Operator Growth in SYK model and Beyond Xiao-Liang Qi Stanford - - PowerPoint PPT Presentation

Operator Growth in SYK model and Beyond

Xiao-Liang Qi Stanford University YITP, Kyoto, 6/28/2019

• Motivation
• Definition of operator size and size distribution
• General results: thermal state and single fermion

excitations

• SYK model
• Large q limit
• Large 𝛾𝐾 limit
• Dual theory: size distribution of bulk fermion
• perators

Overview

Ref.

• 1. Alexandre Streicher, XLQ, arxiv:1810.11958
• 2. Yingfei Gu, Yuri Lensky, XLQ, Pengfei Zhang, in progress

Part I: General results

• Classical chaos:
• 𝜊 0 = 𝑦 0 , 𝑞 0

→ 𝜊 𝑢 = 𝑦 𝑢 , 𝑞 𝑢

• Sensitivity to initial condition
• Lyapunov exponents
• 𝜊 𝑢 is a complicated function
• f 𝜊 0
• Quantum analog: Heisenberg operator evolution

෠ 𝑃 𝑢 = 𝑓𝑗𝑢𝐼 ෠ 𝑃(0)𝑓−𝑗𝑢𝐼

• ෠

𝑃(𝑢) is a complicated function of ෠ 𝑃

• How to define “complicated”?

Motivation

𝜊 𝑢 𝜊 0

• Non-interacting system:

A particle has 𝑂 possible positions. 𝜔𝑦 𝑢 = σ𝑧=1…𝑂 𝜚𝑦 𝑧 ෠ 𝜔𝑧 0 .

• Generic interacting system:

A particle can decay into multi-particle states. Exponentially many final states in the Hilbert space.

• ෠

𝜔𝑦 𝑢 = 𝜚𝑦 𝑧 ෠ 𝜔𝑧 0 + 𝜚𝑦 𝑧1𝑧2𝑧3 ෠ 𝜔𝑧1 ෠ 𝜔𝑧2

+ ෠

𝜔𝑧3 + ⋯

Many-body quantum chaos

• 𝑞𝑚 probability of size 𝑚 (Roberts-Stanford-Susskind ’14, Hosur XLQ ’16,

Roberts, Streicher, Stanford ‘18)

• Example: Majorana fermions
• ෠

𝜔1 𝑢 = 𝑏 ෠ 𝜔3 + 𝑐 ෠ 𝜔4 + 𝑑 ෠ 𝜔2 ෠ 𝜔3 ෠ 𝜔4 + 𝑒 ෠ 𝜔1 ෠ 𝜔3 ෠ 𝜔4 + ⋯

Operator size distribution

𝑏 2 𝑐 2 𝑑 2 𝑒 2

size

𝑞1 = 𝑏 2 + 𝑐 2 𝑞3 = 𝑑 2 + 𝑒 2

• Example: a random fermionic operator
• 𝑞𝑚 = 21−𝑂 𝑂

𝑚 , for odd 𝑚. Average size

𝑂 2

• Doubled formalism:
• Take two systems 𝑀, 𝑆, each consisting of 𝑂

Majoranas.

• Prepare a maximally entangled state 𝐽
• For every operator 𝑃, applying it to the left system, to
• btain a state: 𝑃 = 𝑃𝑀 ⊗ 𝕁𝑆 𝐽
• 𝐵 𝐶 =

1 𝐸 𝑢𝑠 𝐵+𝐶

Operator size distribution

• In the doubled formalism, size is a linear operator

The size superoperator

𝜔1

=1 ⋅

𝜔1

=2 ⋅

𝜔1𝜔2 𝜔1𝜔2

……

• Size superoperator ො

𝑜 depends on reference state 𝐽 . ො 𝑜 𝐽 = 0 is required

• A convenient choice of 𝐽 : 𝜔𝑗𝑀 − 𝑗𝜔𝑗𝑆 𝐽 = 0
• |𝐽⟩ is the vacuum state of fermion with annihilation
• perator 𝑔

