Eternal traversable worm hole in 2D

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1 Eternal traversable worm hole in 2D Xiao-Liang Qi Stanford University Strings2018, OIST, 6/27/2018 Ref: J. Maldacena & XLQ, arxiv:

2 Goal: dual Coupled SYK Global AdS2 Outline: 1. Overview of the AdS2-SYK duality 2. Proposal: Global AdS2 coupled SYK model 3. Low energy effective theory 4. Beyond low energy: large q limit 5. Finite temperature: Hawking-Page type phase transition

3 Jackiw-Teitelboim gravity No bulk graviton. Jackiw 85, Teitelboim 83 Boundary condition φ = φ b Reduction to boundary dynamics Different solutions

4 Sachdev-Ye-Kitaev model H = σ ijkl J ijkl χ i χ j χ k χ l with Gaussian random coupling J ijkl. (Sachdev-Ye, 93, Kitaev 15) Or complex fermion model H = σ ij,kl J ijkl c i + c j + c k c l. Generalization (Maldacena-Stanford 16) : H = σ i1 i 2 i q J i1 i 2 i q χ i1 χ i2 χ iq Averaging over disorder q-body interaction Z n Z n in large N limit. Large N order parameter N fermions G τ 1, τ 2 = 1 N σ i χ i τ 1 χ i τ 2 = + + +

5 Sachdev-Ye-Kitaev model Approximate conformal invariance at low energy G τ 1, τ 2 G f = f τ 1 f Δ τ 2 G f τ1, f τ 2 Low energy manifold Diff(S 1 )/SL(2, R) Effective action S eff [G f ] = α dτ Sch tan f τ, τ, α N J Thermofield double dual to AdS2 black hole for N βj χ il χ ir Analytic continuation An infinite tower of massive matter fields in the bulk χ i t 2 χ i t 2 χ i t 1 f χ i t 1 φ n

6 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 with different time evolution. vs Finding dual of global AdS 2 Finding a Hamiltonian with TFD(t = 0) as ground state

7 Conjecture We conjecture that SYK L + SYK R + relevant coupling has the ground state G TFD Simplest model H = σ J i1 i 2 i q i q 2 χ i1 Lχ i2 L χ iq L + i q 2χ i1 Rχ i2 R χ iq R + iμ σ i χ il χ ir Intuition: small μ couples low energy states more strongly. More precise argument: Variationally minimize free energy among reparameterizations G f Qualitatively similar phenomena in higher dimensional CFT (see XLQ, Katsura, Ludwig 12) + = gap

8 Low energy effective theory Coupled reparameterization modes Repara. of χ il u χ ir u Solution t l u = t r u = 2π β u, β 1 : effective temperature. Also gap of the coupled model T determined by minimizing the free energy F = 2π 2 + [ ]μ 2π 2Δ 1 1β μ 2 2Δ N β β Two-point function periodic in time O u 1 O u 2 2π β 1 cos π β u 1 u 2 Δ. β

9 Physical interpretation on the boundary Coupling μ is relevant. Coupled SYK model has a gap E gap μ/j On comparison, for a free fermion H = μ σ i iχ il χ ir, E free = 2μ ΔJ 1 Since < 1, E 2 2Δ gap E free for small μ The SYK interaction terms help tunneling of fermion SYK with imperfect coupling This enhancement is only possible because identical J in L, R E gap μ if J L J R < J L J L not perfect.

10 Relation to traversable worm hole Boundary point of view: μ term undoes the scrambling and restore the quantum computer to a clean state. exp iδt H μ exp iδt H L + H R By switching on a fine-tuned μ in the blackhole geometry, one can open a worm hole for arbitrarily long time. (compare with Gao, Jafferis, Wall 16, Maldacena, Stanford, Yang 16)

11 Low energy excitations More general solutions SL 2, R gauge symmetry Q 0 = 0, Q ± = 0 General physical solutions are gauge equivalent to t l u = t r u = t u t φ u u = e φ = e 2φ + Δηe 2Δφ V φ = 1 2 e2φ ηe 2Δφ Itinerant states: black hole Bound states: graviton φ

12 Low energy excitations Excitation spectrum Black hole Boundary graviton matter E n = 2 2Δ n E n = n + Δ

13 Beyond low energy: Schwinger-Dyson equation H = σ J i1 i 2 i q i q 2 χ i1 Lχ i2 L χ iq L + i q 2χ i1 Rχ i2 R χ iq R + iμ σ i χ il χ ir With the bilinear coupling, Schwinger-Dyson equation still apply G ω n = iω n I Σ 1 Σ ab τ = J 2 G ab τ q 1 + iμε ab δ τ Numerical solution in general μ, β, J

14 Large q limit Large q limit allows an analytic solution G LL τ 1, τ 2 = 1 g sgn τ LL τ1,τ e q, G LR τ 1, τ 2 = i g e LR τ1,τ2 q 2 S eff g LL, g LR = N dτ q 2 1 dτ 2 ൫ 1 g LL 2 g LL 1 g LR 2 g LR + J 2 e g LL + J 2 e g LR ൯ + μ Ƹ dτg LR τ, τ ൧ μ = μ q Zero temperature solution α 2 e g LL =, J 2 sinh 2 α τ 12 +γ eg LR = α = α μ, γ = γ μ α 2 J 2 cosh 2 α τ 12 +γ For small μ, α μj, γ α. Consistent with AdS2 2 J

15 Hawking-Page-like transition Finite temperature of the coupled model 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 coupled matter field Low T high T

16 Small black hole If the transition is like Hawking-Page, is there a small black hole solution? Yes in large q α 2 e g LL =, J 2 sinh 2 α τ 12 +γ eg LR = identification) For low temperature β = log q 2α σ σ Multiple solution at given β Unstable saddle q α 2 J 2 cosh 2 α τ 12 +γ (+ periodic

17 Small black hole The unstable saddle point is stable in microcanonical ensemble Negative specific heat region 2 S = β2 > 0 E 2 c v

18 Overlap G TFD The overlap of ground state of coupled system and the thermal field double can be computed directly. Bulk picture: quench by switching off μ Low temperature: G TFD e N βj 2 const G TFD Large q: G TFD Finite N numerics e N q 3 const

19 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