The spectral function in a strongly coupled, thermalising CFT
TThe spectral function in a strongly coupled,thermalising CFT
Alice Bernamonti
Instituut voor Theoretische Fysica, KU Leuven,Celestijnenlaan 200D, B-3001 Leuven, BelgiumE-mail: [email protected]
Ben Craps
Theoretische Natuurkunde, Vrije Universiteit Brussel, and International Solvay Institutes,Pleinlaan 2, B-1050 Brussels, BelgiumE-mail: [email protected]
Joris Vanhoof ∗ Theoretische Natuurkunde, Vrije Universiteit Brussel, and International Solvay Institutes,Pleinlaan 2, B-1050 Brussels, BelgiumE-mail: [email protected]
In relation to the fluctuation-dissipation theorem, we discuss a time-dependent notion of spectralfunction and effective temperature. Extending recent results from [1], we work out these quan-tities in a two-dimensional thermalising CFT dual to AdS -Vaidya spacetime which interpolatesbetween a black brane geometry at early times and a higher temperature black brane at late times.The computation is carried out in the gravitational holographic dual in the geodesic approximationand using a non-standard analytic continuation. Proceedings of the Corfu Summer Institute 2012September 8-27, 2012Corfu, Greece ∗ Speaker. c (cid:13) Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licence. http://pos.sissa.it/ a r X i v : . [ h e p - t h ] M a r he spectral function in a strongly coupled, thermalising CFT Joris Vanhoof
1. Introduction and summary
Strongly coupled far-from-equilibrium systems appear in many areas of physics and their the-oretical study is notoriously difficult. The gauge/gravity duality is a novel tool to study certainclasses of strongly coupled field theories, and its extension to far-from-equilibrium situations isa very active area of current research. In order to extract useful field theory information fromcomputations in the dual gravity theory, it is important to identify field theory observables that aremeaningful in far-from-equilibrium states, as well as prescriptions for how to compute them in thegravity dual.In the recent paper [1], to which we refer for a more elaborate introduction as well as referencesto the literature, our collaborators and we studied a notion of time-dependent spectral functions andoccupation numbers (based on Wigner transforms of two-point functions) in a simple holographicmodel (see [2, 3] for related work in other holographic models). Starting with the vacuum stateof a strongly coupled 2d conformal field theory, the sudden, homogeneous injection of energy andits subsequent thermalization was modeled by an AdS -Vaidya geometry that interpolates betweenpure AdS at early times and a black brane geometry at late times. In order to compute two-pointfunctions of high-dimension operators, we used a geodesic approximation [4]. For timelike sepa-rations of the operators, it was necessary to use either a non-standard Euclidean continuation of thebulk spacetime, or to use complex geodesics, and both procedures gave the same result. We alsoworked out an approach that went beyond the geodesic approximation, but will not discuss it here.In the present contribution, we extend some of the results of [1] in two ways. First, insteadof starting with the vacuum, we start with an initial thermal state. After a sudden, homogeneousinjection of energy, the system will evolve towards a thermal state with higher temperature. Second,as in [2], we extract from our results a time-dependent effective temperature, which evolves in anon-monotonic way from the initial to the final temperature.In section 2, we review the notions of time-dependent spectral function [3, 1] and time-dependent effective temperature [2]. In section 3, we present the AdS-Vaidya model that describesthe evolution from a black brane with an initial temperature to a black brane with a larger finaltemperature via the injection of a shell of null dust. In this setting, section 4 describes the com-putation of retarded and time-ordered two-point functions using a geodesic approximation and anon-standard continuation to Euclidean signature. In section 5, these two-point functions are usedto extract time-dependent spectral functions and effective temperatures, which are shown to inter-polate between the initial and final equilibrium values.
