Chaotic Behavior in Matrix Gauge Theories

Guy Gur-Ari Stanford University

In progress, with Masanori Hanada and Steve Shenker

Lawrence Livermore National Laboratory, April 2015

Overview• Chaotic behavior of gauge theories

using real-time dynamics

• Motivations:

• Thermalization in quark-gluon plasma

• Black hole dynamics

• Classical dynamics of 0+1D gauge theory with matrix degrees of freedom

The Gauge Theory

Matrix gauge theory

• Quantum mechanics of N x N Hermitian matrices:

• U(N ) gauge symmetry, SO(9) global symmetry

• Large N limit: keep fixed

X1(t), . . . , X9(t)

� = g2YMN

gauge field, fermions

L =N

2�Tr

0

@9X

i=1

(Xi)2 +

1

2

9X

i,j=1

[Xi, Xj ]2

1

A+ · · ·

Holographic dualities

High temperature

Large N, strong coupling

Black hole

9+1D Gravity (string theory)

Supersymmetric Yang-Mills in 3+1D

GravityMatrix gauge theory in 0+1D

[Banks, Fischler, Shenker, Susskind 1997]X1, . . . , X9

Matrix model holographic duality

x1

x2

xN

Particle

String

9+1D

Matrix model holographic duality

Xi =

0

B@x1

x2

. . .

1

CA

Xi =

0

B@x1 x12

x

⇤12 x2

. . .

1

CA

x1

x2

xN

Particle

String

9+1D

Holographic duality at finite temperature

Thermal state of matrix gauge theory

Black hole in a gravity theory

• Valid at large N, strong coupling • We focus on weak coupling — large stringy corrections • But: no phase transition in the coupling

Xi =� �

N⇥N

U ⇠ Tr [Xi, Xj ]2

Real-time dynamics of matrix gauge theory

Weak coupling limit of matrix model

• Effective dimensionless coupling:

• Large N, weak coupling / high temperature limit → classical dynamics

• Classical observables are functions of

L =N

2�Tr

0

@9X

i=1

(Xi)2 +

1

2

9X

i,j=1

[Xi, Xj ]2

1

A+ · · ·

�e↵ =�

T 3

�e↵ T 4

Equations of motionL =

N

2�Tr

0

@9X

i=1

(Xi)2 +

1

2

9X

i,j=1

[Xi, Xj ]2

1

A+ · · ·

Xi(t) =9X

j=1

[Xj , [Xi, Xj(t)]]

9X

i=1

[Xi(t), Xi(t)] = 0

(Discretize and solve

Lyapunov exponent

�~x(0)

�~x(t)

~x(0)

~x(t)

Positive exponent signals chaotic behavior

Long time, small perturbation:

|�~x(t)| ⇡ e

�Lt|�~x(0)|

Measuring theLyapunov exponent

Fix the energy (micro-canonical

ensemble)

Measuring theLyapunov exponent

Fix the energy (micro-canonical

ensemble)

Evolve for a while to reach a typical state

Measuring theLyapunov exponent

Fix the energy (micro-canonical

ensemble)

Evolve for a while to reach a typical state

Perturb

Measuring theLyapunov exponent

Fix the energy (micro-canonical

ensemble)

Evolve for a while to reach a typical state

Perturb

Measure distance

⇠ exp(�Lt)

Real-time exponential growth

|�X|

t

|�X| ⇠ exp(�Lt)

Perturb

� �� �� �� �� ���

��-�

��-�

�����

�����

⇠pN

(N = 8, 16)

1/N behavior

���� ���� ���� ����������������������������������������������������

�/�

λ �

(N = 6, . . . , 24)�L =

0.293� 0.014

N+O(1/N2)

��1/4e↵ T

Lyapunov exponent: gauge theory vs. black holes

�e↵

No phase transition

Black hole holographic dual

Classical [Shenker, Stanford 2014][Sekino, Susskind 2008]

c = ~ = kB = 1

[Maldacena, Shenker, Stanford 2014]⇠ �1/4

e↵ T

2⇡T

�L

Scrambling time and large N behavior

• Scrambling time = time to completely de-localize a local perturbation

• Numerically:

|�X|

t

t⇤ ⇠ logN

�L

� �� �� �� �� ���

��-�

��-�

�����

�����⇠

pN

exp(�Lt⇤) ⇠pN

Scrambling time and large N behavior

• Scrambling time = time to completely de-localize a local perturbation

• Black holes are ‘fast scramblers’:

• Same behavior in our system:

t⇤ ⇠ logS ⇠ logN

�L ⇠ N0 , t⇤ ⇠ logN

�L⇠ logN

Real-Time Correlators

Lyapunov exponent from correlators

Motivation: Making contact with the quantum theory

hO(0)O(t)i = 1

T

Z T

0dt0 O(t0)O(t0 + t)

