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

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

Part I: General results

Motivation • Classical chaos: • 𝜊 0 = 𝑦 0 , 𝑞 0 → 𝜊 𝑢 = 𝑦 𝑢 , 𝑞 𝑢 • Sensitivity to initial condition 𝜊 𝑢 • Lyapunov exponents 𝜊 0 • 𝜊 𝑢 is a complicated function of 𝜊 0 • Quantum analog: Heisenberg operator evolution 𝑃 𝑢 = 𝑓 𝑗𝑢𝐼 ෠ ෠ 𝑃(0)𝑓 −𝑗𝑢𝐼 • ෠ 𝑃(𝑢) is a complicated function of ෠ 𝑃 • How to define “complicated” ?

Many-body quantum chaos • Non-interacting system: A particle has 𝑂 possible positions. 𝜔 𝑦 𝑢 = σ 𝑧=1…𝑂 𝜚 𝑦 𝑧 ෠ 𝜔 𝑧 0 . • Generic interacting system: A particle can decay into multi-particle states. Exponentially many final states in the Hilbert space. • ෠ 𝜔 𝑦 𝑢 = 𝜚 𝑦 𝑧 ෠ 𝜔 𝑧 0 + + ෠ 𝜚 𝑦 𝑧 1 𝑧 2 𝑧 3 ෠ 𝜔 𝑧 1 ෠ 𝜔 𝑧 2 𝜔 𝑧 3 + ⋯

Operator size distribution • 𝑞 𝑚 probability of size 𝑚 (Roberts-Stanford- Susskind ’14, Hosur XLQ ’16, Roberts, Streicher, Stanford ‘18) • Example: Majorana fermions • ෠ 𝜔 1 𝑢 = 𝑏 ෠ 𝜔 3 + 𝑐 ෠ 𝜔 4 + 𝑑 ෠ 𝜔 2 ෠ 𝜔 3 ෠ 𝜔 4 + 𝑒 ෠ 𝜔 1 ෠ 𝜔 3 ෠ 𝜔 4 + ⋯ 𝑑 2 𝑏 2 𝑐 2 𝑒 2 size 𝑞 3 = 𝑑 2 + 𝑒 2 𝑞 1 = 𝑏 2 + 𝑐 2

Operator size distribution • Example: a random fermionic operator 𝑂 • 𝑞 𝑚 = 2 1−𝑂 𝑂 , for odd 𝑚 . Average size 𝑚 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 obtain a state: 𝑃 = 𝑃 𝑀 ⊗ 𝕁 𝑆 𝐽 1 𝐸 𝑢𝑠 𝐵 + 𝐶 • 𝐵 𝐶 =

The size superoperator • In the doubled formalism, size is a linear operator = 1 ⋅ 𝜔 1 𝜔 1 = 2 ⋅ 𝜔 1 𝜔 2 𝜔 1 𝜔 2 ……

The size superoperator • Size superoperator ො 𝑜 depends on reference state 𝐽 . 𝑜 𝐽 = 0 is required ො • A convenient choice of 𝐽 : 𝜔 𝑗𝑀 − 𝑗𝜔 𝑗𝑆 𝐽 = 0 • |𝐽⟩ is the vacuum state of fermion with annihilation + = 𝜀 𝑗𝑘 1 operator 𝑔 𝑗 = 2 𝜔 𝑗𝑀 − 𝑗𝜔 𝑗𝑆 , 𝑔 𝑗 , 𝑔 𝑘 + 𝑔 • In this basis ො 𝑜 = σ 𝑗 𝑔 𝑗 is fermion number 𝑗 + … |𝐽⟩ + 𝑔 • 𝜔 𝑗𝑀 𝜔 𝑘𝑀 … 𝐽 = 𝑔 𝑗 𝑘 • In Majorana basis 1 = • ො 2 𝑂 − 𝑗 σ 𝑗 𝜔 𝑗𝑀 𝜔 𝑗𝑆 𝑜 =

Generating function • The size superoperator can be used to compute the full size distribution 2 • 𝑞 𝑚 = σ 𝛽 𝑚𝛽 ෠ 𝑃 𝐽 𝑜 ෠ • 𝒣 𝜈 = 𝐽 ෠ 𝑓 −𝜈𝑚 𝑞 𝑚 ෠ 𝑃 + 𝑓 −𝜈 ො 𝑂 𝑃 𝐽 = σ 𝑚=0 𝑃 • For random fermionic operator 𝑠𝑏𝑜𝑒 = σ 𝑚 odd 𝑓 −𝜈𝑚 𝑂 • 𝒣 𝜈 2 1−𝑂 𝑚 𝑂 𝑂 1+𝑓 −𝜈 1−𝑓 −𝜈 = − . 2 2

