April 26, 2017

Memory Loss in Holographic Non-equilibrium Heating

D.S.Ageev and I.Ya.Aref’evaSteklov Mathematical Institute, Russian Academy of Science

We consider the time evolution of entanglement entropy after a global quench in a strongly coupledholographic two-dimensional system with initial temperature Ti. Subsequent equilibration increasesthe temperature up to a final temperature Tf . We describe this process holographically by aninjection of a thin shell of matter in the black hole background with temperature Ti.

In the limit of large regions of entanglement, the evolution of entanglement entropy at largetimes becomes a function of one variable ∆S(τ, `) ∼ M(τ − `). This is a so called memory lossregime well known for the holographic thermalization from zero temperature. We evaluate specificcharacteristics of this regime in nonequilibrium heating and find that the behaviour of the functionM(t−`) during post-local-equilibration and saturation times has similar forms as for thermalizationfrom zero temperature, but the scaling coefficients depend on the temperature difference Tf − Ti.

CONTENTS

I. Introduction 1

II. Setup 2A. Length of the ETEBA geodesics describing

thermalization 2B. Length of the ETEBA geodesics describing

non-equilibrium heating 2C. Different regimes in equilibration 3D. Scaling characteristics of thermalization

regimes 4

III. Critical curve 4

IV. Memory loss regime in non-equilibrium heating 5A. Saturation. Expansion near π/2 6B. Post-local-equilibration linear growth.

Expansion near ϑκ 7C. Late-time memory loss regime.

Interpolation between φ = ϑκ and π/2 8

V. Numerical results 8

VI. Conclusions and discussions 8

Acknowledgements 10

References 10

I. INTRODUCTION

It is universally recognized that one of the most chal-lenging questions in traditional methods of quantum fieldtheories is the description of thermalization process. TheAdS/CFT correspondence [1–3] proposes a powerful toolfor a description of thermalization process in general classof strong coupling quantum theories. In the holographicduality the temperature in the quantum field theory isrelated with the black hole (or black brane) temperature

in the dual background [4]-[6]. The description of thethermalization of conformal field theory after a globalquench is well known [7, 8] and in the dual language thethermalization is described by a black hole formation,see [9, 10] and references therein. Also see [11–13] fordifferent applications of the AdS/CFT duality. The sim-plest description of this process is provided by the Vaidyadeformation of a given background, see [14]-[36] and ref-erences therein.

However not all interesting non-equilibrium processesin strong correlated systems start from zero temperature.In [35] the global quench starting from the thermal ini-tial state has been considered for some particular simplemodels. Many interesting and noteworthy phenomenastart from non-zero temperatures, in particular, all phe-nomena related with biological systems. So it is naturalto try to study these phenomena using the gravity dualdescription. In particular, such approach has been usedin the holographic description of the photosynthesis [36].

In some sense the non-equilibrium heating process ascompared to the thermalization is simpler to study, butholographically the situation is just opposite. It is natu-ral to expect that non-equilibrium heating inherits typ-ical regimes taking place during thermalization. Theregimes in thermalization processes have been studied indetails in the papers by H.Liu and S.Suh [20, 21], see also[37–39], and they are – regimes of pre-local-equilibrationquadratic growth, post-local-equilibration linear growth,a late-time regime in which the evolution does not carryany memory of the size and shape of the entangled region,and the saturation regime. In this paper we show thatthis is really true for 1+1 dimensional systems, whosedual admits the explicit formula [24], but curiously cal-culations of scaling coefficients are more involved, as com-pared to those of the thermalization.

The paper is organized as follows. In Section II we re-mind the dual geometry of the thermalization and non-equilibrium heating processes and present the explicitformulae for the entanglement entropy evolution after theglobal quench in initially thermal system. We also list thespecific regimes during these processes and remind the

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previous results about behavior of the entanglement en-tropy during thermalization. Section III is devoted to thestudy of detailed relations between geometrical charac-teristics of the entanglement minimal surface ( geodesic)in the bulk and physical parameters on the boundary,that are the size ` of the entangled area and thermal-ization time τ of this area. In Section IV we calculatethe scaling characteristics of the memory loss regime inthe process of the global quench in thermal system withtemperature Ti 6= 0 and especially their dependences onthe difference between final and initial temperatures. Inthe final Section VI we conclude and we discuss somepossible generalizations.

