Eternal traversable
worm hole in 2D

Xiao-Liang Qi
Stanford University
Strings2018, OIST, 6/27/2018
Ref: J. Maldacena & XLQ, arxiv: 1804.00491

Goal:
dual

Coupled SYKGlobal AdS2Outline:
1. Overview of the AdS2-SYK duality
2.
3.
4.
5.

Proposal: Global AdS2 ⇔ coupled SYK modelLow energy effective theory
Beyond low energy: large 𝑞 limit
Finite temperature: Hawking-Page type phase transition

Jackiw-Teitelboim gravity• No bulk graviton.
• Boundary condition 𝜙 = 𝜙𝑏
• Reduction to boundary dynamics• Different solutions

Jackiw ‘85, Teitelboim ‘83

Sachdev-Ye-Kitaev model• 𝐻 = σ𝑖𝑗𝑘𝑙 𝐽𝑖𝑗𝑘𝑙 𝜒𝑖 𝜒𝑗 𝜒𝑘 𝜒𝑙 with Gaussian random coupling 𝐽𝑖𝑗𝑘𝑙 .
(Sachdev-Ye, ‘93, Kitaev ‘15)

• Or complex fermion model𝐻 = σ𝑖𝑗,𝑘𝑙 𝐽𝑖𝑗𝑘𝑙 𝑐𝑖+ 𝑐𝑗+ 𝑐𝑘 𝑐𝑙 .

q-body interaction• Generalization (Maldacena-Stanford ‘16) :𝐻 = σ𝑖1𝑖2 …𝑖𝑞 𝐽𝑖1 𝑖2 …𝑖𝑞 𝜒𝑖1 𝜒𝑖2 … 𝜒𝑖𝑞• Averaging over disorder• 𝑍𝑛 ≃ 𝑍

nN fermionsin large 𝑁 limit.
• Large N order parameter• 𝐺 𝜏1 , 𝜏2 =1σ 𝜒 𝜏 𝜒 𝜏
𝑁 𝑖 𝑖 1 𝑖 2+=++⋯

Sachdev-Ye-Kitaev model• Approximate conformal invariance at low energyΔ′′• 𝐺 𝜏1 , 𝜏2 → 𝐺𝑓 = 𝑓 𝜏1 𝑓 𝜏2 𝐺 𝑓 𝜏1 , 𝑓 𝜏2
• Low energy manifold Diff(𝑆 1 )/𝑆𝐿(2, 𝑅)• Effective action 𝑆𝑒𝑓𝑓 [𝐺𝑓 ] = −𝛼∫ 𝑑𝜏 Sch tan 𝑓 𝜏 , 𝜏 , 𝛼 ∝𝑁𝐽11• Thermofield double dual to AdS2 black hole for ≪ ≪ 1𝑁𝛽𝐽

𝜒𝑖𝐿

𝜒𝑖𝑅Analyticcontinuation• An infinite tower of massive matter fields in the bulk
𝜒𝑖 𝑡2

𝜒𝑖 𝑡2
𝜒𝑖 𝑡1𝑓

𝜒𝑖 𝑡1

𝜙𝑛Goal: finding a dual theory of global AdS2
• Two causally connected boundaries• Requires negative averaged null energy
from matter field• An eternal version of traversable WH(Gao, Jafferis, Wall ’16, Maldacena, Stanford, Yang ‘16)

• Basic idea: The same state as
𝑇𝐹𝐷(𝑡 = 0) but withdifferent time evolution.
• Finding dual of global 𝐴𝑑𝑆2
≃ Finding a Hamiltonian
with 𝑇𝐹𝐷(𝑡 = 0) as ground statevs

