The spectral function in a strongly coupled, thermalising CFT

Transcription

1 The spectral function in a strongly coupled, thermalising CFT Vrije Universiteit Brussel and International Solvay Institutes in collaboration with: V. Balasubramanian, A. Bernamonti, B. Craps, V. Keranen, E. Keski-Vakkuri, B. Muller, L. Thorlacius 25 September 2012

2 1 Motivation 2 Spectral function 3 Setup of the model 4 Two point functions 5 Wick rotation 6 Conclusion

3 1 Motivation 2 Spectral function 3 Setup of the model 4 Two point functions 5 Wick rotation 6 Conclusion

4 Heavy ion collision formation of Quark-Gluon Plasma Behaves as near-ideal Fermi liquid after fast thermalisation We want to understand thermalisation process itself Figure: Snapshot of RHIC video: Hot quark soup

5 Heavy ion collision formation of Quark-Gluon Plasma Behaves as near-ideal Fermi liquid after fast thermalisation We want to understand thermalisation process itself Problems: Strongly coupled dynamics Non equilibrium state Difficult computation Goal: Understanding thermalisation process using AdS/CFT correspondence Not just thermalisation time Understand as much as possible

6 Different probes of thermalisation behaviour: Two point functions [Abajo-Arrastia, Aparício and López, arxiv: ] [Balasubramanian, Bernamonti, de Boer, Copland, Craps, Keski-Vakkuri, Muller, Schafer, Shigemori and Staessens, arxiv: ] [Erdmenger, Lin and Ngo, arxiv: ] Spacelike Wilson loops Entanglement entropy [Albash and Johnson, arxiv: ] Mutual and tripartite information [Balasubramanian, Bernamonti, Copland, Craps and Galli, arxiv: ] New probe: Spectral function: follow time evolution during thermalisation [Balasubramanian, Bernamonti, Craps, Keranen, Keski-Vakkuri, Muller, Thorlacius and Vanhoof, (to appear)]

7 1 Motivation 2 Spectral function 3 Setup of the model 4 Two point functions 5 Wick rotation 6 Conclusion

8 Two point functions in equilibrium determined by Occupation number: n(k, ω) Spectral function: ρ(k, ω) Spectral function = weight of free propagator in interacting propagator + ρ(k, ω ˆD ) ω dω R (k, ω) = ω 2 ω 2 + iɛ 2π with + ρ(k, ω ) ω dω 2π = 1 For a free particle deltapeak: ρ(k, ω) = 4πδ(ω 2 ω 2 k ) For an interacting system smearing due to interactions It can be derived from retarded two point function id R (x, t) = θ(t) [O(x, t), O(0, 0)] ρ(k, ω) = 2 Im ˆD R (k, ω) We now want to determine this in the strongly coupled regime.

9 Time dependence: ρ(k, ω, T ) = 2 Im ˆD R (k, ω, T ) { t = t 1 t 2 T = t1+t2 2 { t 1 = T + t 2 t 2 = T t 2 Compare to quenched harmonic oscillator (ω i = 2, ω f = 1) Ω Ω Ω Equal-space 2pnt functions momentum average ( + ) ρ T (ω) ρ T (k, ω)dk = 4π Im dt e iωt D R (x = 0, t, T ) Goal: Determine equal-space two point functions

10 1 Motivation 2 Spectral function 3 Setup of the model 4 Two point functions 5 Wick rotation 6 Conclusion

11 AdS/CFT: CFT on boundary asymptotic AdS spacetime Equilibrium (T = 0) Injection of energy + thermalisation Equilibrium (T 0) Empty AdS Infalling shell of null dust Black hole (T = R 2π ) Vaidya geometry: thin infalling shell

12 AdS/CFT: CFT on boundary asymptotic AdS spacetime Equilibrium (T = 0) Injection of energy + thermalisation Equilibrium (T 0) Empty AdS Infalling shell of null dust Black hole (T = R 2π ) Vaidya geometry: thin infalling shell ds 2 = (r 2 θ(v)r 2 )dv 2 + 2dvdr + r 2 dx 2 v < 0 : t = v + 1 v > 0 : t = v 1 r 2R ln r R r + R ds 2 = dr2 r 2 r2 dt 2 + r 2 dx 2 ds 2 = dr2 r 2 R 2 (r2 R 2 )dt 2 + r 2 dx 2 AdS 3 (vacuum) BTZ (black hole)