II. SETUP

A. Length of the ETEBA geodesics describingthermalization

According to the AdS/CFT prescription, the Vaidya-AdS geometry describes thermalization

ds2 =R2

z2

(−f(v, z)dv2 − 2dvdz + d~x2

), (2.1)

with f(v, z) = 1 − θ(v)g(z). Where the function h(z) ≡1− g(z) describes different final equilibrium states. Theentanglement entropy S of the interval with length ` inthe holographic model [40, 41] of thermalization is givenby the length of the ETEBA geodesics. The ETEBA(this terminology has been introduced in [25]) geodesicsare spacelike geodesics with both endpoints anchored onthe boundary, at the same time τ and located on thedistance `. For the case of g(z) = z2/z2

h this length isdescribed by the following formula [27]

S(`, τ) = log(zh`s

sinhτ

zh

), (2.2)

where s is a geometric parameter that together withadditional parameter ρ specify the top of the ETEBAgeodesics z∗ (its furthest from the boundary point in thebulk) and its intersection point zc with the null shell,

s =zcz∗, ρ =

zhzc, (2.3)

here s and ρ are implicitly related with the time τ andlength ` via the set of two equations

τ

zh= arccoth

(c+ 2ρ2 + 2cρ2

2cρ+ 2ρ

), (2.4)

` = `+ + `−, (2.5)

`−zh

=c

ρs,`+zh

= log2(1 + c)ρ2 + 2sρ− c2(1 + c)ρ2 − 2sρ− c

, (2.6)

where c =√

1− s2, 0 ≤ s ≤ 1 and 0 < ρ. These threeequations (2.2), (2.4) and (2.5) totally describe the evolu-tion of the entanglement entropy during thermalization.

singularity singularity

AdS

BH zh BH zhBH zH

z = 0

v = 0v = 0

FIG. 1. The plot of the Penrose diagrams for the Vaidyametrics (2.1) and (2.7).

B. Length of the ETEBA geodesics describingnon-equilibrium heating

In this paper we consider black hole collapsing fromthe initial state defined by the horizon position zH to thefinal state with horizon zh as a dual background. TheBH-Vaidya metric is given by

ds2 =R

z2

(−f(v, z)dv2 − 2dvdz + d~x2

), (2.7)

where

f(z, v) = θ(v)fh(z) + θ(−v)fH(z), (2.8)

and the functions fH and fh are defined as

fH = 1−(z

zH

)2

, fh = 1−(z

zh

)2

, zh < zH .(2.9)

The time is related with variable v as

v < 0 t = v + zHarctanhz

zH, (2.10)

v > 0 t = v + zharctanhz

zh. (2.11)

The initial and final temperatures, energy and entropyare

Ti =1

2πzH, Tf =

1

2πzh, (2.12)

Ei =R

8πGN

1

z2H

, Ef =R

8πGN

1

z2h

, (2.13)

si =R

zH

1

4GN, sf =

R

zh

1

4GN, (2.14)

where GN is Newton’s constant in the bulk and R is atypical scale in the bulk. In this paper we set 8πGN/R =

3

1. This metric describes the shell located at v = 0 andusually zH > zh. Note, that the case zH < zh corre-sponds to a model of cooling [18] and this model violatesNEC condition [20].

We have solved this model explicitly [24] and we haveobtained the following expression for the entanglemententropy

S = log

(zh

`Sκ(ρ, s)sinh

τ

zh

), (2.15)

where Sκ is given by

Sκ(ρ, s) =cρ+ ∆

∆·

√∆2 − c2ρ2

ρ (c2ρ+ 2c∆ + ρ)− κ2. (2.16)

The entanglement entropy (2.16) has been obtained as alength of the ETEBA geodesic with minimal divergencesubtraction. Both endpoints of these geodesic are an-chored on the boundary points at the same time τ locatedon the distance `. The relations between the bound-ary data τ, ` and bulk characteristics s, ρ of the ETEBAgeodesic are more complicated as compared with formula(2.4), (2.5), but admit the explicit form

τ

zh= arccoth

(−cκ2 + 2cρ2 + c+ 2∆ρ

2cρ+ 2∆

), (2.17)

` = `+ + `−, (2.18)

`+ =zh2

log

(c2γ4 − 4∆

(cs(κ2 − 2ρ2 + 1

)+ ∆ + ∆

(ρ2 − 2

)s2)

c2γ4 − 4∆2(ρs− 1)2

), `− =

zh2κ

log

((cκ+ ∆s)2

ρ2s2 − κ2

), (2.19)

where in addition to (2.3) we introduce

κ =zhzH

< 1. (2.20)

In (2.24) we also use the notations

c =√

1− s2, (2.21)

γ = 1− κ2, (2.22)

∆ =√ρ2 − κ2. (2.23)

The bulk variables define relative position of the geodesictop z∗ and the point where geodesic crosses the null shellzc. Now there are restrictions z∗ < zH and zc < zH , butthere are still 3 types of ETEBA geodesics.