Conjecture• We conjecture that𝑆𝑌𝐾𝐿 + 𝑆𝑌𝐾𝑅 + 𝑟𝑒𝑙𝑒𝑣𝑎𝑛𝑡 𝑐𝑜𝑢𝑝𝑙𝑖𝑛𝑔
has the ground state G ≃ 𝑇𝐹𝐷• Simplest model𝑞2𝑞−2• 𝐻 = σ 𝐽𝑖1 𝑖2 …𝑖𝑞 𝑖 𝜒𝑖1 𝐿 𝜒𝑖2 𝐿 … 𝜒𝑖𝑞𝐿 + 𝑖 𝜒𝑖1 𝑅 𝜒𝑖2𝑅 … 𝜒𝑖𝑞 𝑅 +𝑖𝜇 σ𝑖 𝜒𝑖𝐿 𝜒𝑖𝑅• Intuition: small 𝜇 couples low energy states more strongly.
• More precise argument: Variationally minimize free energy
among reparameterizations 𝐺𝑓• Qualitatively similar phenomena in higher dimensional CFT (seeXLQ, Katsura, Ludwig ‘12)+=gap

Repara. of
𝜒𝑖𝐿 𝑢 𝜒𝑖𝑅 𝑢

Low energy effective theory• Coupled reparameterization modes

2𝜋= 𝑢,𝛽• Solution 𝑡𝑙 𝑢 = 𝑡𝑟 𝑢𝛽−1 : “effective temperature”. Also gap ofthe coupled model• 𝑇 determined by minimizing the free energy
𝐹=−…
𝑁2𝜋𝛽2+ [… ]𝜇2𝜋𝛽2Δ11⇒ ∝ 𝜇 2−2Δ𝛽• Two-point function periodic in time
𝑂 𝑢1 𝑂 𝑢2∝2𝜋𝛽1

cos𝜋𝑢 −𝑢2
𝛽 1Δ.

𝛽Physical interpretation on the boundary• Coupling 𝜇 is relevant. Coupled SYK model has a gap
𝐸𝑔𝑎𝑝 ∝1𝜇/𝐽 2−2Δ 𝐽• On comparison, for a free fermion
H = 𝜇 σ𝑖 𝑖𝜒𝑖𝐿 𝜒𝑖𝑅 , 𝐸𝑓𝑟𝑒𝑒 = 2𝜇.
1• Since< 1, 𝐸𝑔𝑎𝑝 ≫ 𝐸𝑓𝑟𝑒𝑒 for small 𝜇2−2Δ• The SYK interaction terms help “tunneling”
of fermion

SYK with imperfectcoupling• This enhancement is only possible because identical 𝐽 in 𝐿, 𝑅
• 𝐸𝑔𝑎𝑝 ∝ 𝜇 if 𝐽𝐿 𝐽𝑅 < 𝐽𝐿 𝐽𝐿 not perfect.

Relation to traversable worm hole
• Boundary point of view: 𝜇 term “undoes” thescrambling and restore the quantum computer
to a clean state.
exp −𝑖Δ𝑡 𝐻𝜇exp −𝑖Δ𝑡 𝐻𝐿 + 𝐻𝑅

• By switching on a fine-tuned 𝜇 in the blackholegeometry, one can open a worm hole forarbitrarily long time. (compare withGao, Jafferis, Wall ’16, Maldacena, Stanford, Yang ’16)

Low energy excitations• More general solutions• 𝑆𝐿 2, 𝑅 gauge symmetry• 𝑄0 = 0, 𝑄± = 0• General physical solutions are gauge
equivalent to 𝑡𝑙 𝑢 = 𝑡𝑟 𝑢 = t u• 𝑡′ 𝑢 = 𝑒𝜙 𝑢
• 𝜙 ′′ = −𝑒 2𝜙 + Δ𝜂𝑒 2Δ𝜙𝑉 𝜙 =1 2𝜙
𝑒 − 𝜂𝑒 2Δ𝜙2Itinerant states:black hole𝜙Bound states:graviton

Low energy excitations• Excitation spectrum

Black hole

Boundarygraviton

𝐸𝑛 = 2 − 2Δ 𝑛 +matter12෩෪𝑛 = 𝑛 + Δ𝐸

Beyond low energy: Schwinger-Dyson equation𝑞2𝑞2−• 𝐻 = σ 𝐽𝑖1 𝑖2 …𝑖𝑞 𝑖 𝜒𝑖1 𝐿 𝜒𝑖2 𝐿 … 𝜒𝑖𝑞𝐿 + 𝑖 𝜒𝑖1 𝑅 𝜒𝑖2𝑅 … 𝜒𝑖𝑞 𝑅 +𝑖𝜇 σ𝑖 𝜒𝑖𝐿 𝜒𝑖𝑅• With the bilinear coupling, Schwinger-Dyson equation still apply• 𝐺 𝜔𝑛 = 𝑖𝜔𝑛 𝕀 − Σ −1
• Σab 𝜏 = 𝐽2 𝐺𝑎𝑏 𝜏 𝑞−1 + 𝑖𝜇𝜖𝑎𝑏 𝛿 𝜏• Numerical solution in general 𝜇, 𝛽, 𝐽

