SYK, Chaos and Higher-Spins

Cheng Peng

Brown University

March 27, 2019

Based on: JHEP 1812 (2018) 065, 1812.05106 with Ahn

• Strongly coupled Quantum Mechanics model

• Perturbatively solvable in the large N limit

(Sachdev, Ye, 1993; Parcollet, Georges, 1998; Kitaev 2015,2017, Maldacena, Stanford 2016, Polchinski, Rosenhaus 2016, Jevicki, Suzuki, Yoon 2016, Bagrets, Altland, Kamenev 2017... )

The Sachdev-Ye-Kitaev (SYK) model

The Sachdev-Ye-Kitaev (SYK) model

• Strongly coupled Quantum Mechanics model

• Perturbatively solvable in the large N limit

• 2-point function in the IR , conformal

(Sachdev, Ye, 1993; Parcollet, Georges, 1998; Kitaev 2015,2017, Maldacena, Stanford 2016, Polchinski, Rosenhaus 2016, Jevicki, Suzuki, Yoon 2016, Bagrets, Altland, Kamenev 2017... )

• A tower of higher-spin operators in the IR

The Sachdev-Ye-Kitaev (SYK) model

𝑡1 𝑡3

𝑡4 𝑡2

Charm of the SYK-like models

• New insights into old problems

– Perturbatively solvable

– Holographic

– Chaotic

– Tower of operators

• Question: emergence of higher spin symmetry

Large number of symmetries

Another corner of AdS/CFT correspondence

Tensionless/High energy limit of string theory, a very symmetric phase

• One dimensional case

Charm of the SYK-like models

• New insights into old problems

– Perturbatively solvable

– Holographic

– Chaotic

– A tower of operators

• Question: emergence of higher spin symmetry

Large number of symmetries

Another corner of AdS/CFT correspondence

Tensionless/High energy limit of string theory, a very symmetric phase

• One dimensional case

Charm of the SYK-like models

• New insights into old problems

– Perturbatively solvable

– Holographic

– Chaotic

– A tower of operators

• Question: emergence of higher spin symmetry

Large number of symmetries

Another corner of AdS/CFT correspondence

Tensionless/High energy limit of string theory, a very symmetric phase

• One dimensional case

(CP, 2017)

Higher dimensions

• Why higher dimensions ?

– Sensible notion of spins

– Have better studied higher-spin/string models

– Simplest example is 1+1D

Supersymmetric models

• SUSY is important to reach the SYK-like fix point

• Reduce the number of supersymmetries for flexibility

• Look for connections with other established models

(Murugan,Stanford, Witten, 2017)

An N=(0,2) model

•

Chiral: Fermi:

• , with fixed (but tunable)

IR solution

(CP, 2018)

Range of m

• Convergence of the FT

• In this range the model flows to the SYK-like fixed point

Numerical Confirmation

4-point function

h = h

…

…

• , , , …

= ℎ ℎ

ℎ ℎ

1-ℎ 1-ℎ

ℎ ℎ

1-ℎ 1-ℎ

4-point function

whose eigenvalue x satisfies

=

• , , , …

Solve x=1 to get the spectrum of 𝑂ℎ ,ℎ, spin s = ℎ − ℎ .

ℎ ℎ

ℎ ℎ

1-ℎ 1-ℎ

ℎ ℎ

1-ℎ 1-ℎ

Lightest scalar operators

(Murugan,Stanford, Witten, 2017)

Find λ𝐿 by solving x=1

The Lyapunov exponent

(Kitaev 2015, Maldacena Stanford, 2016) Out-of-Time-Ordered Correlators

𝑧1

𝑧3

𝑧2

𝑧4

The Lyapunov exponent

Two interesting limits

Fix q, check the m dependence of the model

• The Lyapunov exponent for the whole range

Two interesting limits

Two interesting limits

• Lyapunov exponent drops to zero

• “Integrablity” takes over ?

• Large symmetries ?

looking for the smallest g for each m

Lightest operators with spins

• Emergent higher-spin

operators in the two limits!

• Generate large symmetry

nonchaotic

Dispersion relation

• How does the anomalous dimension g depend on spin s ?

Dispersion relation

• How does the anomalous dimension g depend on spin s ?

Relations with Higher-spin theory

• Higher-spin perturbation computation

• Rotating folded closed long string in AdS

logarithmic due to the AdS geometry

(Gaberdiel, CP, Zadeh, 2015)

(Gubser, Klebanov, Polyakov, 2002)

Relations with Higher-spin theory

• Logic reversed

• Regarded as deformation away from higher-spin point

• Consistent with previous results

• A toy model that mimics the process of turning off the string

tension where the tuning is explicit

Comments on the higher-spin limits

• A tower of higher-spin operators, generate a higher-spin-type algebra

for each q, similar to the algebra in higher-spin holography

• The model is not free in this limits. Special property is from some delicate

scaling/screening in the IR dynamics

• Singular: cannot simply plug in , rather take

• The other limit is similar, although not identical

Backup slides

Can we understand the observed

higher-spin symmetry better?

In general hard due to the strong coupling, but…

For a single realization

• It is more convenient to work with a single realization

(although SYK is self-averaging.)

• Structure that is independent of realization

• 𝑄 -cohomological algebra ( 𝑄 2 =0 )

(Ahn, CP, 1812.05106)

For a single realization

• The algebra does not depend on the value of the random

coupling.

• Admits a free field realization

• Strategy: looking for any hint of higher-spin symmetry in

this chiral algebra.

(Witten, 1994)

For a single realization

• At generic parameter m and 𝑞, we do not find any higher-spin subalgebra.

• The higher-spin extension leads to a very larger W-algebra

where “conventional” higher-spin operators mix with (all)

other operators.

• Consistent with the expectation that all of them are lifted away

from the cohomology.

(Ahn, CP, 1812.05106)

The cohomological chiral algebra

• In the special limit

• We do find a conventional higher-spin subalgebra that is

quadratic in the fundamental fields

• We find more…

• A second higher spin subalgebra that is mulitlinear in the

fundamental fields with minimal derivatives

The cohomological chiral algebra

• Actually we find more…

• A second higher-spin subalgebra that is higher order in the

fundamental fields

+…

(Ahn, CP, 1812.05106)

𝑊4𝑣

𝑊3𝑣

𝑊3ℎ 𝑊4

ℎ 𝑇

𝑊22

𝑊21

𝑊4𝑣

𝑊3𝑣

𝑊3ℎ 𝑊4

ℎ 𝑇

+…

…

…

(Ahn, CP, 1812.05106, in progress)

𝑊22

𝑊21

𝑊4𝑣

𝑊3𝑣

𝑊3ℎ 𝑊4

ℎ 𝑇

+…

…

…

(Ahn, CP, 1812.05106, in progress)

The cohomological chiral algebra

• Observe a similar structure as stringy “higher-spin-square”

• Another connection to higher-spin/string theories

(Gaberdiel, Gopakumar 2015)

Comments

• The same special value of the parameter where the IR higher-spin operators appear.

(Higher spin is every where, just need to go to the

• Another realization of higher-spin-square – from a strongly coupled model

(Although our model, and more generally many of the SYK-like

models, are largely generalized free theories.)

– with a free parameter

(The two higher-spin subalgebras are all at =1 with no other

parameter. On the other hand, we do observe q-dependence in the

full higher-spin square.)

• The computation is done at finite N

Thank you !