For τ < 0 our ETEBA geodesic (first type) lies entirelyin the BTZ bulk with temperature Ti. The entanglemententropy is independent on τ and is equal (after the min-imal renormalization) to

Si = log

(zH`

sinh`

zH

). (2.24)

The top of the corresponding ETEBA geodesic z∗ < zHand z∗ → zH when ` increases in the correspondence with[37]. Let us fix ` and start to increase the time.

At very small τ > 0, the ETEBA geodesic starts in-tersecting the null shell and for τ � zh the point of in-tersection is close to the boundary, zc � zh. This is thesecond type of the ETEBA geodesics. When τ is of orderzh the ETEBA geodesic (third type) intersects the shellbehind the horizon, i.e. zc > zh. At some time τ = τsthe ETEBA geodesic lies entirely in the black hole (withthe temperature Tf ) region.

The role of the second horizon located at z = zH isthat it pulls out of the ETEBA geodesic with the topz∗ > zh to the first horizon, and this effect is stronger asthe difference of two temperatures decrease, i.e. κ→ 1.

C. Different regimes in equilibration

The regime which we consider are specified by the ratioof `/zh, τ/zh and τ/`. It is interesting that the choiceof the regime does not depend on `/zH , τ/zH , but thescaling parameters in these regimes do depend on zh/zH ,see Section IV. One can study

• Pre-local-equilibration growth regime with small τ .

– Both τ and ` are small

τ < ` < zh (2.25)

– Only τ is small and ` is large

τ < zh < ` (2.26)

• Memory loss. Both τ and ` are larger then zh

zh � τ ≤ `. (2.27)

Depending on the ratio of `− t to `, τ , zh there arethe following more detailed cases

– saturation regime:

`− τ � zh (2.28)

– late-time memory loss regime:

`� `− τ � zh (2.29)

– Post-local-equilibration linear growth regimewith large `

zh � τ � `, (2.30)

and therefore, `− τ ∼ `

4

D. Scaling characteristics of thermalization regimes

Characteristics of different regimes during thermaliza-tion have been studied in numerous studies [27, 31, 32,37, 42], and in more details in the paper by H.Lui andJ.Sui [20]. In this paper different models have been stud-ied and as one can see from these studies that the mainproperties are already seen in 1+1 example, described insect.A. To mention possible generalizations of our resultto arbitrary dimension, see VI, below we remind their re-sults for d-dimensional case. According to these studiesin the pre-local-equilibration growth regime the entan-glement entropy grows as

∆SΣ(τ) =π

d− 1EAΣ

(τ

zh

)2

+ · · · , (2.31)

where E is the energy density and AΣ is the area ofΣ. This result is independent of the shape of Σ, thespacetime dimension d, and the specific form of h(z)[27, 37, 42].

In the memory loss regime

S(`, τ)− Seq(`) = −seqλ (ts(`)− τ) , (2.32)

i.e. ∆S(`, τ) depends only on the difference ts(`)−τ , andnot on τ and ` separately. Here λ is some function thatdepends on h(z) and which can be determined explicitlyonly for d = 2 case, eq.(2.34) below,

λ(y) =[y + zh log

(sinχ−1(y/zh)

)], (2.33)

with y = `− τ and

χ(φ) =

(cot

φ

2− 1

)+ log tan

φ

2. (2.34)

Note λ(y) interpolates between the linear behavior for

large y/zh and the critical behavior y32 near saturation

as y/zh → 0. For d = 2, the behaviour (2.32) has beenobserved also in [16, 42].

Formula (2.32) interpolates between the post-local-equilibration linear growth

∆SΣ(t) = vEseqAΣτ

zh+ · · · (2.35)

and the saturation regime, ts − τ � zh,

S(`, τ)− S(eq)(`) ∝ −(ts − τzh

)γ , γ =d+ 1

2. (2.36)

In (2.35) vE is a dimensionless number which is inde-pendent of the shape of Σ, but does depend on the finalequilibrium state seq = c/(6zh) and in the particular cased=2 vE = 1.

