OTOinQM

HETjournalclub17May2017

KojiHashimoto

1703.09435 w/ K. Murata, R. Yoshii

(Out-of-Time-Order)

OTO研究会 5/26（金）13:00-@南部陽一郎ホール13:00-13:05橋本幸士  趣旨説明 13:05-13:50西田充宏  AdS/CFTとOTO(40+5)13:50-14:20村田佳樹  量子力学のOTO(25+5)14:20-14:40沼澤宙朗  2次元CFTのOTO(15+5)14:50-15:35辻直人    物性とOTO(40+5)15:35-16:05段下一平  冷却原子とOTO(25+5)16:15-17:00藤井啓祐  量子情報とOTO(40+5)17:00-17:30根来誠    実験とOTO(25+5)17:30-17:35北川勝浩 おわりに

DoesOTOmeasurequantumchaos?

Notalways.

Q

5 pages

A

7 pages

1703.09435 w/ K. Murata, R. Yoshii

�x(t)p(0)x(t)p(0)�T

StadiumbilliarddoesnotgiveLyapunov.

DoesOTOmeasurequantumchaos?Q

5 pages

�x(t)p(0)x(t)p(0)�T

Chaos:sensi+vetoini+alcondi+onsExample:Stadiumbilliard

QuantumanalogueisOTOCOTOCandLyapunovupperbound

Chaos:sensi+vetoini+alcondi+ons

Classicalchaos=Non-periodicboundedorbitssensiYvetoiniYalcondiYonsinnon-lineardeterminisYcdynamicalsystems

LyapunovexponentL,posiYve

PoincaresecYon,sca]ered

Q-1

�x(0)�x(t)

�x(t) � �x(0) exp[Lt]

4

Example:Stadiumbilliard

5

LyapunovexponentL,posiYve

PoincaresecYon,sca]ered

Q-2

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

6

QuantumanalogueisOTOCQ-3

SchrodingerequaYonislinear,thusnoquantumchaos!?

SoluYon:Out-of-Yme-order(OTO)correlators[Larkin,Ovchinnikov`69][Kitaev`14][Maldacena,Shenker,Stanford`15]

ExponenYalgrowthofOTOcorrelatorcandefine“QuantumLyapunovexponent”L

�x(t)p(0)x(t)p(0)� ��

�x(t)�x(0)

�2

�x(0) �x(t)

�x(t) � �x(0) exp[Lt]

7

OTOCandLyapunovupperboundQ-4

[Maldacena,Shenker,Stanford`15]

Lyapunovupperbound(conjecture)forthermalOTOC

SYK(Sachdev-Ye-Kitaev)model

SolvableatLargeNandstrongcoupling

H =�14!

N�

i,j,k,l=1

j[ijkl]�i�j�k�l

[Kitaev`15][Maldacena,Stanford`16]

[Kitaev`15][Sachdev,Ye`95]

L � 2�T

(1+0dim.,NMajoranafermions,disorderedinteracYon)�

�N�

j,k,l=1

�jijkljijkl� = 6J2

�

�

�J ��

SuggestedfromAdS/CFTwithblackholes

Seeformodifiedmodels[Gross,Rosenhaus`16][Wi]en`16]

SYKmodelsaturatesthebound

DoesOTOmeasurequantumchaos?

Notalways.

Q

5 pages

A

7 pages

1703.09435 w/ K. Murata, R. Yoshii

�x(t)p(0)x(t)p(0)�T

StadiumbilliarddoesnotgiveLyapunov.

Notalways.A

7 pages

1703.09435 w/ K. Murata, R. Yoshii

StadiumbilliarddoesnotgiveLyapunov.

OTOCinquantummechanics,easyOTOCofintegrablesystems

Quantumbilliard,numericallysolvedMicro-canonicalOTOC

ThermalOTOCUniversalmaximumvalueofOTOCNoexp.growthinStadiumbilliards

OTOCinquantummechanics,easy

10

A-1

Micro-canonicalOTOC:

ThermalOTOC:

Step1)

Step2)

OTOCofintegrablesystems

11

A-2

OTOC:

Harmonicoscillator

ParYclein1Dbox

OTOCisobtainedbyanumericalsum.

