Workshop “String theory, gravity and cosmology (SGC2021), APCTP Korea (online), Nov 17 2021
Workshop “ApplicaFon of Quantum InformaFon in QFT and Cosmology”, U.Lethbridge (online), Nov 22 2021
Workshop “TheoreFcal Physics at CQUeST: Past, Present, and Future, CQUeST, Korea (online), Feb 10 2022
The 3rd Colloquium of Extreme Universe collaboraFon (online), Feb 28 2022

Chaos and Holography
Koji Hashimoto(Kyoto U)
2112.11163 w/ K. Murata, N. Tanahashi, R. Watanabe
Based also on
2007.04746, 1903.04718, 2004.04381, 1903.04718, 1804.01737, 1803.06756,
1703.09435, 1610.06070, 1605.08124 w/ T. Akutagawa, K. B. Huh, K. Y. Kim,
T. Miyazaki, K. Murata, T. Ota, T. Sasaki, N. Tanahashi, K. Yoshida, R. Yoshii, R. Watanabe)

24-year history of AdS/CFT[Polchinski]
1995

Low energy

SUSY QFT[Douglas]1996

Low energy

Soliton[Maldacena]
1998

AdS = CFT

[Sakai Sugimoto]
2004

AdS = QCD
AdS = Entanglement
Horizon = Chaos
Inside BH = Complexity
Worm hole = InformaWon recovery[Ryu Takayanagi] 2006
[Kitaev] 2014
[Susskind] 2014
[Penington] 20192

22-year history of AdS/CFT

What kind of QFT allows a gravity dual?

AdS = Entanglement
Horizon = Chaos
Inside BH = Complexity
Worm hole = InformaQon recovery[Ryu Takayanagi] 2006
[Kitaev] 2014
[Susskind] 2014
[Penington] 20193

4

Chaos and Holography

1

Quantum chaos, redundantly defined2

BH horizon is universally chaotic

3

How to look for “BH-like” QM

4Conjecture: Chaos energy bound5

1

Quantum chaos, redundantly defined

1-1

Chaos : sensitivity to initial conditions

1-2

Example : Stadium billiard

1-3

No quantum chaos?

1-4

Two deﬁniBons of quantum chaos

1-1Chaos : sensitivity to initial conditions

Classical chaos= Non-periodic bounded orbits
sensitive to initial conditions
in non-linear deterministic dynamical systems
Poincare section,
scattered

Lyapunov exponent
posi;ve<latexit sha1_base64="/cpIy7UAJZZ0ZpHSNrhE5nFM6rU=">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</latexit><latexit sha1_base64="9OCgXiZ9/zYiXx9MJrqM7ZKiidc=">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</latexit>,

x(t) ⇠ x(0) exp[ t]
x(0)

x(t)
7

1-2

Example : Stadium billiard

Poincare section,
scattered

Lyapunov exponent L,
positive

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

8

1-3

No quantum chaos?

Reason 1 : Schroedinger eq. is linear.
Reason 2 : Infinitesimal change in initial condition
does not exist in quantum systems.
Quantum chaos, seen by energy level spacings

Atomic spectra of Lithium in electric field[Courtney, Spellmeyer, Jiao, Kleppner, 95]

Hadron spectra spacings[Pando-Zayas, 00]
91-4

Two definitions of quantum chaos

Def 1 “Chaos” : Quantization of classically chaotic system.
Energy level spacings follow Wigner distribution[Muller, Heusler, Altland, Braun, Haake `09]

Def 2 “Scrambling” : Exponential growth of
Out-of-time-order (OTO) correlator.

hx(t)p(0)x(t)p(0)i / exp[2 t]<latexit sha1_base64="3HeapIS68nTrgD6JhIQcOUnj+8Y=">AAACoXichVFBSxtBFP7cqtW0aqwXwctisMRLeLEtltKD1IveYmxUyIawu07i4uzusDsJ2uAf8A946ElBRPozevGq4MGfID1a6KUHXzYL2or6lp355nvve/PNjKOkF2uiqz7jRf/A4Muh4cyr1yOjY9nxN2tx2IpcUXFDGUYbjh0L6QWioj0txYaKhO07Uqw724vd/HpbRLEXBl/1rhI1324GXsNzbc1UPfvZknbQlMLcyetZU+Vp9h6yol7OUlGodGhaYkdV51jhO5u2qWv1bI4KlIT5EBRTkEMapTB7AgubCOGiBR8CATRjCRsxf1UUQVDM1dBhLmLkJXmBPWRY2+IqwRU2s9s8NnlVTdmA192ecaJ2eRfJf8RKEzN0Sad0Q2f0g67p76O9OkmPrpddnp2eVqj62P7k6p9nVT7PGlt3qic9azTwMfHqsXeVMN1TuD19+9vBzeqn8kznLR3RL/Z/SFf0k08QtH+7xyui/B0ZfoDi/9f9EKzNFYrvCh9W3ucWvqRPMYQpTCPP9z2PBSyhhArve4QznOPCyBnLRsko90qNvlQzgX/CqN4C0fKeyw==</latexit><latexit sha1_base64="MO688p7wo9D/hOtponpq877i9Sg=">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</latexit>: Quantum Lyapunov exponent