2. A notion of time-dependent spectral function and effective temperature
We probe the thermalisation of the field theory with time-ordered (Feynman) two-point func-tions, iG F ( t , x ; t , x ) ≡ (cid:104) T { O ( t , x ) O ( t , x ) }(cid:105)≡ θ ( t − t ) (cid:104) O ( t , x ) O ( t , x ) (cid:105) + θ ( t − t ) (cid:104) O ( t , x ) O ( t , x ) (cid:105) , (2.1)and retarded two-point functions (which are real in position space) iG R ( t , x ; t , x ) ≡ θ ( t − t ) (cid:104) [ O ( t , x ) , O ( t , x )] (cid:105) . (2.2)2 he spectral function in a strongly coupled, thermalising CFT Joris Vanhoof
From these definitions, using the hermiticity of the operator O ( t , x ) , we can deduce the relation G R ( t , x ; t , x ) = θ ( t − t ) (cid:2) G F ( t , x ; t , x ) + ( G F ( t , x ; t , x )) ∗ (cid:3) , (2.3)which we will employ later on. When the field theory is in equilibrium, the system is translationinvariant in space and time. The correlators are then functions of the differences only, i.e.G R , F ( t , x ; t , x ) = G R , F ( t − t , x − x ; 0 , ) ≡ G R , F ( t , x ) , (2.4)with t = t − t and x = x − x , and their Fourier transforms are given by G R , F ( ω , k ) = (cid:90) + ∞ − ∞ d t (cid:90) + ∞ − ∞ d x e i ω t e − ikx G R , F ( t , x ) . (2.5)In momentum space, the retarded correlation function is in general complex valued, and its imagi-nary part defines the spectral function, ρ ( ω , k ) = − G R ( ω , k ) . (2.6)When the system is at thermal equilibrium, the retarded and time-ordered correlation functions arerelated by the Fluctuation-Dissipation Theorem, [ + n ( ω )] ρ ( ω , k ) = − G F ( ω , k ) , (2.7)where the occupation number equals the Bose-Einstein distribution: n ( ω ) = ( e ω / θ − ) − for sometemperature θ . Note that using the previous expressions, we have θ =
12 Im G F ( ω ) (cid:18) ∂ Im G R ( ω ) ∂ ω (cid:19) − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ω = , (2.8)where G R , F ( ω ) = (cid:90) + ∞ − ∞ d k π G R , F ( ω , k ) = (cid:90) + ∞ − ∞ d t e i ω t G R , F ( t , x = ) (2.9)is the momentum average of the correlators.In a thermalising field theory, time translational invariance is generically broken. Therefore asin [1] we introduce an average time T and a relative time t by (cid:40) T = t + t t = t − t ⇔ (cid:40) t = T − t t = T + t , (2.10)and we define time-dependent correlators by keeping T fixed and Fourier transforming with respectto the relative time t G R , F ( ω , T , k ) = (cid:90) + ∞ − ∞ d t (cid:90) + ∞ − ∞ d x e i ω t e − ikx G R , F ( t , T , x ) , (2.11)and G R , F ( ω , T ) = (cid:90) + ∞ − ∞ d k π G R , F ( ω , T , k ) = (cid:90) + ∞ − ∞ d t e i ω t G R , F ( t , T , x = ) . (2.12)3 he spectral function in a strongly coupled, thermalising CFT Joris Vanhoof
We can then define a time-dependent spectral function ρ ( ω , T , k ) = − G R ( ω , T , k ) , (2.13)and a time-dependent temperature θ ( T ) =
12 Im G F ( ω , T ) (cid:18) ∂ Im G R ( ω , T ) ∂ ω (cid:19) − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ω = . (2.14)This notion of effective temperature was also considered in [2] to assess thermalisation followingan anisotropic deformation of the boundary.
3. Setup of the model
As a holographic model for a thermalising field theory, we consider a three-dimensional, thinshell AdS-Vaidya spacetime. For R > R >
0, it has the metric ds = − [( r − R ) − θ ( v )( R − R )] dv + dvdr + r dx . (3.1)The spacetime structure is depicted in Fig. 1 and it represents an infalling shock wave of ‘null’ dustin a black brane background that collapses to form a heavier black brane. The coordinate v is anEddington-Finkelstein coordinate such that v = r is a radial coordinate such that r = ∞ is the planar boundary of the asymptoticallyAdS spacetime and r = A dSb o und a r y ( r = ∞ ) singularity ( r = 0) s h o c k w a v e ( v = ) r = R r = R BTZ ( v <
0) BTZ ( v > Figure 1:
The causal structure of the thin shell AdS-Vaidya spacetime.