Note that in the saturation regime the evolution is be-yond the linear regime and it depends on the shape, thespacetime dimension d, and may also depend on the fi-nal equilibrium state [20]. In particular case of 1+1, thesaturation time is

ts = `+ o(`), (2.37)The richer structure of the saturation regime takes placein higher dimensions and in the case of RN and Kerrblack holes [26, 27].

III. CRITICAL CURVE

Formulae (2.19) and (2.17) give the explicit depen-dence for ` and τ from the geometry of the entangle-ment surface specified by parameters ρ and φ. For any0 ≤ κ ≤ 1 there is a critical curve in the parameter spaceρ and φ, so that only on the right of this line (the redline in Fig.2) one can perform the change of variables(ρ, φ) → (`, τ). To visualize this change of variables itis convenient to draw the lines of fixed values of ` and τ(brown and blue lines in Fig.2). From Fig.2 we see thatlarge values of ` and τ are located near the critical line.The explicit form of the critical line ρ = ρ∗(φ) is definedby the following formula

ρ∗ =1

2V(κ, φ), (3.1)

where we define

V(κ, φ) ≡ 1 + (1−Q) cscφ, (3.2)

Q(κ, φ) ≡√

(1− 2κ2) cos2 φ+ 2κ2(1− sinφ). (3.3)

Substituting (3.1) into (2.17) and (2.19) one can check,that τ and `+ are equal to infinity on the critical line.

Near the critical line we have

−cκ2 + 2cρ2 + c+ 2ρ√ρ2 − κ2

2cρ+ 2√ρ2 − κ2

∣∣∣ρ=ρ∗+ε

= 1 + 2εKτ (φ, κ) +O(ε2), (3.4)

where

K(κ, φ) (3.5)

=c2∆

(κ2 + 2ρ2 − 1

)+ cρ

(−3κ2 + 4ρ2 − 1

)+ 2∆3

4∆(cρ+ ∆)2

∣∣∣ρ=ρ∗

,

5

0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

0.0

0.5

1.0

1.5

0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

0.0

0.5

1.0

1.5

0.8 1.0 1.2 1.4 1.6 1.8 2.0

0.0

0.5

1.0

1.5

φφ φ

ρ ρρ

κ = 0.25κ = 0 κ = 0.667

FIG. 2. The contour plots for ` and τ as functions of ρ and φ for different κ = 0, 0.25 and 0.667. These plot show that in the leftof the critical line (the darker red line) for ` > τ we can always find solution to the system of equations: τ = τ(ρ, φ); ` = `(ρ, φ).Lines of the equal times τ = 0.7, 1, 1.5, 2, 2.5 in all panels are shown by blue color with increasing thickness from right to left.Lines of the equal ` = 0.7, 1, 1.5, 2, 2.5 in all panels are shown by brown with increasing thickness from up to bottom.

and, therefore, we get near the critical curve the followingtime asymptotic

τ = −zh2

log ε− zh2

logK +O(ε). (3.6)

The same expansion for `+ is

`+ = −zh2

log ε+zh2

logK+ +O(ε), (3.7)

where

K+ =N

K, (3.8)

and

K = 2√

2Q sinφ− 4 cos2 φ+ 4κ2(2 sinφ(1− sinφ)−Q),

(3.9)

N = 4(κ2 − 1

)2cos2 φ−

(V2 − 4κ2

) ((V2 − 8

)sin2 φ+ 4

)4

+1

2

√V2 − 4κ2

(V2 − 2

(κ2 + 1

))sin(2φ), (3.10)

where V and Q are defined by (3.2) and (3.3), respec-tively.

The expression for `− is not singular on the critical line

`− =zh2κ

logK−, (3.11)

and

K− =2 sinφ

√V2 − 4κ2 + 4κ cosφ

V2 sin2 φ− 4κ2. (3.12)

In the limit of κ→ 0 we have

K(κ, φ) →κ→0

1 + cotφ

2, (3.13)

K+(κ, φ) →κ→0

cot φ2 + 1

cot φ2, (3.14)

K−(κ, φ) ≈κ∼0

1 + 2κ(cotφ

2− 1). (3.15)

IV. MEMORY LOSS REGIME INNON-EQUILIBRIUM HEATING

Using formulae (3.6), (3.11) and (3.7) we get, that nearthe critical curve the difference

T− ≡τ − `zh

(4.1)

is finite and can be expressed in the form

T− = −χκ(φ), (4.2)

where function χκ is defined as

χκ(φ) ≡ −1

2logK(κ, φ)K+(κ, φ)− 1

2κlogK−(κ, φ).