� ⇡ �L ??

hO(0)O(t)i � hOi2 ⇠ exp(�˜�t)

Lyapunov exponent from correlators

• Choose operators with vanishing 1-point functionsDTr(XiXj)(0) Tr(XiXj)(t)

E

loghO

Oi

� � � � � ��-�-�-�-�-�-�

�

(i 6= j)

Lyapunov exponent from correlators

• Choose operators with vanishing 1-point functionsloghO

Oi

� � � � � ��-�-�-�-�-�-�

�

DTr(XiXj)(0) Tr(XiXj)(t)

E(i 6= j)

Lyapunov exponent vs. correlator exponents

Lyapunov

Correlators

���� ���� ���� ������������������������������������

�/�

��������

Lyapunov Spectrum

Lyapunov spectrum• Generic perturbation grows as:

• Tuned perturbations have different exponents

• There is a spectrum of Lyapunov exponents:

no. exponents = dimension of phase space

• Lyapunov exponent is the largest eigenvalue�L

|�~x(t)| ⇡ exp(�Lt)|�~x(0)|

Lyapunov spectrum

~x(0)

|�~x(t)| ⇠ e

�Lt

�~x

0

�~x

|�~x0(t)| ⇠ e

�0Lt

Lyapunov spectrum motivation

• Detailed information about chaotic behavior

• Kolmogorov-Sinai entropy measures rate of entropy production

• Approximated by sum of positive exponents

[Kunihiro, Müller, et al. 2010]

Measuring the spectrum• Lyapunov spectrum = spectrum of transfer matrix

✓�X�V

◆= U(t)

✓�X�V

◆

�Xi = �Vi , �Vi = [�Xj , [Xi, Xj ]] + · · ·

Global Lyapunov spectrum• Measure spectrum of U(t) at late times

✓�X�V

◆= U(t)

✓�X�V

◆

�L ⇡ 0.3

Partial Results-���-���-��� ��� ��� ��� ����

���

���

���

��� N = 4

Local Lyapunov spectrum• Spectrum of U(t) at short time

• Lyapunov exponent is not the largest eigenvalue

�L ⇡ 0.3Partial Results-� � � ��

����

����

����

���� N = 4

Fast eigenvector evolution

�x(0)

e(0) Largest eigenvector

�x(t)e(t)

Fast eigenvector evolution

��� ��� ��� ��� ���-�-�-�-�-��

�

��������

Largest eigenvector overlap

Perturbation overlap

(normalized)

Perturbation cannot catch up with evolving eigenvector!

(e(t), e(0))

(�x(t), �x(0))

Random matrix toy model

• Toy model: Evolve with random matrices

• Spectrum + time step taken from gauge theory

• Measure exponent

��� ��� ��� ��� ���-�-�-�-�-��

�

��������

is typical decay time

dt

�x(t) = (Un)dt · · · (U1)

dt�x(0) ⇠ e

�t�x(0)

Random matrix exponents

Gaugetheory

Randommatrices

���� ���� ���� ������������������������������������

�/�

��������

Summary• Real-time chaotic behavior in classical gauge theory

• Lyapunov exponent converges at large N

• Fast scrambling behavior, similar to black holes

• Relation to real-time correlators

• Going to higher dimensions:

• UV cascade, classical approx. may break down

Thank You!

� �� �� �� �� ���

��-�

��-�

�����

�����

Discretization

Xi(t+ dt) = Xi(t) + Vi(t)dt+ Fi(t)dt2

2

Vi(t+ dt) = Vi(t) + [Fi(t) + Fi(t+ dt)]dt

2

Vi(t) = X(t)

Fi(t) =9X

j=1

[Xj(t), [Xi(t), Xj(t)]]

Sprott’s algorithm

• Make a small perturbation X -> X+dX

• Evolve one time step dX -> dX’

• Record log(|dX’| / |dX|)

• Rescale |dX’| -> |dX|

Dlog

|�X 0||�X|

E! �L

Error accumulation

�� �� �� �� ����

��

��

��

��

��

��

�

|�|

N = 8

�t = 5 · 10�4�t = 10�4

Exponent vs. thermalization time

0 500 1000 1500 2000 2500 3000 35000.2890

0.2895

0.2900

0.2905

0.2910

0.2915

0.2920

t

λ L

N = 8

Classical coupling dependence

�L = f(�T ) = f(�e↵T4) ⇠ T

�E =N

2Tr

⇣Xi + [Xi, Xj ]2

⌘

�L ⇠ �1/4e↵ T