Holographic thermalization of 2D CFT Shu Lin ( 林树 ) Sun Yat-Sen University ICTS, USTC, 9/8/2016 SL, Shuryak, PRD 2008 SL, PRD 2016
Outline Phenomenon of thermalization in physics Holographic description of thermalization Equal-time probes and geodesic approximation Beyond geodesic approximation How pure state can evolve into thermal state Summary
Thermalization in physics Thermalization: off-equilibrium state thermal equilibrium Thermalization in Heavy ion collisions Provide initial condition for Quark Gluon Plasma & Cosmological evolution Reheating in Cosmology
Difficulty in theoretical studies of thermalization Systems displaying thermalization phenomenon often strong interaction, weak coupling methods e.g. kinetic theory not applicable. Need strong coupling methods: Holography CFT techniques
Probe of thermalization in strongly coupled theory No quasi-particle picture, we look at thermalization from correlation functions One point function: VEV of operators Two point function: correlator of operators contains fluctuation/dissipation of operators contains more generic out-of-equilibrium physics, e.g. photon emission in history of heavy ion collisions
Holographic thermalization Consider simple problem in holography: AdS-Vaidya background dual to: thermalization of CFT
AdS-Vaidya background v 0 0, m v Mθ(v) CFT state: t < 0, vacuum t = 0, injection of energy density m t >0, thermalizing state. t, thermal state
Other similar models Gravitational collapse of massive shell: Initial state introduces additional scale BTZ black hole initial position SL, Shuryak, PRD 2008 pure AdS horizon Gravitational collapse of gravitational pulse: deformation of boundary metric Initial state with tiny temperature Chesler, Yaffe, PRL 2009 horizon
Equal-time (spacelike) probes Correlation function large scaling dimension, geodesic approximation x x Wilson loop Entanglement entropy Balasubramanian et al, PRL 2011 PRD 2011 colors: separation of points
Unequal-time (timelike) probes Euclideanize Vaidya metric, compute geodesics Balasubramanian et al, JHEP 2013 Confirmed in independent CFT computation Anous et al, 1603.04856
2D CFT computation Confirmed holographic results Anous et al, 1603.04856
Beyond geodesic approximation In this work, focus on two point function of O: O scaling dimension not large, geodesic approximation not applicable Wightman correlator for thermalizing state Retarded correlator is state independent Wightman correlator is state dependent Caron-Huot, Chesler, Teaney, PRD 2011
Dictionary Wightman correlator for thermalizing state Extrapolated dictionary (good for out-of-equilibrium) φ bulk field dual to boundary operator O mass dimension Δ = 2 Skenderis, van Rees, JHEP 2010 Keranen, Kleinert, JHEP 2015
Recipe of bulk correlator in AdS-Vaidya Initial data Caron-Huot, Chesler, Teaney, PRD 2011 Propagators 3 4 boundary z=0 AdS 1 2 BTZ shell v=0
AdS3 and BTZ correlators BTZ is equivalent to AdS locally Euclidean AdS bulk-bulk correlator By analytic continuation:
Initial data in AdS Singular as v 1, v 2 0 and x 21 0 (ξ 1, lightcone singularity)! Regularize by subtracting BTZ counterpart free of singularity vacuum thermal
Propagator in BTZ Non-vanishing only on the lightcone
Regularized CFT correlator vacuum thermal thermalizing thermal CFT correlator in thermalizing state thermal correlator As v 3, v 4 0, vacuum correlator thermal correlator As v 3, v 4, 0
Properties of CFT correlator in vacuum and thermal state ig > 2 1 O(t 2, x 2 )O(t 1, x 1 ) O scalar operator with mass dimension d t 21 = t 2 t 1 x 21 = x 2 x 1 ig 0 > 2 1 = 2 1 π ( t 21 iε 2 + x 2 21 ) d vacuum: long range correlation in space and time (power law) ig th > 2 1 = 1 2π (2πT) 2d ( cosh 2πTt 21 iε + cosh (2πTx 21 )) d thermal: screening in space and time (exponential)
Equal-time correlator O(v, x)o(v, 0) v 0, O v, x O v, 0 2 π 1 x 4 1 2π 1 ( 1 + cosh x) 2 1 x 4, x 1 vacuum thermal x 2v, x 1, O v, x O v, 0 = 1 x 4 F(v) x = 15 Analytic expression SL PRD 2016