(4.3)

The second lightcone variable,

T+ ≡ τ/zh + `/zh, (4.4)

is singular near the critical line

T+ ∼ − log ε. (4.5)

The function χκ(φ) plays an important role in what fol-lows, controlling the behaviour of the lightcone variableT−. In the limit κ→ 0 we have

χκ(φ) →κ→0

χ0(φ), (4.6)

where χ0(φ) = cot φ2 − 1 − log cot φ2 in accordance with

(2.34). The inverse function χ−1 can be expressed interms of Lambert function W

φ = χ−10 (T−) = 2arccot

(−W−1

(−eT−−1

)). (4.7)

6

−χκφ

0.5 1.0 1.5

-5

-4

-3

-2

-1

FIG. 3. The plot of functions −χκ(φ) (solid lines) for differ-ent κ. The vertical dashed lines correspond to φ = ϑκ forcorresponding κ. The magenta lines correspond to κ = 0.1,darker red line to κ = 0.25 and blue lines to κ = 0.667. Thegreen line represents −χ0(φ).

Now let us consider the entanglement entropy be-haviour near the critical line

S − Seq = log1

S

sinh τzh

sinh `zh

(4.8)

≈ τ − `zh− 2 sinh

τ − `zh

e− τ+`zh − logS(φ, ρ∗H , κ).

Note, that S(φ, ρ, κ) is not singular on the critical line.Neglecting the exponentially small terms in (4.8) and in-troducing Sκ(φ) = S(φ, ρ∗H , κ) we get

∆S ≡ S − Seq ≈ T− − logSκ(φ). (4.9)

Formula (4.2) gives the representation of φ in term of T−

φ = χ−1κ (−T−), (4.10)

and we get that near the critical line the entanglemententropy has the wave spreading

∆S ≈ T− − logSκ(χ−1κ (−T−)). (4.11)

The dependence of entanglement entropy propagationonly on T−, i.e. realization of the memory loss regime, isbased on the exponential suppression of T+ dependence,see (4.8), and dependence of S only on φ near the criticalline.

A. Saturation. Expansion near π/2

One of limiting forms of memory loss regime is calledthe saturation regime and it corresponds to T− → −0.This corresponds to time scales near the final equilibta-tion time ts ≈ ` In parametric space it corresponds tothe values φ→ π/2.

Let us consider asymptotics of τ , `− and `+ given by(3.7), (3.11) and (3.6) when φ→ π/2

−χκ φ0.5 1.0 1.5

-5

-4

-3

-2

-1

FIG. 4. The plot of −χκ(φ) (solid lines) for different κ com-pared with the approximate formula (4.13) (dashed lines) nearφ = π/2. The green lines correspond to κ = 0, the magentalines to κ = 0, the darker red line to κ = 0.25 and the bluelines to κ = 0.667.

τ

zh≈ − log ε

2+

log 2

2+

(π2 − φ

)4√

1− κ2+

(−1 + 2κ2

) (φ− π

2

)232 (−1 + κ2)

,

`−zh≈(π2 − φ

)√

1− κ2+

(φ− π

2

)22 (1− κ2)

, (4.12)

`+zh≈ − log ε

2+

log 2

2+

3(π2 − φ

)4√

1− κ2+

(−1 + 2κ2

) (φ− π

2

)216 (−1 + κ2)

.

and we have (we compare this formula with exact one inFig.5)

`− τzh

= χκ(φ) ≈ (4.13)

zhδ2

2 (1− κ2)+

(zh + 2κ2zh

)δ3

6 (1− κ2)3/2

−(5zh + 7κ2zh

)δ4

24(1− κ)2(1 + κ)2.

where δ ≈ π/2− φ.The inverse function χ−1 is expressed in the form

δ ≈ −√

2(1− κ2)(`− τ

zh

)1/2

+

√1− κ2(1 + 2κ2)

3

`− τzh

−(20κ4 − κ2 − 10

)√1− κ2

18√

2

(`− τzh

)3/2

. (4.14)

Finally, expanding logS near φ→ π/2

logS ≈ δ2

2 (1− κ2)+ (4.15)

κ2δ3

2 (1− κ2)3/2

+

(2 + 7κ2 + 3κ4

)δ4

24 (1− κ2)2 , (4.16)

we get

∆S ≈ −fκ(`− τzh

)3/2

− nκ

(ts − τzh

)2

(4.17)

fκ =

(√2(1− κ2

))3

, nκ =(1− κ2)2

6.