Quantumbilliard,numericallysolved

12

A-3

1

10

100

1000

0 0.5 1 1.5 2 2.5 3

Micro-canonicalOTOC

13

A-4

1

10

100

0 0.5 1 1.5 2 2.5 3

1

2

5

10

20

40

100

200

400

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

ParYcleinabox

Circlebilliard

Stadiumbilliard

ChaoYcexampleissmoother,non-recursive

0.1

1

10

100

0 0.5 1 1.5 2 2.5 3

1

10

0 0.05 0.1 0.15

ThermalOTOC

14

A-5

1

10

0 0.5 1 1.5 2 2.5 3

1

2

5

10

20

40

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

ParYcleinabox

Circlebilliard

Stadiumbilliard

ChaoYcexampleissmoother,non-recursive,andsaturatestoaconstant

15

UniversalmaximumvalueofOTOCA-6

(Typicalmomentum)2(Typicalsize)2� mT (a + R)2CT �

IntuiYvely,

NumericalCheck:

16

Noexp.growthinStadiumbilliardsA-7

TooshortEhrenfestYme!

EhrenfestYme:aYmescalewhenaparYclewavefuncYonisdelocalizedtothesystemsize

:OriginalsizeofthewavefuncYon,givenbythermaldeBloglielength

tE � L�1 log � � T�1/2 log T

� � �/�

T

tE

1 = �eLtE

:Lyapunovexponent,proporYonaltovelocityandthusto

L ��

T �E

( T � 400 � tE � 0.3 )

DoesOTOmeasurequantumchaos?

Notalways.

Q

5 pages

A

7 pages

1703.09435 w/ K. Murata, R. Yoshii

�x(t)p(0)x(t)p(0)�T

StadiumbilliarddoesnotgiveLyapunov.

18

QuantumProblem:Lyapunovwashedout?App

Quantumchaos=QuanYzingclassicallychaoYcsystem

AtomicspectraofLithiumunderelectricfield[Courtney,Spellmeyer,Jiao,Kleppner,95]

Character:EnergylevelspacingisWigner,notPoisson

Spacingofhadronspectra[Pando-Zayas,00]

SchrodingerequaYonislinear,thusnochaos!?

19

BlackholehorizonandLyapunovApp

Blackholeisafastscrambler?[Sekino,Susskind`08]

Shockwavedelay

horizon

[Shenker,Stanford`13,`14]

UniversalchaosinparYclemoYonnearBH [Tanahashi,KH`16]

r

separatrix

t2

t1�E

�t2 =�E

8�TMe2�T (t2�t1)

2ddilatongravitydualtoSYK[Almheiri,Polchinski`14][Engelsoy,Martens,Verlinde`16]

� C

�r2 +

1(2�T )2

(r � r0)2�

r0

Pot.

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

sn

p�n

a/R = 0 a/R = 1

Figure 2. Studium billiard.

easily restore dimensional parameters notifying

Time ⇠ 2mA

~ , Energy ⇠ ~22mA

, Length ⇠pA .

2 Dynamical studium billiyards

2.1 Classical billiyards

We consider the Sinai studium billiard [13–16] shown in Fig.2 and study classica and

quantum dynamics. We denote radii of semicircles as R and the length of straignt

lines as 2a. Before quantum dynamics, we revisit the chassical dynamics of particles

in the billiard. Inside the studium, the particle moves freely with a constant velocity.

At the boundary of the studium, the partcle is reflected elastically. In Fig.3(a), we

show a typical trajectory of the particle in the studium. We can find the chaotic

behaviour.

One of the most characteristic behaviour in chaotic systems is the sensitivity to

initial conditions: A tiny di↵erence of initial conditions causes significant di↵erence

in future. The Lyapunov exponent is a useful quantity to mesure the strength of

the sensitivity to intial conditions. Denoting the phase space variable as X(t), we

consider its linear perturbation: X(t) ! X(t) + �(t). If X(t) is a chaotic solution,

because of the sensitivity to initial consitions, the perturbation expands exponentially

as �(t) ⇠ e�t. The growing rate � is called Lyapunov exponent. Positive Lyapunov

exponent is the signal of chaos.

In Fig.3(b), we show the Lyapunov exponent as a function of the deformation

parameter a/R.1 Here, we took the unit of v = A = 1, where v is the velocity of

the particle and A = ⇡R2 + 4aR is the area of the studium. From the dimensional

analysis, we can easily restore v and A by replacing � ! pA�/v. The Lyapunov

exponent is zero at the integrable limit a/R = 0. For positive a/R, � increases

1 The boundary condition for the perturbation �(t) at elastic hard collisons has been obtainedin Ref.[20]. We computed the time evolution of �(t) using the boundary condition.

– 5 –

n-thbouncepoint

:Distancefromtheorigintothen-thbouncepointalongtheboundary(whereonecycleisnormilizedto1)

sn

p�n :TangenYalmomentumat

then-thbouncepoint

Origin

Poincaresec+onofstadiumbilliardApp