𝑗 = 1 2 𝜔𝑗𝑀 − 𝑗𝜔𝑗𝑆 , 𝑔 𝑗, 𝑔 𝑘 + = 𝜀𝑗𝑘

• In this basis ො

𝑜 = σ𝑗 𝑔

𝑗 +𝑔 𝑗 is fermion number

• 𝜔𝑗𝑀𝜔𝑘𝑀 … 𝐽 = 𝑔

𝑗 +𝑔 𝑘 + … |𝐽⟩

• In Majorana basis
• ො

𝑜 =

1 2 𝑂 − 𝑗 σ𝑗 𝜔𝑗𝑀𝜔𝑗𝑆

The size superoperator

=

• The size superoperator can be used to compute the

full size distribution

• 𝑞𝑚 = σ𝛽 𝑚𝛽 ෠

𝑃 𝐽

2

• 𝒣𝜈 = 𝐽 ෠

𝑃+𝑓−𝜈 ො

𝑜 ෠

𝑃 𝐽 = σ𝑚=0

𝑂

𝑓−𝜈𝑚𝑞𝑚 ෠ 𝑃

• For random fermionic operator
• 𝒣𝜈

𝑠𝑏𝑜𝑒 = σ𝑚 odd 𝑓−𝜈𝑚 𝑂

𝑚 21−𝑂

=

1+𝑓−𝜈 2 𝑂

−

1−𝑓−𝜈 2 𝑂

.

Generating function

• Independent from details, this formalism already tells us

interesting information about operator size.

• Consider the TFD state:
• 𝜍 =

1 𝑎 𝑓−𝛾𝐼

• 𝑈𝐺𝐸 = 𝜍

1 2 = 𝑎−1 2𝑓−𝛾𝐼𝑀 𝐽 =

• Size of 𝜍

1 2:

• 𝑜 𝜍1/2 = 𝑈𝐺𝐸 ො

𝑜 𝑈𝐺𝐸 = 𝑂 2 − 𝑗 2 ෍

𝑗

𝑈𝐺𝐸 𝜔𝑗𝑀𝜔𝑗𝑆 𝑈𝐺𝐸 = 𝑂 2 − 1 2 ෍

𝑗

𝜔𝑗 𝛾 2 𝜔𝑗 0 ≡ 𝑂 2 − 𝑂 2 𝐻 𝛾 2

General results

• Distance to scrambling (average size of random
• perator 𝑜∗ = 𝑂/2):
• 𝜀𝛾 ≡ 1 −

𝑜 𝑜∗ = 𝐻 𝛾 2

• Usually, 𝐻

𝛾 2 decays to zero

at 𝛾 → ∞ ➔ Size approaches

𝑂 2

• For example, 𝜍𝛾→∞ has size

𝑂 2 if

the system has unique ground state and a gap, or if the system is a CFT.

General results 1: thermal state

• At infinite temperature, a single fermion operator 𝜓𝑗

has size 1. In our doubled language, this means ො 𝑜𝜔𝑗𝑀 𝐽 = 𝜔𝑗𝑀 𝐽

• As a generalization, we can consider 𝜔𝑗𝑀 𝑈𝐺𝐸
• 𝜀𝑜𝛾 𝜔𝑗 = 𝑈𝐺𝐸 𝜔𝑗𝑀 ො

𝑜𝜔𝑗𝑀 𝑈𝐺𝐸 − 𝑈𝐺𝐸 ො 𝑜 𝑈𝐺𝐸 = 𝑈𝐺𝐸 𝜔𝑗𝑀, ො 𝑜 𝜔𝑗𝑀 𝑈𝐺𝐸 .

• Fermion anticommutation gives
• 𝜀𝑜𝛾 𝜔𝑗 = −𝑗 𝑈𝐺𝐸 𝜔𝑗𝑆𝜔𝑗𝑀 𝑈𝐺𝐸
• 1

𝑂 σ𝑗 𝜀𝑜 𝜔𝑗 = 𝐻 𝛾 2

= 𝜀𝛾

General results 2: single fermion excitation

𝑜∗ = 𝑂/2 𝑜∗ 𝑜∗ 1 − 𝜀𝛾

• The effect of (chaotic) dynamics

is described by studying the size

• f Heisenberg operators.
• 𝜀𝑜𝛾[𝜔𝑗 𝑢 ] = 𝑈𝐺𝐸 𝜔𝑗𝑀 𝑢 , ො

𝑜 𝜔𝑗𝑀(𝑢) 𝑈𝐺𝐸 .