7

We see that there is essential dependence of the coef-ficient fκ in front of scaling low on the initial tempera-ture. Note, that at κ = 1 all the expansion for saturationregime vanishes identically.

B. Post-local-equilibration linear growth.Expansion near ϑκ

First let us consider `−. The new feature of behaviourof `− for κ > 0, is that it is singular near φ = ϑκ,

ϑκ = arccos

(√1 + 2κ− 3κ2

√1 + 2κ+ κ2

). (4.18)

When φ = ϑκ + δ we get

`−zh≈ − log δ

2κ+ ζ−κ , (4.19)

where

ζ−κ =1

2κlog

(4κ(−2κ2 + κ+ 1

)(κ+ 1)2

√(2− 3κ)κ+ 1

). (4.20)

Since time τ and `+ are not singular near this point weget

−T− ≡ `− τ = χκ(ϑκ + δ) = − 1

2κlog δ + ζκ,(4.21)

ζκ = ζ−κ +1

2log

κ2

(1 + κ)2. (4.22)

Using the following properties of factors defined S, seeequation (2.16),

Sκ(ρ, s) =cρ+ ∆

∆·

√∆2 − c2ρ2

ρ (c2ρ+ 2c∆ + ρ)− κ2, (4.23)

∆

cρ+ ∆

∣∣∣ρ=ρcr,φ=ϑκ

=1

2, (4.24)

∆2 − c2ρ2∣∣∣ρ=ρcr,φ=ϑκ+δ

= q2κδ, (4.25)

ρ(c2ρ+ 2c∆ + ρ

)− κ2

∣∣∣ρ=ρcr,φ=ϑκ

= r2κ, (4.26)

where

q2κ =

0.16δ for κ = 0.10.39δ for κ = 0.250.86δ for κ = 0.66,

(4.27)

r2κ = 1.331 − 3(κ− 0.334)2, (4.28)

we get that near the critical line and when φ ≈ ϑκ + δ

Sκ(ρ, s)∣∣∣ρ=ρcr,φ=ϑκ+δ

=2qκrκ

δ1/2. (4.29)

Hence

∆S ≈ T− − log2qκrκ

e−κ(−T−−ζκ)) (4.30)

= (1− κ)T− − κζκ − log2qκrκ

. (4.31)

Therefore, the linear coefficient is changed as compare tothe case of the initial zero temperature, and

kκ = 1− κ, (4.32)

that is in agreement with numerical calculations pre-sented in Fig. 7.

0.2 0.4 0.6 0.8 1.0 1.2 1.4

-25

-20

-15

-10

-5

0

−χκ

FIG. 5. The plot of −χκ(φ) (solid lines) for different κ com-pared with the approximate formula (4.21),(4.22) (dashedlines) near φ = ϑκ. The magenta lines correspond to κ = 0.1,darker red line to κ = 0.25 and blue lines to κ = 0.667.

0.2 0.4 0.6 0.8 1.0 1.2 1.4

-10

-8

-6

-4

-2

0

−xκ(φ)

FIG. 6. The plots of the best fit −xκ(φ) (dashed lines) givenby (4.33) compare to the exact −χκ(φ) (solid lines) at differ-ent κ. The magenta lines correspond to κ = 0.1, darker redlines to κ = 0.25 and blue lines to κ = 0.667. The green linerepresents −χ0(φ).

C. Late-time memory loss regime.

8

Interpolation between φ = ϑκ and π/2

As interpolation functions xκ(φ) to χκ(φ) on the inter-val (ϑκ, π/2) we take, see Fig.6,

xκ = aκ

(cot

φ− ϑκ2bκ

− cotπ2 − ϑκ

2bκ

− logcot φ−ϑκ2bκ

cotπ2−ϑκ2bκ

), (4.33)

where the numerical values of aκ and bκ are

a0.1 = 0.15, b0.1 = 2.5, (4.34)

a0.25 = 0.07, b0.25 = 2.8,

a0.667 = 0.014, b0.667 = 3.8.

The inverse functions to xκ are given by

x−1κ ≡ φ = ϑκ (4.35)

+2bκarccotW−1

(− e−

xκaκ−cotφκ,0 cotφκ,0

),

where

φκ,0 ≡π2 − ϑκ

2bκ. (4.36)

Substituting (4.35) to (4.30) we get

∆S ≈ T− (4.37)

− logSκ

(ϑκ + 2bκarccotW−1

(− e

Taκ−cotφκ,0 cotφκ,0

)).

