Black Holes and Random Matrices II:
New Universality at Early Time?
Masanori Hanada
花田 政範

June 29, 2018 @ OISTHrant Gharibyan

Hidehiko Shimada

Masaki Tezuka
Brian Swingle

Characterization of classical chaos•

Sensitivity to a small perturbation.
Lyapunov exponent λL>0.

~x(t)

~x(0)
~x(0)

~x(t)

| ~x(t)| ⇠ exp( L t)

Characterization of quantum chaos
Early time•Sensitivity to a small perturbation.
Lyapunov exponent λL>0.
(Out-of-time-order correlation functions)

Late time•‘Universal’ energy spectrum.
Fine-grained energy spectrum should
agree with Random Matrix Theory (RMT).
‘Black Holes and Random Matrices’@ STRINGS 2017 by S. Shenker
D. Stanford’s talk in this conferenceCharacterization of quantum chaos
Early time•

Late time•Sensitivity to a small perturbation.
Lyapunov exponent λL>0.
(Out-of-time-order correlation functions)
RMT is hidden here as well

‘Universal’ energy spectrum.
Fine-grained energy spectrum should
agree with Random Matrix Theory (RMT).
‘Black Holes and Random Matrices’@ STRINGS 2017 by S. Shenker
D. Stanford’s talk in this conferenceCharacterization of quantum chaos
Also in classical chaos
Early time•

Late time•Sensitivity to a small perturbation.
Lyapunov exponent λL>0.
(Out-of-time-order correlation functions)
RMT is hidden here as well

‘Universal’ energy spectrum.
Fine-grained energy spectrum should
agree with Random Matrix Theory (RMT).
‘Black Holes and Random Matrices’@ STRINGS 2017 by S. Shenker
D. Stanford’s talk in this conferenceCharacterization of quantum chaos
Also in classical chaos
Early timeSensitivity to a small perturbation.
Lyapunov exponent λL>0.
(Out-of-time-order correlation functions)•Late time•

RMT is hidden here as well

‘Universal’ energy spectrum.
Fine-grained energy spectrum should
agree with Random Matrix Theory (RMT).
‘Black Holes and Random Matrices’@ STRINGS 2017 by S. Shenker
D. Stanford’s talk in this conference

Interesting connection to quantum gravity

Lyapunov exponents(Lyapunov spectrum)

Lyapunov Spectrum in Classical Chaos•

Classical phase space is multi-dimensional.

•Perturbation can grow or shrink to various directions.

singular value si(t)eigenvalue si(t)2finite-time Lyapunov exponents

Largest Exponent is not enough
λ1=100λ1=λ2=…λ1000=1λ2=λ3=…λ1000=0

Which is more chaotic?

Coarse-grained entropy andKolmogorov-Sinai Entropy# of cells to cover the region 〜 Π exp(λt)λ>0eλtCoarse-grained entropy＝ log[# of cells to cover the region]〜 (sum of positive λ) × t

KS entropy ＝ (sum of positive λ)＝ entropy production rate

Largest Exponent is not enough
λ1=100λ1=λ2=…λ1000=1λ2=λ3=…λ1000=0

Which is more chaotic?

λ1++λ2+…+λ1000=100

λ1++λ2+…+λ1000=1000

More chaotic

Bigger black hole is colder.

Bigger black hole is less chaotic?

D0-brane matrix modelNNBH

2 BHsgas

N/2N

N/2N

(@high-T region)

N/2

λ=0

N

N/2
N

λ=0
(@high-T region)

N/2

λ=0

N

N/2
N

λ=0
(@high-T region)

Similar calculation is doable at low-T and also for other theories (Berkowitz-MH-Maltz 2016)

More chaotic

N/2

λ=0N

N/2
N

λ=0
(@high-T region)

Similar calculation is doable at low-T and also for other theories (Berkowitz-MH-Maltz 2016)

Plan•

Universality of classical Lyapunov spectrum
MH, Shimada, Tezuka, PRE 2018

•Universality of quantum Lyapunov spectrum
Gharibyan, MH, Swingle, Tezuka, in progressLyapunov Spectrum→ singular value si(t)→ eigenvalue si(t)2finite-time Lyapunov exponents

Lyapunov Spectrum

Easily to calculate with good precision

D0-brane matrix model
negligible at high-T

black holein IIA string

equivalenthigh-T = ‘stringy’string

BH

D0-brane matrix model

high-T = classicalGur Ari-MH-Shenker, JHEP2016
t=20.7

NN

Fitting ansatz

T=1

RMT vs Classical Chaos• The correlation of the finite-time Lyapunov
exponents may have a universal behavior?