General results • Independent from details, this formalism already tells us interesting information about operator size. • Consider the TFD state: 1 𝑎 𝑓 −𝛾𝐼 • 𝜍 = 1 2 = 𝑎 − 1 2 𝑓 −𝛾𝐼 𝑀 𝐽 = • 𝑈𝐺𝐸 = 𝜍 1 • Size of 𝜍 2 : • 𝑜 𝜍 1/2 = 𝑈𝐺𝐸 ො 𝑜 𝑈𝐺𝐸 = 𝑂 2 − 𝑗 2 ෍ 𝑈𝐺𝐸 𝜔 𝑗𝑀 𝜔 𝑗𝑆 𝑈𝐺𝐸 𝑗 = 𝑂 2 − 1 𝛾 ≡ 𝑂 2 − 𝑂 2 𝐻 𝛾 2 ෍ 𝜔 𝑗 2 𝜔 𝑗 0 2 𝑗

General results 1: thermal state • Distance to scrambling (average size of random operator 𝑜 ∗ = 𝑂/2 ): 𝑜 𝛾 • 𝜀 𝛾 ≡ 1 − 𝑜 ∗ = 𝐻 2 𝛾 • Usually, 𝐻 2 decays to zero 𝑂 at 𝛾 → ∞ ➔ Size approaches 2 𝑂 • For example, 𝜍 𝛾→∞ has size 2 if the system has unique ground state and a gap, or if the system is a CFT.

General results 2: single fermion excitation • At infinite temperature, a single fermion operator 𝜓 𝑗 has size 1 . In our doubled language, this means 𝑜𝜔 𝑗𝑀 𝐽 = 𝜔 𝑗𝑀 𝐽 ො • As a generalization, we can consider 𝜔 𝑗𝑀 𝑈𝐺𝐸 • 𝜀𝑜 𝛾 𝜔 𝑗 = 𝑈𝐺𝐸 𝜔 𝑗𝑀 ො 𝑜𝜔 𝑗𝑀 𝑈𝐺𝐸 − 𝑈𝐺𝐸 ො 𝑜 𝑈𝐺𝐸 = 𝑈𝐺𝐸 𝜔 𝑗𝑀 , ො 𝑜 𝜔 𝑗𝑀 𝑈𝐺𝐸 . • Fermion anticommutation gives • 𝜀𝑜 𝛾 𝜔 𝑗 = −𝑗 𝑈𝐺𝐸 𝜔 𝑗𝑆 𝜔 𝑗𝑀 𝑈𝐺𝐸 1 𝛾 • 𝑂 σ 𝑗 𝜀𝑜 𝜔 𝑗 = 𝐻 = 𝜀 𝛾 2 𝑜 ∗ = 𝑂/2 𝑜 ∗ 1 − 𝜀 𝛾 𝑜 ∗ 0

General results 3: Time evolution and OTOC • The effect of (chaotic) dynamics is described by studying the size of Heisenberg operators. • 𝜀𝑜 𝛾 [𝜔 𝑗 𝑢 ] = 𝑈𝐺𝐸 𝜔 𝑗𝑀 𝑢 , ො 𝑜 𝜔 𝑗𝑀 (𝑢) 𝑈𝐺𝐸 . • This is related to the out-of-time-ordered correlation function (OTOC): 𝛾 • 𝜀𝑜 𝛾 𝜔 𝑗 𝑢 = σ 𝑘 𝜔 𝑘 𝜔 𝑗 𝜗 + 𝑗𝑢 , 𝜔 𝑘 0 𝜔 𝑗 −𝜗 + 𝑗𝑢 . 2 𝛾 • Infinite temperature case Roberts, Streicher, Stanford ’17 • Finite temperature XLQ- Streicher ‘18

Part II: SYK model

Sachdev-Ye-Kitaev model • 𝐼 = σ 𝑗𝑘𝑙𝑚 𝐾 𝑗𝑘𝑙𝑚 𝜓 𝑗 𝜓 𝑘 𝜓 𝑙 𝜓 𝑚 with Gaussian random coupling 𝐾 𝑗𝑘𝑙𝑚 . ( Sachdev- Ye, ‘93, Kitaev ’15, Maldacena- Stanford ‘16) • Or complex fermion model q-body interaction + 𝑑 + 𝑑 𝑙 𝑑 𝑚 . 𝐼 = σ 𝑗𝑘,𝑙𝑚 𝐾 𝑗𝑘𝑙𝑚 𝑑 𝑗 𝑘 • Generalization (Maldacena- Stanford ‘16) : 𝐼 = σ 𝑗 1 𝑗 2 …𝑗 𝑟 𝐾 𝑗 1 𝑗 2 …𝑗 𝑟 𝜓 𝑗 1 𝜓 𝑗 2 … 𝜓 𝑗 𝑟 • Averaging over disorder n in large 𝑂 limit. • 𝑎 𝑜 ≃ 𝑎 N fermions • Large N order parameter 1 • 𝐻 𝜐 1 , 𝜐 2 = 𝑂 σ 𝑗 𝜓 𝑗 𝜐 1 𝜓 𝑗 𝜐 2 = + ⋯ + +