Classical analogue:
x(t)p(0)x(t)p(0)

x(t)
x(0)

[Larkin, Ovchinnikov `69]
[Kitaev `14] [Maldacena,
Shenker, Stanford `15]

2

x(0)

x(t)
<latexit sha1_base64="9Y0RpjgtLVPoz3sk7gUEHdE7rKI=">AAACbnichVHLSsNAFD2Nr1pfVUEEEYulUjdl6gPFVdGNyz6sLbRFkjhqME1CMi3W4g+4FxeCoiAifoYbf8BFP0HcCApuXHibBkSLesNkzpy5586ZO4qla45grOGTOjq7unv8vYG+/oHBoeDwyKZjVmyVZ1VTN+28Ijtc1wyeFZrQed6yuVxWdJ5T9tea+7kqtx3NNDZEzeKlsrxraDuaKguiCsVtrgs5dBAVs1vBMIsxN0LtIO6BMLxImsEbFLENEyoqKIPDgCCsQ4ZDXwFxMFjElVAnziakufscRwiQtkJZnDJkYvfpv0urgscatG7WdFy1SqfoNGxShhBhj+yWvbIHdsee2MevtepujaaXGs1KS8utraHj8cz7v6oyzQJ7X6o/PQvsYNn1qpF3y2Wat1Bb+urh6WtmJR2pz7Ar9kz+L1mD3dMNjOqbep3i6TME6AHiP9vdDjbnYvH52GJqIZxY9Z7CjwlMI0r9XkIC60gi63bsBOe48L1IY9KkNNVKlXyeZhTfQop+Ahmyja4=</latexit>

2

BH horizon is universally chaotic

2-1 Chaotic motion around BH, known, but…
2-2

Lyapunov is Hawking temperature

2-3

Quantum Lyapunov upper bound2-4 Chaotic motion near horizon is universal

2-1 Chaotic motion around BH, known, but…
Motion of a spinning particle
going around Kerr BH[Suzuki, Maeda ‘96]

But 1 ) the motion is far from horizon
But 2 ) Geodesic motion around Schwarzschildor Kerr is integrable
14

2-2

Lyapunov is Hawking temperature

Black hole is the fastest scrambler?

[Sekino, Susskind `08]

Shock wave delay : Lyapunov = Hawking temperature[Shenker, Stanford `13,`14]

horizont2 2d dilaton gravity dual to SYK[Almheiri, Polchinski `14] [Engelsoy, Martens, Verlinde `16]E

t1

E 2 T (t2 t1 )
e
t2 =
8 TM

15

2-3

Quantum Lyapunov upper boundQuantum Lyapunov upper bound for thermal OTOC
<latexit sha1_base64="y9BKDSLAzCuBeHe/ReyD6XEF8EE=">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</latexit>[Maldacena, Shenker, Stanford `15]

 2⇡T

Suggested from AdS/CFT with black holes
SYK model saturates the bound [Kitaev `15] [Maldacena, Stanford `16]
SYK (Sachdev-Ye-Kitaev) model [Kitaev `15][Sachdev,Ye `95]
(1+0 dim., N Majorana fermions, disordered interaction)
1
H=
4!