• This is related to the out-of-time-ordered correlation

function (OTOC):

• 𝜀𝑜𝛾 𝜔𝑗 𝑢

= σ𝑘 𝜔𝑘

𝛾 2

𝜔𝑗 𝜗 + 𝑗𝑢 , 𝜔𝑘 0 𝜔𝑗 −𝜗 + 𝑗𝑢

𝛾

.

• Infinite temperature case Roberts, Streicher, Stanford ’17
• Finite temperature XLQ-Streicher ‘18

General results 3: Time evolution and OTOC

Part II: SYK model

• 𝐼 = σ𝑗𝑘𝑙𝑚 𝐾𝑗𝑘𝑙𝑚𝜓𝑗𝜓𝑘𝜓𝑙𝜓𝑚 with Gaussian random coupling 𝐾𝑗𝑘𝑙𝑚.

(Sachdev-Ye, ‘93, Kitaev ’15, Maldacena-Stanford ‘16)

• Or complex fermion model

𝐼 = σ𝑗𝑘,𝑙𝑚 𝐾𝑗𝑘𝑙𝑚𝑑𝑗

+𝑑 𝑘 +𝑑𝑙𝑑𝑚.

• Generalization (Maldacena-Stanford ‘16) :

𝐼 = σ𝑗1𝑗2…𝑗𝑟 𝐾𝑗1𝑗2…𝑗𝑟𝜓𝑗1𝜓𝑗2 … 𝜓𝑗𝑟

• Averaging over disorder
• 𝑎𝑜 ≃ 𝑎

n in large 𝑂 limit.

• Large N order parameter
• 𝐻 𝜐1, 𝜐2 =

1 𝑂 σ𝑗 𝜓𝑗 𝜐1 𝜓𝑗 𝜐2

=

N fermions q-body interaction

Sachdev-Ye-Kitaev model

+ + + ⋯

𝜐

Sachdev-Ye-Kitaev model

𝑀 𝑆

• Critical correlation function (Δ = 1/𝑟)

𝜓𝑗 𝜐 𝜓𝑗 0 ∝ sin

𝜐 𝛾 𝜌 −2Δ

sgn(𝜐).

• Real time: thermal double state

𝑈𝐺𝐸(𝑢) = 𝑎−1

2 σ𝑜 𝑓−𝐹𝑜 𝛾 2+𝑗𝑢 𝑜 𝑀 𝑜 𝑆.

• 𝜓𝑗𝑀 𝑢 𝜓𝑗𝑆 𝑢

∝ cosh

𝜌𝑢 𝛾 −2Δ

• Chaos. Maximal Lyapunov exponent

𝜇 = 2𝜌𝑈 (Kitaev ’15, Maldacena-Stanford ’16,

Maldacena-Shenker-Stanford ‘15)

• (Approximately) dual to

Jackiw-Teitelboim gravity

• Infinite temperature case

(Roberts-Stanford-Streicher ‘18)

• For finite temperature, we study

the size distribution ⟨𝑈𝐺𝐸 𝑓−𝜈 ො

𝑜 𝑈𝐺𝐸⟩ and

⟨𝑈𝐺𝐸 𝜔𝑗𝑀𝑓−𝜈 ො

𝑜𝜔𝑗𝑀 𝑈𝐺𝐸⟩

• 𝑈𝐺𝐸 𝑓−𝜈 ො

𝑜 𝑈𝐺𝐸 ≡ 𝑎𝜈 is a

partition function of SYK with twisted boundary condition

Operator growth in the SYK model

𝜔 𝛾 4 + 𝜗 = cosh 𝜈 𝜔 𝛾 4 − 𝜗 − sinh 𝜈 𝜔 3𝛾 4 − 𝜗

• Two-point function with 𝜈 term can be obtained by

Schwinger-Dyson equation

• 𝐻 = 𝜖𝜐 − Σ𝜈

−1

Σ𝜈 𝜐1, 𝜐2 = 𝐾2𝒣𝜈

𝑟−1 𝜐1, 𝜐2 .