Large q limit• Large 𝑞 limit allows an analytic solution
𝑔

𝜏 ,𝜏𝑔

𝜏 ,𝜏

𝐿𝐿 1 2𝐿𝑅 1 21𝑖• 𝐺𝐿𝐿 𝜏1 , 𝜏2 = sgn 𝜏12 𝑒 𝑞 , 𝐺𝐿𝑅 𝜏1 , 𝜏2 = 𝑒 𝑞22𝑁
• 𝑆𝑒𝑓𝑓 𝑔𝐿𝐿 , 𝑔𝐿𝑅 = 2 ൣ∫ 𝑑𝜏1 𝑑𝜏2 ൫𝜕1 𝑔𝐿𝐿 𝜕2 𝑔𝐿𝐿 − 𝜕1 𝑔𝐿𝑅 𝜕2 𝑔𝐿𝑅 +𝑞
𝐽2 𝑒 𝑔𝐿𝐿 + 𝐽2 𝑒 𝑔𝐿𝑅 ൯ + 𝜇∫
Ƹ 𝑑𝜏𝑔𝐿𝑅 𝜏, 𝜏 ൧ෝ𝜇• 𝜇=𝑞• Zero temperature solution• 𝑒𝑔𝐿𝐿𝛼2𝛼2𝑔𝐿𝑅= 2,𝑒= 2𝐽 sinh2 𝛼 𝜏12 +𝛾𝐽 cosh2 𝛼 𝜏12 +𝛾• 𝛼 = 𝛼 𝜇 ,𝛾 = 𝛾 𝜇• For small 𝜇, 𝛼 ≃𝜇𝐽𝛼, 𝛾 ≃ . Consistent with AdS22𝐽

Hawking-Page-like transition• Finite temperature of the coupledmodel• First order phase transition for all 𝑞• Low T phase: thermal gas in AdS2
• High T phase: 𝜇 term not important,
two black holes, with coupledmatter field

Low Thigh T

Small black hole
• If the transition is like Hawking-Page, is there a small black holesolution?
• Yes in large 𝑞• 𝑒 𝑔𝐿𝐿 =෥2𝛼2𝛼𝑔, 𝑒 𝐿𝑅 = 222
෥ 𝜏12 +෥𝐽 sinh 𝛼 𝜏12 +𝛾
𝐽 cosh2 𝛼𝛾identification)𝑞𝑞• For low temperature 𝛽 =log
2෥𝛼 𝜎𝜎• Multiple solution at given 𝛽
Unstable saddle(+ periodic

Small black hole
• The unstable saddle point is stable in microcanonical ensemble

Negative specific heat𝜕2 𝑆𝛽2region 𝜕𝐸2 = − 𝑐 > 0𝑣

Overlap 𝐺 𝑇𝐹𝐷• The overlap of groundstate of coupled systemand the thermal field
double can be computed
directly.
• Bulk picture: quench byswitching off 𝜇𝐺𝑇𝐹𝐷• Low temperature: 𝐺 𝑇𝐹𝐷 ≃ 𝑒• Large 𝑞: 𝐺 𝑇𝐹𝐷 ≃ 𝑒• Finite 𝑁 numerics

𝑁𝑞− 3 ×𝑐𝑜𝑛𝑠𝑡−𝑁×𝑐𝑜𝑛𝑠𝑡
𝛽𝐽 2Summary• AdS2 JT gravity (+matter) with general boundary are dual to SYK
model with relevant coupling.
• Eternal traversable worm hole in 2D• Low energy spectrum fixed by approximate conformal
symmetry• Hawking-Page transition and small black hole phase• Open questions:- Higher dimensional generalization- Evaporation of small black hole
- Possible implication to the physics of black hole interior