N

N

j[ijkl]i,j,k,l=1ijkjijkl jijkl = 6J 2l
j,k,l=1

Generalizations: [Gross, Rosenhaus `16] [Witten `16][Murugan Stanford Witten `17]

Solvable at Large N and strong couplingJ16

2-4 Chaotic motion near horizon is universal[Tanahashi, KH `16]r

Potential V(r)Gravityr0
Universal Lyapunov exponent = Hawking temperature
Potential

Separatrix
1
C ṙ +(r2(2 T )
2

r0

r

r0 )2
17

22-year history of AdS/CFT

What kind of QFT allows a gravity dual?
Look for a quantum system with<latexit sha1_base64="IL8uoeniJfv/0L7RfdFNXs3d2a8=">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</latexit>= 2⇡T

AdS = Entanglement
Horizon = Chaos
Inside BH = Complexity
Worm hole = Information recovery[Ryu Takayanagi] 2006
[Kitaev] 2014
[Susskind] 2014
[Penington] 201920

3

How to look for “BH-like” QM

3-1

Computing OTOC is difficult

3-2

How to look for “BH-like” QM

3-3

Quantum billiards : no scrambling

3-4

Coupled harmonic oscil. is scrambled

3-5 Inverted harm. oscil., scrambled w/o chaos
3-6Caution! Scrambling 6= Chaos<latexit sha1_base64="dR8cqXybAcGP1XNUpaZWKuvdaaQ=">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</latexit>3-1Computing OTOC is difficult

OTOC in N=4 Supersymmetric Yang-Mills theory?
-- Difficult, because…
1) Energy eigenstate wave function, unknown2) Perturbation theory does not give long time behavior
3) It’s out-of-time order, so path integral does not workEx) OTOC of

4<latexit sha1_base64="2KSP9kk6pG/avhEc6355vcG+azg=">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</latexit>

matrix theory. [Stanford `15]

Ex) OTOC of IP model and IOP model.
[Michel, Polchinski, Rosenhaus, Suh `16]

22

3-2

How to look for “BH-like” QM

1. Prepare your Hamiltonian and solve energy eigenstates

H|Ei = E|Ei
<latexit sha1_base64="P/2Bt8gjs/2dmRJIvATxlc18Sqk=">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</latexit>

2. Calculate microcanonical OTOC

f (t; E) ⌘ hE|x(t)p(0)x(t)p(0)|Ei<latexit sha1_base64="36vHxCUD1z9CPwDrygZiXptefDU=">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</latexit>

3. Convert it to thermal OTOC

f (t; T ) ⌘<latexit sha1_base64="G6zl1bheRHuaY8KF6q1wn7QZKgY=">AAACmXichVHLSiNBFD22jo84atSN4KYwOETQUPGB4mx8EJBZqTEq2Bq6OxUt7HS33ZWAE/wBf2BgXCmIiD8x4GbGvQs/QWbpgBsXc7vTMIyi3qbrnjp1z61TVaZny0BxftekNbd8aG1r70h0fuzq7kn29q0HbtW3RMFybdffNI1A2NIRBSWVLTY9XxgV0xYb5v5iuL5RE34gXWdNHXpiu2LsOrIsLUMRVUxOltPq89oI08VBVdaYLh3FSjmmjzLd33PTuREWFlASO/Ux3RTKYLmjYjLFMzwK9hJkY5BCHMtu8gI6SnBhoYoKBBwowjYMBPRtIQsOj7ht1InzCcloXeAICdJWqUpQhUHsPo27NNuKWYfmYc8gUlu0i02/T0qGYX7LL/kD/8mv+D1/erVXPeoRejmkbDa0wiv2HA/kH99VVSgr7P1TvelZoYyZyKsk717EhKewGvra128P+dnV4fonfsZ/k/9Tfsev6QRO7Y91viJWT5CgB8g+v+6XYH08k53ITK1MpuYW4qdoxyCGkKb7nsYclrCMAu37HT/wCzfaoDavLWlfGqVaU6zpx3+h5f8CzQWa2g==</latexit>ZdE ⇢(E)f (t; E)eE

4. Read your thermal Lyapunov and check!
<latexit sha1_base64="SV+3mhPSs1JnzAxHQOlfPAM/fh4=">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</latexit>

1
log f (t; T )thermal (T ) = lim
t:large t

23

3-3

Quantum billiards : no scrambling[Murata, Yoshii, KH `17]

OTOC

Particlein a box

Circlebilliard

Stadiumbilliard

40

100

20
10

10

10

5

1

10

2

1
1

1
0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0

0.5

1

1.5

2

2.5

3

0.1

0

0

0.5

1

1.5

0.05

2

0.1

2.5

Harmonic oscillator
OTOC :

Periodic, no scrambling.