(Some hints found in the previous study by Gur-Ari, MH, Shenker)(different from si = exp(λit), sorry for using the same letter!)• N→∞ before t→∞(In chaos community, often t→∞ is taken first.)

GOE-distribution at any time
t=0

t=10

Lyapunov exponents are described by RMT
M.H.-Shimada-Tezuka, PRE 2018

with a mass term (→no gravity interpretation),
GOE is gone, at t=0.

m=3, t=0

But GOE is back at later timet=0t=3

Summary of numerical observations• Universality beyond nearest-neighbor can be checked.
(Spectral Form Factor)• D0-brane matrix model — RMT already t=0
Maybe a special property of quantum gravitational systems?

• Other systems — not RMT at t=0, but graduallyconverges to RMT.

Likely to be a universal property in classical chaos.
Generalization to quantum theory?

• So far we have looked at only the bulk of the spectrum;
not the edge.

Early-time universality in quantum chaos

Gharibyan, MH, Swingle, Tezuka, in progress•

There is no consensus for the definition of ‘quantum
Lyapunov spectrum’•

Let’s try the simplest choice:grows exponentiallycannot capture the growth

SYK model

maximally chaotic

integrable

Lyapunov growth
20%80%

(T=∞)

λNt

λNt

tt
Preliminary

Lyapunov growth
20%

λi

80%

20%80%λi

Preliminary

RMT behavior
Poisson
1GUE
0.8

N = 14, K = 0.01
t = 0.02
t=1t = 100
GOE
GUE
PoissonP(s)0.60.4

Preliminary
0.2

000.5

11.5s22.5

RMT behavior
K > 0 → chaotic at high energy, non-chaotic at low energy(Garcia-Garcia, Loureiro, Romero-Bermudez, Tezuka, 2017)

Our numerical data suggests:
Chaotic states → RMT
non-chaotic states → Poisson
Brownian circuit version is consistent with this interpretation.

Spin chain (XXZ model)

XXX model

random magnetic field• Ergodic at small• Many-body localized (MBL) at large

Spin chain (XXZ model)
Preliminary
N=7
N=6
N=5
N=4

w=0.5 (ergodic phase), T=∞Spin chain (XXZ model)
power growth(ballistic growth of operator size?)

Preliminary
N=7
N=6
N=5
N=4

power growth(diﬀusion?)
exponentialgrowth

w=0.5 (ergodic phase), T=∞

RMT vs Lyapunov spectrum in XXZ model
1

PoissonN = 10, W = 0.5
t = 0.2
t=1
t=2
GOE
GUE
Poisson

0.9

GUE

0.8
0.7

P(s)

0.6
0.5

Preliminary

0.4
0.3
0.2
0.1
000.5

11.5
Normalized gap s22.5

N=10, w=0.5 (ergodic phase)

RMT vs Lyapunov spectrum in XXZ model
1Poisson
GUE0.8N = 10, W = 4t = 0.2
t=1t=2GOE
GUEPoisson

P(s)

0.6

Preliminary
0.4

0.2

000.5

11.5
Normalized gap s22.5

N=10, w=4.0 (MBL phase)Summary of numerical observations•Classical chaos
D0 matrix model — ‘strongly’ universal
Other chaotic systems — universal•

Quantum chaos
SYK — ‘strongly’ universal
Other chaotic systems — universal
MBL — not universal (Poisson-like)•Lyapunov growth can be seen precisely.

Conclusion & Outlook
The largest Lyapunov exponent is not enough.
Lyapunov spectrum captures physics more precisely.
New universality.
Black hole is (probably) special.
What is the mechanism?
How can we formulate the spectrum in gravity side?
Relation to the late time universality (energy spectrum)?
‘KS entropy’ vs EE growth rate?
Generalization of the chaos bound to KS entropy?