4 -2M0

-2

-1

1

2

f0

◉

shorter time span

initial state

growing amplitude

Holographic quench


ds

2 =1z

2

✓�A(t, z)e�2�(t,z)

dt

2 +dz

2

A(t, z)+ dx

2

◆, �(t, z)

ds

2 =1z

2

✓�A(t, z)e�2�(t,z)

dt

2 +dz

2

A(t, z)+ dx

2

◆, �(t, z)

⇧0 =

✏

↵

1

cosh

2 t/↵

⇧ ⌘ A�1e��̇

⇧0 =

✏

↵

1

cosh

2 t/↵ → time dependent wall profile :

M = M0 + MΦ = const

⤷

⤶

energy exchange

M<1 : ever bouncing


ËË

-12 -10 -8 -6 -4 -2M0

-2

-1

1

2

f0

◉

shorter time span

initial state

growing amplitude

Revivals versus bounces

when to expect revivals? interactions disfavor revivals, except



energy density < ( infrared scale )

# fieldsd




# fieldsd

global AdS # fields ~ 1/G , infrared scale ~ 1/R M R < 1d




# fieldsd


AdS3 hard wall

Mcollapse : mass leading to horizon formation for bc {M0 ,Φ0}




# fieldsd


√Mcollapse - Mgs ≲ infrared scale

AdS3 hard wall





# fieldsd


√Mcollapse - Mgs ≲ infrared scale

AdS3 hard wall


M0=-∞◉

M0=-8

M0=1/2

0.5 1.0 1.5 2.0f0

0.1

0.2

0.3

0.4

0.5

DM êw

Decoding oscillating geometriesassociated field theory state mechanism behind revivals


parameters of the wall action : α (time span), ε (amplitude)



α = √gtt α~ __↓



α = √gtt α~ __↓

α >1: adiabatic change~-20 -15 -10 -5M0

-2

-1

1

2

f0

-20 -15 -10 -5M0

-2

-1

1

2

f0

⤴••



α = √gtt α~ __↓


-2

-1

1

2

f0

-20 -15 -10 -5M0

-2

-1

1

2

f0

⤴••

α <1: radial localization~

MΦ=∫ρdz



α = √gtt α~ __↓


-2

-1

1

2

f0

-20 -15 -10 -5M0

-2

-1

1

2

f0

⤴••

α <1: radial localization~

MΦ=∫ρdz

α ≈1: standing wave~

α = 0.8

α = 0.2

Energy density

M-Mgs

M-Mgse = _

energy density above vacuum normalized to that for fast

thermalization

Energy density

M-Mgs

M-Mgse = _


thermalization

α ≈ 1: e ≪ 1

α < 1: e ≲ 1~

~

Energy density

M-Mgs

M-Mgse = _


thermalization

α ≈ 1: e ≪ 1

α < 1: e ≲ 1~

~

Periodicity

α ≈ 1: periodicity of lowest normal mode

~2πωτ ≈

Energy density

M-Mgs

M-Mgse = _


thermalization

α ≈ 1: e ≪ 1

α < 1: e ≲ 1~

~

Periodicity

α ≈ 1: periodicity of lowest normal mode

~2πωτ ≈

α < 1: different periodicity~

larger far from stability threshold

Tensor networks

variational ansatz for QFT states based on entanglement properties

• ↑↓| i =X

ci1i2. . . iN |i1i2. . . iN i • • ••• • ••• ••••

Tensor networks


• ↑↓| i =X

ci1i2. . . iN |i1i2. . . iN i


• • ••• • ••• ••••

A | | | | | |

A A A A A (❨ (❨

area law

2 → 2 (rank Ai) N 2

Schwinger model ground state, low lying spectrum

(Pichler et al 2015; Buyens et al, 2014,2016)

(Banuls,Cichy,Jansen,Cirac, 2013; Rico et al, 2013)

real time simulations: Schwinger mechanism, quenches

Schwinger model quench massive Schwinger model + quench: turn on abruptly an electric field

(Buyens, Haegeman, van Acoleyen, Verschelde, Verstraete, 2014)