Sachdev-Ye-Kitaev model • Critical correlation function ( Δ = 1/𝑟 ) 𝜐 −2Δ 𝜐 𝜓 𝑗 𝜐 𝜓 𝑗 0 ∝ sin 𝛾 𝜌 sgn(𝜐) . 0 • Real time: thermal double state 𝛾 𝑈𝐺𝐸(𝑢) = 𝑎 − 1 2 +𝑗𝑢 𝑜 𝑀 𝑜 𝑆 . 2 σ 𝑜 𝑓 −𝐹 𝑜 −2Δ 𝜌𝑢 • 𝜓 𝑗𝑀 𝑢 𝜓 𝑗𝑆 𝑢 ∝ cosh 𝛾 • Chaos. Maximal Lyapunov exponent 𝜇 = 2𝜌𝑈 (Kitaev ’15, Maldacena - Stanford ’16, Maldacena-Shenker- Stanford ‘15) 𝑀 𝑆 • (Approximately) dual to Jackiw-Teitelboim gravity

Operator growth in the SYK model • Infinite temperature case (Roberts-Stanford- Streicher ‘18) • For finite temperature, we study the size distribution 𝑜 𝑈𝐺𝐸⟩ and ⟨𝑈𝐺𝐸 𝑓 −𝜈 ො ⟨𝑈𝐺𝐸 𝜔 𝑗𝑀 𝑓 −𝜈 ො 𝑜 𝜔 𝑗𝑀 𝑈𝐺𝐸⟩ 𝑜 𝑈𝐺𝐸 ≡ 𝑎 𝜈 is a • 𝑈𝐺𝐸 𝑓 −𝜈 ො partition function of SYK with twisted boundary condition 𝜔 𝛾 = cosh 𝜈 𝜔 𝛾 − sinh 𝜈 𝜔 3𝛾 4 + 𝜗 4 − 𝜗 4 − 𝜗

Operator growth in the SYK model • Two-point function with 𝜈 term can be obtained by Schwinger-Dyson equation −1 • 𝐻 = 𝜖 𝜐 − Σ 𝜈 𝑟−1 𝜐 1 , 𝜐 2 . Σ 𝜈 𝜐 1 , 𝜐 2 = 𝐾 2 𝒣 𝜈 • Note that 𝒣 𝜈 𝜐 1 , 𝜐 2 = 𝑈𝐺𝐸 𝜔 𝑗𝑀 𝜐 1 𝑓 −𝜈 ො 𝑜 𝜔 𝑗𝑀 𝜐 2 𝑈𝐺𝐸 𝑜 𝑈𝐺𝐸⟩ ⟨𝑈𝐺𝐸 𝑓 −𝜈 ො • This is a ratio of two generating functions. Expansion of 𝒣 𝜈 = σ 𝑛 𝑓 −𝜈𝑛 𝐿 𝑛 Coefficient 𝐿 𝑛 satisfies 1 1 • 𝑞 𝑚 𝜔 𝑗 𝑢 𝜍 2 = σ 𝑛 𝐿 𝑛 𝑞 𝑚−𝑛 𝜍 2 . • 𝐿 𝑛 is the “transition probability” that the fermion operator increases 1 2 by 𝑛 size of 𝜍

Large-q solution • In large 𝑟 , the Schwinger-Dyson equation simplifies 𝐾 2 𝑟 G q−1 ∗ G = 𝕁 , 𝜖 𝜐 − 𝑕 • 𝐻 = 𝐻 0 𝑓 𝑟 , ➔ Liouville equation 𝜖 𝜐 1 𝜖 𝜐 2 𝑕 = −2𝐾 2 𝑓 𝑕 𝜈 ෝ • Define 𝜈 = 𝑟 , ො 𝜈 changes the boundary condition of 𝑕 𝜐 1 𝜐 2

Large-q solution • Size of thermal state 1 𝑂 • 𝑜 𝜍 2 = 2 1 − 𝜀 𝛾 , 2 𝛽 𝑟 • 𝜀 𝛾 = 𝒦 𝜌 • At low temperature 𝛽 ≃ 𝛾 . 2 𝒦 • Fermion size 𝜀𝑜 𝛾 𝜔 1 𝑢 = 1 + 2 𝛽 sinh 𝛽𝑢 • Lyapunov exponent 2𝛽 • Consistency at 𝑢 → 0: conjecture 2 𝒦 𝜀𝑜 𝛾 𝜔 1 𝑢 = 𝜀 𝛾 1 + 2 𝛽 sinh 𝛽𝑢 .

Large-q solution • Full size distribution can be obtained by expanding 𝒣 𝜈 • The entire distribution is the same as infinite temperature case except a renormalization 𝒦 of parameter sinh 𝒦𝑢 → 𝛽 sinh 𝛽𝑢 2 −4 𝑟𝛽𝑢 exp − 2𝛽 𝑓 −2𝛽𝑢 𝑜 ∝ 𝑓 𝒦