• Note that

𝒣𝜈 𝜐1, 𝜐2 = 𝑈𝐺𝐸 𝜔𝑗𝑀 𝜐1 𝑓−𝜈 ො

𝑜𝜔𝑗𝑀 𝜐2 𝑈𝐺𝐸

⟨𝑈𝐺𝐸 𝑓−𝜈 ො

𝑜 𝑈𝐺𝐸⟩

• This is a ratio of two generating functions. Expansion of

𝒣𝜈 = σ𝑛 𝑓−𝜈𝑛𝐿𝑛 Coefficient 𝐿𝑛 satisfies

• 𝑞𝑚 𝜔𝑗 𝑢 𝜍

1 2 = σ𝑛 𝐿𝑛 𝑞𝑚−𝑛 𝜍 1 2 .

• 𝐿𝑛 is the “transition probability”

that the fermion operator increases size of 𝜍

1 2 by 𝑛

Operator growth in the SYK model

• In large 𝑟, the Schwinger-Dyson equation simplifies

𝜖𝜐 −

𝐾2 𝑟 Gq−1 ∗ G = 𝕁,

• 𝐻 = 𝐻0𝑓

𝑕 𝑟, ➔ Liouville equation 𝜖𝜐1𝜖𝜐2𝑕 = −2𝐾2𝑓𝑕

• Define 𝜈 =

ෝ 𝜈 𝑟 , ො

𝜈 changes the boundary condition of 𝑕

Large-q solution

𝜐1 𝜐2

Large-q solution

• Size of thermal state
• 𝑜 𝜍

1 2 =

𝑂 2 1 − 𝜀𝛾 ,

• 𝜀𝛾 =

𝛽 𝒦

2 𝑟

• At low temperature 𝛽 ≃

𝜌 𝛾.

• Fermion size 𝜀𝑜𝛾 𝜔1 𝑢

= 1 + 2

𝒦 𝛽 sinh 𝛽𝑢 2

• Lyapunov exponent 2𝛽
• Consistency at 𝑢 → 0: conjecture

𝜀𝑜𝛾 𝜔1 𝑢 = 𝜀𝛾 1 + 2

𝒦 𝛽 sinh 𝛽𝑢 2

.

• Full size distribution can be
• btained by expanding 𝒣𝜈
• The entire distribution is the

same as infinite temperature case except a renormalization

• f parameter sinh 𝒦𝑢 →

𝒦 𝛽 sinh 𝛽𝑢

Large-q solution

∝ 𝑓

−4 𝑟𝛽𝑢 exp − 2𝛽

𝒦

2

𝑓−2𝛽𝑢𝑜

• At large 𝛾𝐾, the dynamics of the SYK model can be

described by reparameterization modes 𝑔 𝜐 , defined by 𝐻𝑔 𝜐1, 𝜐2 = 𝑔′ 𝜐1 𝑔′ 𝜐2

Δ𝐻𝑇 𝑔 𝜐1 , 𝑔 𝜐2

• Effective theory describes the breaking of

reparameterization symmetry.

• Schwarzian action 𝑇 =

𝑂 𝐾 𝛽 ׬ 𝛾 𝑒𝜐 Sch tan 𝜌 𝛾 𝑔 𝜐

, 𝜐

• Preserves SL(2,R) gauge symmetry
• Bulk picture 𝑇 = 𝑂𝛽(𝑀 − 𝐵). 𝑀 = 𝛾𝐾,

maximize 𝐵.

Low temperature solution

A

• 𝑓−𝜈 ො

𝑜 term adds an interaction

between the time 𝜐 = 0 and 𝜐 =

𝛾 2 .

• 𝜀𝑇 ∝ 𝑂 𝜔

𝛾 2 𝜔 0 𝑔

= 𝑂 cosh 𝐸Δ

• Solution: two arcs (similar to

Gu, Lucas XLQ ‘17)

• Analytic continuation 𝜐1 → 𝜗 + 𝑗𝑢1,

𝜐2 → −𝜗 + 𝑗𝑢2 to determine 𝒣𝜈 𝑢1, 𝑢2 .

