On the relation between the magnitude and exponent of OTOCs (Based - PowerPoint PPT Presentation

On the relation between the magnitude and exponent of OTOCs (Based on the paper with Alexei Kitaev [1812.00120] ) Yingfei Gu Harvard University YITP, June 24, 2019

Outline ◮ Introduction: OTOC in SYK ◮ Kinetic equation approach: an algorithm that computes the Lyapunov exponent ◮ Ladder identity and branching time ◮ Applications of the ladder identity

Introduction ◮ OTOCs in large N systems with all-to-all interaction: OTOC t β t scr e λ L t � A ( t ) B (0) A ( t ) B (0) � connected ∼ 1 Early time: N C ◮ Convention β = 2 π . MSS chaos bound 0 < λ L � 1. ◮ Main example: SYK H SYK = � jklm J jklm χ j χ k χ l χ m

OTOC in SYK ◮ At strong coupling J ≫ 1, SYK is almost maximally chaotic (Maldacena-Stanford, 2016) e λ L t OTOC( t ) ∼ 1 1 λ L ∼ 1 − c 1 C ∼ c 2 J , J , N C ◮ Prefactor 1 C is big: pseudo-Goldstone mode (Schwarzian mode). ◮ The ratio 1 − λ L has a finite limit at J → ∞ . C ◮ This talk is about an identity relating λ L to C . [Corollary: although c 1 , c 2 individually depends on the UV data, the product c 1 c 2 is universal.] ◮ Hope: improve the understanding of fast scrambling.

Four point function and kinetic equation Average over fermion indices, in general depends on four times. ) = 1 � OTOC( t 1 , t 2 � χ j ( t 1 ) χ k ( t 3 ) χ j ( t 2 ) χ k ( t 4 ) � + G ( t 12 ) G ( t 34 ) , t 3 , t 4 N 2 ���� ���� jk past future ◮ At large N limit, four-point functions are dominated by ladder diagrams (Kitaev, Polchinski-Rosenhaus, Maldacena-Stanford ...) + + + . . . ◮ Euclidean four-point function: Bethe-Salpeter equation � F = F n , F n = · F n − 1 ⇒ F = F 0 + K F n ◮ OTOC: deform the contour to double Keldysh ⇒ Kinetic equation R 1 3 OTOC ≈ K R · OTOC , K R ( t 1 , t 2 , t 3 , t 4 ) : W 2 4 R

Single-mode ansatz How does OTOC( t 1 , t 2 , t 3 , t 4 ) look like? ���� ���� past future ◮ We are interested in time regime t 1 ≈ t 2 ≫ t 3 ≈ t 4 : large separation between the perturbations in the past and probes in the future. (But still before the scrambling time.) ◮ Single-mode ansatz (“scramblon”) (Kitaev-Suh, 2017) e λ L ( t 1 + t 2 − t 3 − t 4 ) / 2 OTOC( t 1 , t 2 , t 3 , t 4 ) ≈ 1 Y R ( t 12 ) Y A ( t 34 ) N C 1 3 Y R / A : vertex function R A 2 4

Eigenvalue k R ( α ) The kinetic equation OTOC ≈ K R OTOC means OTOC is an eigenvector of K R with eigenvalue k R = 1. In general finding OTOC is a complicated search. We can reduce it to a shooting problem: ◮ Guess a growing exponent α for scramblon e α t ; ◮ Adjust vertex function such that the trial OTOC( α ) is an eigenvector of K R with general eigenvalue k R ( α ); ◮ Find α ∗ such that k R ( α ∗ ) = 1 ⇒ λ L = α ∗ .

Prefactor This algorithm is useful in finding λ L (as well as vertex functions), but the prefactor is ambiguous ◮ Because the kinetic equation OTOC ≈ K R OTOC is a homogeneous linear equation ◮ Origin of the ambiguity: we dropped the constant term F 0 in the exact equation OTOC = K R OTOC + F 0 ◮ We would like to get the correct prefactor with the data ( λ L , Y R / A , k R ( α )) we already have

Ladder identity e λ L ( t 1 + t 2 − t 3 − t 4 ) / 2 OTOC( t 1 , t 2 , t 3 , t 4 ) ≈ 1 Y R ( t 12 ) Y A ( t 34 ) N C ◮ Ladder identity: 2 cos λ L π � Y A , Y R � · ( − k ′ 2 R ( λ L )) · = 1 . C ◮ � Y A , Y R � : inner product of vertex functions: � � Y A , Y R � � G W ( t ) � q − 2 Y R ( t ) . = ( q − 1) J 2 dt Y A ( t ) = ◮ Branching time t B := − k ′ R ( λ L ). Average separation between adjacent rungs. t B t 1 t 3 t 2 t 4

Idea of the derivation Idea: cut a long ladder into pieces and find a consistency condition. ◮ Cut in the middle, find adjacent t 5 + t 6 < t 0 < t 7 + t 8 2 2 t 0 t 5 t 7 t 1 t 3 t 2 t 4 t 6 t 8 ◮ Cut-gluing consistency condition 1 3 1 3 ≈ · · 2 4 2 4 ◮ Left hand side contains one copy of C − 1 , right hand side has two copies of C − 1 .