Correlator from CFT calculation O v2, x O v1,0 ~ e π(v2+v1)/τ e 2πx/τ τ~ screening length initial state short range correlation Smooth transition occurs when x > v2 + v1 v2 v1 < x < v2 + v1 Calabrese, Cardy, PRL 2006 x v2 v1 τ Thermalization time of distance x: x + O(1)
Correlator O(v4, x)o(v3,0) O(v4, x)o(v3,0) = 18 πx 4 H(v4, v3) x v4, v3, x 1 reminiscent of long range correlation in initial state Aparicio, Lopez, JHEP 2009 O(v4, x)o(v3,0) = 18 v4 coth v4 1 v3 coth v3 1 πx 4 sinh 2 v4 sinh 2 v3 v4, v3 0, O(v4, x)o(v3,0) vacuum correlator v4, v3 1, O v4, x O v3,0 ~ 1 x4 v4 v3 e 2v4 2v3 Enhanced correlation SL PRD 2016
Initial state w/wo long range correlation τ correlation length long range correlation v v x x O v, x O v, 0 ~ e 4v O v, x O v, 0 ~ 1 x 4 v2 e 4v Calabrese, Cardy, PRL 2006 Effect of long range correlation
Equal-time correlator O(v, x)o(v, 0) v 0, O v, x O v, 0 2 π 1 x 4 1 2π 1 ( 1 + cosh x) 2 1 x 2, x 1 vacuum thermal x, v 1, blue purple brown SL PRD 2016
Equal-space correlator O(v + δv, 0)O(v, 0) δv 1, O v + δv, 0 O v, 0 = 1 δv 2 g( v δv ) δv = 1 10, 1 20, 1 50 v 0, O v + δv, 0 O v, 0 vacuum - thermal correlator 2 1 π δv 4 1 1 2π ( 1 + cosh δv) 2 δv 0, O v + δv, 0 O v, 0 finite Need to resolve time of energy injection?
Equal-space correlator O(v + δv, 0)O(v, 0) δv 1, v 1, O v + δv, 0 O v, 0 ~v δv e 2(2v+δv) O v + δv, 0 O v, 0 ~ thermal ~e 2δv Thermalization time v 1 O v + δv, 0 O v, 0 ~e πδv/τ v 1 Calabrese, Cardy, PRL 2006 Thermalization time ~1, independent of interval δv. In contrast to equal-time case.
Spatially integrated correlator (k=0 mode) k=0 mode dx ΔO(v4, x)o(v3,0) exact, numerical results also by Keranen, Kleinert, JHEP 2015 Two analytic limits: v 3, v 4 1 dx ΔO(v4, x)o(v3,0) 3π 512 v4 v3 v 3 1, v 4 1 dx ΔO(v4, x)o(v3,0) 3π 128 v3 v4e 2v4 SL PRD 2016
Small momentum mode long distance? dx ΔO(v, x)o(v, 0) ~ 1 v when v 1 x, v 1, O v, x O v, 0 = 1 x 2 f(v x ) dx ΔO(v, x)o(v, 0) ~ dx 1 x 2 f v x ~ 1 v At early time, dorminant contribution to small momentum mode from short distance physics! In equilibrium, small momentum = long distance Not true in out-of-equilibrium state? SL PRD 2016
Radiation spectrum of particle weakly Analogy with heavy ion collisions: O electromagnetic current Particle coupled to O dilepton coupled to O Particle emission from energy injection to time v Normalized by thermal emission rate SL PRD 2016 Faster thermalization of lower frequency modes
Pure state to thermal state? Holography SUGRA approximation: 1/N expansion CFT: 1/c expansion t and N not commuting? Anous et al, 1603.04856 Explicit example: quantum revival in harmonic oscillator Cardy, 1603.08267
Summary Holographic thermalization of CFT using geodesic approximation. Beyond geodesic approximation: enhanced correlation from initial long range correlation. Evidence on the inequivalence between small momentum physics and long distance physics away from equilibrium. Subtlety of late time and large central charge limits.
No logarithmic correction to x v4 v3 τ: thermalization time O(v4, x = v4 + δ)o(v3,0) ~e 2v4 h(δ) v3 1, v4 1, δ~1 h(δ) monotonous decaying function O(v4, x = v4 + δ)o(v3,0) ~e 2v4 h δ ~ thermal ~e 2x Thermalization time: v4 + O 1 ~x + O(1) No ln x correction to thermalization time! SL 1511.07622
Holographic thermalization of 2D CFT BTZ black hole boundary z=0 ds 2 = 1 z 2 f z dt2 + dx 2 + dz2 f z f z = 1 m z 2 massless shell falling horizon : z = m 1/2 pure AdS ds 2 = 1 z 2 dt2 + dx 2 + dz 2 Shell starts falling from the boundary z = 0 at t = 0, dual to: t < 0, vacuum t = 0, injection of energy density m t >0, thermalizing state. t, thermal state
AdS3-Vaidya background set m = 1, temperature T~O(1) Eddington-Finkelstein Coordinate v=t on the boundary z=0 pure AdS BTZ shell v=0 Shell starts falling from the boundary z = 0 at t = 0: t < 0, vacuum t = 0, injection of energy density m t >0, thermalizing state. t, thermal state