24

0.15

3

3-4

Coupled harmonic oscil. is scrambled[Akutagawa, Sasaki, Watanabe, KH `20]

Hamiltonian is essence of BFSS matrix model
1 2
2
2
H = px + py + (x + y 2 ) +<latexit sha1_base64="ukU3tWKOpO7+LsD9W4C8iwswO4A=">AAACn3ichVHBShtRFD2OWjVqjboRuhkNFiUQ7kRFCRSCXdRViYlRi6ZhZnzRIZOZYWYSTIM/4A+4cKMFKaWfIYg73bjwE8SlghsX3pkMLVZq7zDvnnfePfed957mmIbnE113SJ1d3W96evti/QODb4fiwyOrnl13dVHUbdN21zXVE6ZhiaJv+KZYd1yh1jRTrGnVj8H6WkO4nmFbK37TEaWaum0ZFUNXfabK8cyS/EF2yrtf03KSczPMmxVX1ZVZeYrpJFPTv7mWQntyUMxsOZ6gFIUhvwRKBBKIImfHf2ATW7Cho44aBCz4jE2o8PjbgAKCw1wJLeZcRka4LrCHGGvrXCW4QmW2yuM2zzYi1uJ50NML1TrvYvLvslLGJF3RT7qjc/pFN/T4z16tsEfgpclZa2uFUx7aHys8/FdV4+xj54/qVc8+KlgIvRrs3QmZ4BR6W9/4dnBXyOQnW+/pO92y/2O6plM+gdW410+WRf4QMX4A5e/rfglW0yllJjW3PJvILkZP0Yt3mMAU3/c8slhCDkXe9whnuMClNC59kj5LuXap1BFpRvEspC9PwcubhQ==</latexit>

4

Classical Poincare sections

1 2 2
x y
10

OTOC

25

3-5 Inverted harm. oscil., scrambled w/o chaos[Huh, Kim, Watanabe, KH `20]

Classical one-dim. system does not give chaos

Quantum energy levels

OTOCCf) Analyses of pure IHO [Bha;acharyya, Chemissany, Haque, Murugan, Yan `20]

26

3-6

Caution! Scrambling 6= Chaos<latexit sha1_base64="dR8cqXybAcGP1XNUpaZWKuvdaaQ=">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</latexit>

Def. 1“Chaos”(ClassicalHarm. oscil.
Billiard
Coupled harm. oscil.
Inverted harm. oscil.

<latexit sha1_base64="kNdg6GnxKmTwq+DhvpE/7Gtj0k0=">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</latexit>

Def. 2“Scrambling”> 0 ) (Quantum<latexit sha1_base64="kNdg6GnxKmTwq+DhvpE/7Gtj0k0=">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</latexit>

>0)

4

Conjecture: Chaos energy bound4-1

Conjecture

4-2

General billiards satisfy the bound4-3 General 2d mechanics sa9sﬁes the bound
4-4

Saturation necessary for black holes

4-5Discrimina9on of BSM models?!

1-4

How to look for “BH-like” QM

1. Prepare your Hamiltonian and solve energy eigenstates

H|Ei = E|Ei
<latexit sha1_base64="P/2Bt8gjs/2dmRJIvATxlc18Sqk=">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</latexit>

2. Calculate microcanonical OTOC

f (t; E) ⌘ hE|x(t)p(0)x(t)p(0)|Ei<latexit sha1_base64="36vHxCUD1z9CPwDrygZiXptefDU=">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</latexit>

3. Convert it to thermal OTOC

f (t; T ) ⌘<latexit sha1_base64="G6zl1bheRHuaY8KF6q1wn7QZKgY=">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</latexit>ZdE ⇢(E)f (t; E)eE

4. Read your thermal Lyapunov and check!
<latexit sha1_base64="SV+3mhPSs1JnzAxHQOlfPAM/fh4=">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</latexit>

1
log f (t; T )thermal (T ) = lim
t:large t

31

4-1

Conjecture[Murata, Tanahashi, Watanabe, KH `21]

Chaos energy bound: Lyapunov exponent for energy E
3. Convert it to thermal OTOC

f (t; T ) ⌘<latexit sha1_base64="G6zl1bheRHuaY8KF6q1wn7QZKgY=">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</latexit>ZdE ⇢(E)f (t; E)eE

When the microcanonical Lyapunov grows faster thanlinearly in energy, you cannot go to canonical!
32

4-1

Conjecture
Chaos energy bound: Lyapunov exponent for energy E“Lyapunov does not grow faster than linearly in energy
in the high energy limit”

33

4-1

Conjecture
Chaos energy bound: Lyapunov exponent for energy E

Satisfied by :Double penduram, Henon-Heiles system,
Particle in black hole geometries, Nonlinear sigma models,Fermi-Pasta-Uram β-model, large N coupled rotors,
Chaotic string in AdS soliton geometry,
Homogeneous Yang-Mills mechanics,
Yang-Mills theories on a lattice, ...
34