From Fig. 7 we see that in the region of large of L ≡−T− we have the linear growth of ∆S.

V. NUMERICAL RESULTS

In the previous sections we have shown, that for somevalues of τ and ` the function ∆S depends only on theirdifference τ − `. From Fig.9 we see that for large valuesof ` and τ and ` > τ , the function ∆S(zH , `, τ) in factdepends only on the T− light-cone variable

T− = τ − `, (5.1)

and the dependence on ` is negligible away from the lineT− = −`, i.e. for large times, and for large length `.Also this effect is presented in coordinates ` and τ inthe top panels in Fig.9. It is instructive to present thedependence on T− in the 2D plot keeping ` fixed, seeFig.8.

In Fig.8 we present the explicit dependence of∆S(zH , `, T− + `) on T− for the different values of κand for different values of `. Solid lines corresponding

2 4 6 8 10 12 14

-20

-15

-10

-5

0

{-dtl(1, 0.005, phi), dtl(1, 0.005, phi) +Scal(1, 0.005, phi)}

{-dtl(1, 0.1, phi), dtl(1, 0.1, phi) +Scal(1, 0.1, phi)}

{-dtl(1, 0.25, phi), dtl(1, 0.25, phi) +Scal(1, 0.25, phi)}

{-dtl(1, 0.66, phi), dtl(1, 0.66, phi) +Scal(1, 0.66, phi)}

{4.2× 2 phi, 1 - 3 × 2.6 phi}

{4.2× 2 phi, 1.2 - 3 × 2.55 phi}

{4.2× 2 phi, 0.56 - 3 × 2.05 phi}

{15 phi, -5 phi}

{10 phi, -10 phi}

0 2 4 6 8 10 12 140

2

4

6

8

L

L

∆S L

κ = 0.0

κ = 1.0

κ = 0.25

κ = 0.66

FIG. 7. The plot of functions L(L) = − logSκ(χ−1κ (+L)) and

∆S as function on L for κ = 0.66, 0.25, 0.1, 0, the blue, red,magenta and green solid lines, respectively. We see the lineardependence of ∆S on L for large L with the scope equal to kκ.The values of kκ = 0.34, 0.75, 0.9, 1, for κ = 0.66, 0.25, 0.1, 0respectively the blue, red, magenta and green dashed linesrespectively. The cyan line is a test line with the scope equalto 1.

to κ = 0.25 in the bottom panel of Fig.8 shows strongdeviation from dotdashed lines corresponding to κ = 0for large enough T−. Also in Fig.8 one can see that forlarger values of ` function ∆S for κ = 0 tends to be closerto the red line in the bottom panel. The middle panelof Fig.8 shows, that the larger values of ` for different κmake the dependence closer to the linear (see dotdashedand solid lines in this panel) for large enough T−.

VI. CONCLUSIONS AND DISCUSSIONS

In this paper we considered the evolution of entangle-ment entropy during equilibration after the global quenchof an initial thermal state at temperature Ti followed bynon-equilibrium heating to temperature Tf , in an exam-ple of the simplest quenched holographic model. In thebulk this equilibration process is described by the Vaidya

9

-1.0 -0.8 -0.6 -0.4 -0.2

-0.5

-0.4

-0.3

-0.2

-0.1

-4 -3 -2 -1

-5

-4

-3

-2

-1

1

∆S(zH , `, T− + `)

∆S(zH , `, T− + `)

T−

T−

-4 -3 -2 -1

-5

-4

-3

-2

-1

1

FIG. 8. The plot of the dependence of ∆S(zH , `, T− + `) onT− for ` fixed and equal to 3, 4, 5, 6, 7, 9, 12; zH = 1.5, zh = 1(dotted lines), zH = 4 , zh = 1 (solid lines), zH =∞ , zh = 1(dotdashed lines). The red line represent ∆S(∞, `, T− + `)given by (2.33).

shell in the black hole background. This is in contrast tothe Vaidya shell in the AdS background, that correspondsto thermalization. In both cases the holographic entan-glement entropy is defined as the area of an extremalsurface anchored on a given area on the boundary.