Schwinger model quench

pair production: energy density ~ fermion mass

massive Schwinger model + quench: turn on abruptly an electric field (Buyens, Haegeman, van Acoleyen, Verschelde, Verstraete, 2014)



oscilations of electric field & particle number (out of phase)

small energies : revivals

(Buyens et al)






(Buyens et al)


oscilations of entanglement entropy



small energy limit : perfect revivals

τ ≈ 2πΔm

Δm : mass of lowest stable excitation



(Buyens et al)





small energy limit : perfect revivals

τ ≈ 2πΔm

Δm : mass of lowest stable excitation

|�i =X an

pn!

e�in�mt|ni



(Buyens et al)



On the holographic interpretation

holographic model: # fields >>1

Schwinger model: # fields ~1


BUT revivals depend on energy density

# fields




pairs of entangled quasiparticles

(Cardy,Calabrese, 2005)

stronger correlations when emitted at close points

entangled pairs stream away at c=1

xt=0

t

simple picture of massive to massless quench


# fields








xt=0

t


l>2t

γ

l


# fields



(Abajo,Aparicio Lopez, 2010;Balasubramanian et al, 2011)






xt=0

t


l>2t

γ

l

l

γ

l<2t


# fields





shell infall drift away of entangled excitations





xt=0

t


l>2t

γ

l

l

γ

l<2t


# fields




Comparing with AdS-HW

lowest excitation Δm → lowest normal mode



small energy quench maps to standing scalar pulse




momentum zero coherent state

→ no radial displacement






~α ≈ 1






~α ≈ 1

electric field ↓

vev of O






~α ≈ 1

2 4 6 8t

-6

-4

-2

2

4

6

XO\

α < 1 ~

electric field ↓

vev of O

Increasing the quench energy

resulting state should contain finite momentum modes

oscillating pulse must involve radial displacement




check 1

ò

ô

0.0 0.2 0.4 0.6 0.8 1.0z

2

4

6

8

10r

e=0.28

e=0.7

α = .2~




check 1

ò

ô

0.0 0.2 0.4 0.6 0.8 1.0z

2

4

6

8

10r

e=0.28

e=0.7

α = .2~

check 2 Φ=Φsol + ε lowest normal mode




check 1

ò

ô

0.0 0.2 0.4 0.6 0.8 1.0z

2

4

6

8

10r

e=0.28

e=0.7

α = .2~radial localization increases with energy

e=0.5ô

ò

0.0 0.2 0.4 0.6 0.8 1.0z

2

4

6

8

10r

check 2 Φ=Φsol + ε lowest normal mode

The holographic dual of a quench

quench: sudden action abrupt time variation of boundary conditions



varying bc at AdS3 boundary shell mass ∝ εδt

2

2 shell thickness ∝ δt




2


δt → 0 : singular configuration

(Buchel, Myers, van Niekerk, 2013;Das, Galante, Myers, 2014, 2015)




2


same for a wall variation of bc






2


finite energy quench → radial localization ~ energy







2



rising energy ~ increasing radial localization




different dynamics of thin and broad shells




2



revivals ~ standing waves rising energy ~ increasing radial localization




different dynamics of thin and broad shells

“Typical” shell profileswhat bulk configurations can represent “typical” QFT out of equilibrium states?


no fine tuning involved↲


no fine tuning involved↲thin shells of small

mass are not typical



able to resolve scales smaller than IR gap

thin shells of small mass are not typical




energy density < gap

# fields2






# fields2 not enough for Schwinger mechanism

virtual pairs can not separate further than ξ






# fields2 not enough for Schwinger mechanism

virtual pairs can not separate further than ξ


high energy

low energy

Conclusions

discuss general aspects of the holographic modeling of quenches using a simple setup

Conclusions


qualitative but detailed map between out of equilibrium QFT and the gravitational dynamics of collapsing shells

Conclusions


needed a further QFT understanding of small mass localized pulses


Conclusions




real time simulations have become recently accessible with tensor network techniques; holography provides a conceptually and perhaps also practical interesting alternative

Conclusions




real time simulations have become recently accessible with tensor network techniques; holography provides a conceptually and perhaps also practical interesting alternative

Thanks

HOLOGRAPHIC QUENCHES WITH A GAPentangle2016/PLopez.pdfAdS boundary AdS3 with an IR cutoff + massless...

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