13 1 Motivation 2 Spectral function 3 Setup of the model 4 Two point functions 5 Wick rotation 6 Conclusion

14 AdS/CFT: two point function = sum over all paths P O(x 1 )O(x 2 ) = DP e i L(P) with L(P) = g µν Ẋ µẋν dλ P Spacelike path P: L(P) imaginary Geodesic approximation O(x 1 )O(x 2 ) geodesics Better for large e L Implicit expression for equal-time two point functions [Balasubramanian et al., arxiv: ]

15 Extend this calculation for timelike separated points Problem 1: Timelike path P: L(P) real geodesic approximation? Problem 2: real timelike geodesics do not extend to boundary Solution: analytic continuation Wick rotation to Euclidean signature (!!!) ds 2 = dr2 r 2 +r2 ( dt 2 +dx 2 ) complexified geodesics r = r(λ) x = x(λ) t = t(λ) y=it ds 2 = dr2 r 2 +r2 (dy 2 +dx 2 ) r = r(λ + iβ) x = x(λ + iβ) t = t(λ + iβ)

16 1 Motivation 2 Spectral function 3 Setup of the model 4 Two point functions 5 Wick rotation 6 Conclusion

17 Geodesic approximation for Euclidean two point functions D(x 1, x 2 ) = O(x 1 )O(x 2 ) = Euclidean BTZ: ds 2 = dr2 r 2 R 2 + (r2 R 2 )dy 2 + r 2 dx 2 Euclidean 2pnt function: DP e L(P) geodesics e L D thermal (x 1, x 2 ) = 1 [ ( ( ) ( ))] 4 R 2 sinh 2 Rδx + sin 2 Rδy 2 2 Limit R 0 recovers vacuum result D vacuum(x 1, x 2 ) = lim R 0 D thermal(x 1, x 2 ) = 1 (δx 2 + δy 2 )

18 Problem: No straightforward continuation of Vaidya metric ds 2 = (r 2 θ(v)r 2 )dv 2 +2dvdr+r 2 dx 2 Solution: Shell of null matter as limit of spacelike matter (E ) ds 2 = (r 2 θ(v)r 2 )dv 2 2Edvdr + r 2 θ(v)r 2 + E + dr 2 2 r 2 θ(v)r 2 + E 2 +r2 dx 2 Double Wick rotation (z = iv, S = ie) ds 2 = (r 2 θ(z)r 2 )dz 2 2Sdzdr r 2 θ(z)r 2 S + dr 2 2 r 2 θ(z)r 2 S 2 +r2 dx 2

19 Back to Minkowski signature: Wick rotation (y = it) gives time-ordered two point function D F (x 1, t 1 ; x 2, t 2 ) = id(x 1, y 1 = it 1 ; x 2, y 2 = it 2 ) Retarded two point function can be found from D R (x 1, t 1 ; x 2, t 2 ) = θ(t 1 t 2 ) [D F (x 1, t 1 ; x 2, t 2 ) + (D F (x 1, t 1 ; x 2, t 2 )) ] Final result: Explicit expression for equal-space 2pnt function 1 t 1 t 2 2 if 0 > t 1 > t 2 1 D R (t 1, t 2 ) = 2 sin(π ) 2 sinh( R R 2 (t 1 t 2 )) 2 if t 1 > t 2 > 0 1 ) 2 if t 1 > 0 > t 2 )t 2 ( R 2 sinh Rt1 2 cosh( Rt1 2

20 Time evolution of (momentum averaged) spectral function: + ( + ) ρ T (ω) ρ T (k, ω)dk = 4π Im dt e iωt D R (x = 0, t, T )

21 Use AdS/CFT to probe thermalisation strongly coupled CFT We found explicit expression for equal-space 2pnt function Non-standard analytic continuation Complexified geodesics result agrees Notion of time dependent spectral function Collaborators use different method: results seem to match Outlook: (Numerically) find general two-point functions? Occupation numbers? ρ(k, ω) = 2 Im ˆD R (k, ω) (1 + 2n(k, ω))ρ(k, ω) = 2 Im ˆD F (k, ω) Thank you for your attention!