Using the explicit representation for the evolution ofthe holographic entanglement entropy we have shown theexistence of the memory loss regime in equilibration pro-cess started from a thermal state and followed by non-equilibrium increasing of the temperature. The mem-ory loss regime occurs long after the system has achievedlocal equilibration at scales of order zh and the holo-graphic entanglement entropy evolution in this regime isdescribed by the function depending only on one variable

∆S(`, τ) ≈Mκ(`− vτ), (6.1)

for our simplest model v = 1. The form of the function inthe LHS of (6.1) depends on κ. As for the thermalization

model [20], there are two special cases of the memory lossregime, namely

• Post-local-equilibration linear growth regime withlarge `, zh � τ � `, `− τ ∼ `

Mκ(`− τ) ≈ −kκ(`− τ) + ... (6.2)

where the scaling parameter kκ depends on κ askκ = 1− κ;

• saturation regime, `− τ � zh,

Mκ(`− τ) ≈ −fκ(`− τzh

)3/2

+ ... (6.3)

where the scaling parameter fκ depends on κ as

fκ =√

23

(1− κ2

).

For more complicated model we expect that the formof the function Mκ depends on the model, but the de-pendence on (`− vτ) will survive for more general initialstates, changing only the value of v. In particular, for thelinear growth regime we expect that the speed v, char-acterizing properties of the equilibrium state, is solelydetermined by the metric of the black hole describingthe final state. The form (6.1) manifests itself the localnature of entanglement propagation.

Note, that the 3/2 law in saturation regime exhibitsalso the temporal evolution of causal holographic infor-mation [37].

Let us also note, that the pre-local-equilibration stagein non-equilibrium heating is very similar to the one dur-ing thermalization. As has been noticed in [21] the pre-local-equilibration stage in thermalization is rather sen-sitive to the nature of initial states, including the valueof the sourcing interval δt. In our case the early growthtime dependence is proportional to the difference of theenergy densities

∆S = 2π (Ef − Ei) τ2 + ..., (6.4)

that is consistent with studies of the entanglement en-tropy of excited states at Ti = 0 in [20, 21, 43–46].

We have considered 1+1 dimension case, but using ex-plicit integral representations for the holographic entan-glement entropy and the others nonlocal observables forhigher dimensional cases describing the non-equilibriumheating, one can derive, similar to the case of holo-graphic thermalization process [21], general scaling be-haviour in the corresponding boundary theories duringinstantaneous heating.

It is interesting to consider non-equilibrium heatingprocess initiated by more general quenches, in particularthose with inhomogeneous states, compare with [47–49],as well to consider different infalling shells, in particu-lar massive infalling shells, charged shells and shells withangular momentum, corresponding thermalization pro-cess have been studied in [5, 50–53], [28, 29] and in [19],respectively, as well as the case of corresponding thick

10

shells. Thick shells infalling in black hole backgroundin higher dimensional cases have been already used tostudy numerically the holographic non-equilibrium heat-ing [18]. When the thickness of the shell is less then typ-ical sizes of intervals which we deal with, the evolution of

the entanglement entropy for large intervals also showsthe memory loss regime numerically, the wave front of en-tanglement develop also a finite spread and the pictureis similar to the thermalization picture, but the corre-sponding wave amplitudes M are suppressed as the initialtemperature increases.

1 2 3 4 5 6 7

1

2

3

4

5

6

7

1 2 3 4 5 6 7

1

2

3

4

5

6

7

1 2 3 4 5 6 7

1

2

3

4

5

6

7

-5

-3

-1

0

τ τ τ

` ` `

A B C

1 2 3 4 5 6 7

-7

-6

-5

-4

-3

-2

-1

-5

-3

-1.5

-0.2

1 2 3 4 5 6 7

-7

-6

-5

-4

-3

-2

-1

-5

-3

-1.5

-0.2

1 2 3 4 5 6 7

-7

-6

-5

-4

-3

-2

-1

-3

-2

-1.5

-0.7

-0.2

T− T− T−

` ` `

A B C

FIG. 9. Top: Contour plot for ∆S(`, τ). We see the wave character of the exact ∆S(τ, `) = S−Seq for large τ, ` .The contourplot of ∆S(`, τ): A) zH = 4, zh = 1; B) zH = 1.5, zh = 1. Bottom: The contour plot of ∆S(`, T− + `) as function of ` andT−. A. zH =∞, B. zH = 4, C. zH = 1.5. We see the wave character of the exact ∆S(`, τ) = S − Seq for large τ and `.

ACKNOWLEDGEMENTS

The authors are grateful to M. Khramtsov for usefuldiscussions. This work is supported by the Russian Sci-

ence Foundation (project 14-50-00005, Steklov Mathe-matical Institute).

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