4-2

General billiards sa,sfy the boundH=p( > 0)<latexit sha1_base64="Urxdq1oiJN8+WRlNhF0gpzRAaco=">AAACb3ichVHLSsNAFD2N7/po1YWCIMGi6KZMfKAIQtFNl9XaKtgqSZzWYF4kaaEWf8AP0IULHyAifoYbf8BFP0FcSQU3LrxNA6JFvcPMnDlzz50zM4qta67HWC0ktLV3dHZ194R7+/oHItHBoaxrlRyVZ1RLt5xtRXa5rpk842mezrdth8uGovMt5XCtsb9V5o6rWeamV7F53pCLplbQVNkjKpcUV0R7N1eUDUPei8ZYnPkhtgIpADEEkbKit8hhHxZUlGCAw4RHWIcMl9oOJDDYxOVRJc4hpPn7HMcIk7ZEWZwyZGIPaSzSaidgTVo3arq+WqVTdOoOKUVMsid2x+rskd2zZ/bxa62qX6PhpUKz0tRyey9yMpp+/1dl0Ozh4Ev1p2cPBSz5XjXybvtM4xZqU18+Oqunlzcmq1Psmr2Q/ytWYw90A7P8pt6s841zhOkDpJ/P3Qqys3FpLr6wPh9LrAZf0Y0xTGCa3nsRCSSRQobOtXGKC1yGXoURYVwQm6lCKNAM41sIM5+bOI3n</latexit><latexit sha1_base64="rH1lhl2MEr3JA7BxHisrjyffecM=">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</latexit>)<latexit sha1_base64="2r9G/Wj8zvALzrin82kbCzgm7xY=">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</latexit>35

4-3 General 2d mechanics sa/sﬁes the boundGeneral 2d Hamiltonian :Consistent motion needs
Bounded orbits need) Scaling symmetry determines energy exponent c,<latexit sha1_base64="2r9G/Wj8zvALzrin82kbCzgm7xY=">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</latexit>

36

4-4

Saturation necessary for black holes
The chaos bound(quantum large N)

Chaos energy bound(quantum / classical,
finite / large N)[Maldacena, Shenker, Stanford `15]c<latexit sha1_base64="trIC8M3OB/o7bqvZtpcUgg8sQRM=">AAACgXichVHLShxBFD2279HoqBshLpoMhggy3PaBIW7EILj0NSo4OnTXlNpY09121wzo4MalP+DClYKouNJfcOMPZOEniEsD2bjwTk9DSCTxFlV16tQ9t05VOYFyI0300GA0NjW3tLa1pzo6P3R1p3t6lyO/HAqZE77yw1XHjqRyPZnTrlZyNQilXXKUXHF2vtf2VyoyjFzfW9J7gVwv2Vueu+kKWzNVSA/kFScXbTMfhH6gfXNmQwybIq/krmkV0hnKUhzmW2AlIIMk5vz0BfIowodAGSVIeNCMFWxE3NZggRAwt44qcyEjN96XOECKtWXOkpxhM7vD4xav1hLW43WtZhSrBZ+iuIesNDFIP+iKnumerumRXv5ZqxrXqHnZ49mpa2VQ6D7qX/z1rqrEs8b2b9V/PWts4mvs1WXvQczUbiHq+sr+8fPit4XB6mc6oyf2f0oPdMc38Co/xfm8XDhBij/A+vu534Llkaw1mh2fH8tMTSdf0YaP+IQv/N4TmMIs5pDjcw9xiRvcGo3GkEHGSD3VaEg0ffgjjMlXKyCTaA==</latexit>/ E ,c  1
SaturaHon<latexit sha1_base64="7ZQt4eNrgQYOGyPoZy/FcFyLyaw=">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</latexit>

Large N/E

<latexit sha1_base64="tU92cHK7EcYvZ7W92S5Y2kHpp5Y=">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</latexit>(Density of states

T /T

)
37

4-5

Discrimina)on of BSM models?!
Fig.1.1 of “Origin of
Symmetries” by
Froggatt and Nielsen

Lyapunov is Kolmogorov-Sinai entropy,
and saturation means the fastest
entropy production at high energy.
QFT saturating the chaos energy boundis selected by entropic principle, as
the theory of extreme universe?!
38

Chaos and Holography

1

Quantum chaos, redundantly defined2

BH horizon is universally chaoAc

3

How to look for “BH-like” QM

4Conjecture: Chaos energy bound40