Low temperature solution

A

𝜐1 𝜐2

(Yingfei Gu, Yuri Lensky, XLQ, Pengfei Zhang, in progress)

• For 𝑢1 = 𝑢2,
• Schwarzian
• Large 𝑟
• Agreement for long time requires to take the UV

cutoff 𝜗 =

𝜌 𝛾𝐾, and small ො

𝜈 = 𝑟𝜈 ≪ 1

Low temperature solution

(𝛽 ≃ 𝜌 + 𝑑𝜈)

Part III: Holographic dual theory

• The SYK is approximately dual to Jackiw-Teitelboim

gravity coupled with matter fields.

• Gravitational dynamics Repara. Modes
• Matter field Conformal fields
• In particular, fermion

𝜓𝑗𝛽 𝜍, 𝑢 𝜔𝑗(𝑢)

• Bulk fermion mass

𝑛 = Δ −

1 2 = 1 𝑟 − 1 2.

• HKLL construction

𝜓𝑗𝛽 𝜍, 𝑢 = ׬

𝑢− 𝑢+ 𝑒𝑢′𝐿𝛽 𝜍, 𝑢 − 𝑢′ 𝜔𝑗 𝑢′ .

• Size of 𝜓𝑗𝛽 𝜍, 𝑢 can be computed.

The dual theory: bulk operator size

Yingfei Gu, Yuri Lensky, XLQ, Pengfei Zhang, in progress

• Generating function
• 𝒣𝜈

𝐶 𝜍, 𝑢 ≡ 𝑈𝐺𝐸 𝜓𝑗𝛽 𝜍,𝑢 𝑓−𝜈ෝ

𝑜𝜓𝑗𝛽 𝜍,𝑢 𝑈𝐺𝐸

𝑈𝐺𝐸 𝑓−𝜈ෝ

𝑜 𝑈𝐺𝐸

= ׬ 𝑒𝑢1𝑒𝑢2𝐿𝛽 𝜍, 𝑢 − 𝑢1 𝐿𝛽 𝜍, 𝑢 − 𝑢2 ⋅ 𝑈𝐺𝐸 𝜔𝑗 𝑢1 𝑓−𝜈 ො

𝑜𝜔𝑗 𝑢2 𝑈𝐺𝐸

𝑈𝐺𝐸 𝑓−𝜈 ො

𝑜 𝑈𝐺𝐸

= ׬ 𝑒𝑢1𝑒𝑢2𝐿𝛽 𝜍, 𝑢 − 𝑢1 𝐿𝛽 𝜍, 𝑢 − 𝑢2 𝒣𝜈 𝑢1, 𝑢2

• Simplification: contribution is dominated by the light

cone 𝑢 = 𝑢±

• 𝒣𝜈

𝐶 𝜍, 𝑢 = 1 2 𝒣𝜈 𝑢+, 𝑢+ + 𝒣𝜈 𝑢−, 𝑢−

± 𝑆𝑓 𝒣𝜈 𝑢+, 𝑢−

The dual theory: bulk operator size

• Bulk fermion size is almost determined by the size of

boundary fermions at 𝑢±.

• Approximately,
• 𝑜 =

𝛾𝐾 2 2𝜌2

cosh 𝑢 coth 𝜍 − 1 +

𝛾𝐾 2𝜌 cosh 𝜍 − sinh 𝜍 cosh 𝑢 + 𝑃

𝛾𝐾 0 .

• Size grows exponentially along radial

and temporal direction with proper distance.

• Operator size diverges near the horizon
• The calculation only applies to 𝑢± < 𝑢∗

(before scrambling time)

The dual theory: bulk operator size

• Operator size growth characterizes chaos (at early time)
• The same general results apply to qubit systems.
• SYK model operator size growth

can be computed for low temperature or large 𝑟

• Size growth of bulk operator

provides an interpretation of emergent bulk direction 𝜍

• Generalization to global AdS2?
• Does the size of bulk operator

depend on the representation?

• Relation to other works, such as
• A. Brown et al 1804.04156

Conclusion and further discussion

Thanks!