Applications The ladder identity: 2 cos λ L π � Y A , Y R � 2 · t B · = 1 . C Applications: ◮ Computational shortcuts: C ⇔ λ L ; ◮ In a 1D model, exact maximal chaos.

Computational shortcuts ◮ C ⇒ λ L : for SYK at strong coupling, the prefactor C is determined by the coefficient of the Schwarzian action α S . However, the Lyapunov exponent will receive small correction from the conformal matters (Maldacena-Stanford, 2016) 6 α S δλ L ≈ R ( − 1)∆(1 − ∆)(1 − 2∆) tan( π ∆) . Jk ′ ◮ λ L ⇒ C : for q -body interacting SYK model at q → ∞ limit, the exact value of λ L is easy to find using the kinetic equation (Maldacena-Stanford, 2016) , λ L = J . 2 cos πλ L 2 But the prefactor is hard to compute. The ladder identity gives a result consistent with the computation using Liouville equation (Qi-Streicher, 2018) . e λ L ( t 1 + t 2 − t 3 − t 4 ) / 2 1 OTOC( t 1 , t 2 ; t 3 , t 4 ) ≈ � . � � � N cos λ L π 2 cosh λ L t 12 2 cosh λ L t 34 2 2 2

Maximal chaos in a 1D model ◮ Idea: regard λ L and C as analytic functions of some parameter, 2 cos λ L π � Y A , Y R � 2 · t B · = 1 . C then the analytical properties of λ L and C are locked by the ladder identity. ◮ A concrete example: SYK chain (YG-Qi-Stanford, 2016) J ′ J jklm , x − 1 jklm , x j l m j k l k m

Operators at two different locations Spatial propagation of chaos: measured by � A ( x , t ) B (0 , 0) A ( x , t ) B (0 , 0) � t t = x / v B x ◮ Fourier transform: � dp 2 π e ipx OTOC( p , t ) OTOC( x , t ) = ◮ Each OTOC( p , t ) can be solved using retarded kernel approach.

Fourier Transform ◮ The ladder identity holds for each OTOC( p , t ): C ( p ) = 2 cos λ L ( p ) π · t B · ( Y A , Y R ) , 2 ◮ The dependence of t B and ( Y A , Y R ) on p is not important (analytic and do not vanish in the domain of interest). � + ∞ e λ L ( p ) t + ipx Y R Y A OTOC( x , t ) ∼ 1 dp · t B ( Y A , Y R ) . 2 cos πλ L ( p ) N 2 π −∞ 2 � �� � u ( x , t )

Butterfly wavefront � + ∞ e λ L ( p ) t + ipx dp u ( x , t ) = 2 π 2 cos λ L ( p ) −∞ 2 ◮ Butterfly wavefront: u ( x , t ) ∼ 1. ◮ For large x > 0 and t , we can estimate the asymptotics by saddle point of the exponent. λ ′ L ( p ) t + ix = 0 , p = i | p | . ◮ The relevant saddle point p s = i | p s | is purely imaginary: deform the integral contour to pass. Warning: pole contribution! cos λ L ( p 1 ) π = 0 , λ L ( p 1 ) = 1 , p 1 = i | p 1 | . 2

Pole contribution: maximal chaos ◮ When J > J c , the pole dominates the butterfly wavefront: u 1 ( x , t ) ≈ e t −| p 1 || x | 1 L ( p 1 ) , v 1 = π i λ ′ | p 1 | ◮ Four regions with different OTOC behavior. t t = x / v B + t scr 3 t = x / v ∗ t = x / v B 2 1 4 x

Summary and discussion An identity relates C and λ L ; 2 cos λ L π � Y A , Y R � 2 t B = ( − k ′ · t B · = 1 , R ( λ L )) C Some lessons from the applications: ◮ Computational shortcuts: in SYK model we find the correction δλ L ∝ t − 1 B . If we call λ L = 1 the “coherent scrambling” (Schwarzian, gravity...), then the branching time t B measures the “decoherence” effect. ◮ Maximal chaos: spatial locality provides a mechanism to enhance the Lyapunov exponent. Ladder identity: the enhancement cancels the correction exactly Thank you!

Derivation: cos λ L π factor 2 ◮ Naively, we would have a formula: OTOC ≈ OTOC L · BOX · OTOC R ◮ Subtlety: multiple choices on the double Keldysh contour t 0 t 5 t 7 t 1 t 3 t 2 t 6 t 8 t 4 ◮ Sum of two choices: � � λ L π λ L π e i + e − i OTOC ≈ OTOC L · BOX · OTOC R 2 2

Derivation ◮ Gluing OTOC L and OTOC R through a box: � Y A , Y R � = t B ◮ Now inserting the single mode ansatz to the consistency condition � � λ L π λ L π e i + e − i OTOC ≈ OTOC L · BOX · OTOC R 2 2 1 3 1 3 ≈ · · 2 4 2 4 ◮ Compare two sides (stripping off the vertex functions at two ends) C = 2 cos λ L π 1 · 1 · 1 � Y A , Y R � C · t B · 2 C