The metric (3.1) solves Einstein’s equations with a negative cosmological constant and a deltasource stress-energy tensor on the shock wave. Outside the shock wave ( v > he spectral function in a strongly coupled, thermalising CFT Joris Vanhoof by the BTZ solution with radius R ds = − ( r − R ) dt + dr r − R + r dx with t = v − R ln (cid:12)(cid:12)(cid:12)(cid:12) r − R r + R (cid:12)(cid:12)(cid:12)(cid:12) , (3.2)while inside ( v < R ds = − ( r − R ) dt + dr r − R + r dx with t = v − R ln (cid:12)(cid:12)(cid:12)(cid:12) r − R r + R (cid:12)(cid:12)(cid:12)(cid:12) . (3.3)The dual picture of this spacetime on the boundary CFT is that of a field theory that is initially inequilibrium at a temperature θ = R π , which corresponds to the Hawking temperature of the initialblack brane. After a sudden injection of energy at the time t =
0, there is a period of thermalisationuntil the field theory reaches a new equilibrium at a higher temperature θ = R π .We will now probe the equilibration dynamics of this holographic model using a geodesicapproximation of two-point functions. While here, as in [1], we will focus on timelike two-pointfunctions in order to be able to derive the spectral function, spacelike Green’s functions as probesof thermality have been considered in [5, 6, 7]. Spacelike geodesics connecting two equal timeboundary points separated by a distance (cid:96) are conjectured to also compute the entanglement entropyof a spatial region of size (cid:96) in the boundary two-dimensional CFT [8, 9, 5, 6, 10]. Other non-localprobes of thermalisation derived from the entanglement entropy, namely the mutual and tripartiteinformation, were discussed in Federico Galli’s talk at this workshop [11] and are studied in [12,13].
4. Geodesic approximation of two-point functions
Consider a scalar operator O ( t , x ) with conformal dimension ∆ in the dual CFT. The time-ordered two-point function (cid:104) T { O ( t , x ) O ( t , x ) }(cid:105) is given by a path integral over all paths P that connect the two insertion points ( t , x ) and ( t , x ) on the boundary: (cid:104) T { O ( t , x ) O ( t , x ) }(cid:105) = (cid:90) D P e − ∆ L ( P ) with L ( P ) ≡ (cid:90) P (cid:114) g µν dx µ d λ dx ν d λ d λ . (4.1)For large ∆ , we can use a saddle point approximation, in which the sum over all paths can beapproximated by a sum over all geodesics connecting the boundary endpoints [4] (cid:104) T { O ( t , x ) O ( t , x ) }(cid:105) ∼ ∑ geodesics e − ∆ L , (4.2)where L denotes the geodesic length. However, due to contributions near the asymptoticallyAdS boundary, the geodesic length between two boundary points contains a universal divergenceand needs to be renormalized. Throughout this section, we will define a renormalized length as δ L ≡ L − r , in terms of the bulk cut-off r , by removing the divergent part of the geodesiclength in pure AdS. The renormalized two point function can then be approximated by (cid:104) T { O ( t , x ) O ( t , x ) }(cid:105) ren ∼ e − ∆ δ L , (4.3)where δ L is the renormalised length of the geodesic that connects the points ( t , x ) and ( t , x ) on the boundary. 5 he spectral function in a strongly coupled, thermalising CFT Joris Vanhoof
While for static spacetimes a Wick rotation to Euclidean signature is straightforward, this is notthe case for time-dependent geometries as the AdS-Vaidya background (3.1). Therefore we performa non-standard analytic continuation of the metric first proposed in [1]. Let E > R > R > ds = − [( r − R ) − θ ( v / E )( R − R )] dv + Edvdr (cid:113) ( r − R ) − θ ( v / E )( R − R ) + E + dr ( r − R ) − θ ( v / E )( R − R ) + E + r dx . (4.4)Through the coordinate transformation v = t − R arctanh (cid:18) R E (cid:113) + ( E − R ) r (cid:19) + R arctanh (cid:0) R E (cid:1) for v > t − R arctanh (cid:18) R E (cid:113) + ( E − R ) r (cid:19) + R arctanh (cid:0) R E (cid:1) for v < , (4.5)we recover the BTZ metric (3.2) with radius R for v > R for v <
0. The coordinate transformations are such that v = E > θ ( v / E ) = θ ( v ) , so thatin the E → ∞ limit (4.4) reduces to the Lorentzian ‘null’ Vaidya metric (3.1).On the Lorentzian ‘spacelike’ Vaidya metric (4.4), we can now perform an analytic continua-tion on the time coordinate w = iv , as well as on the parameter Q = iE . Without loss of generality,we can take Q > ds = [( r − R ) − θ ( w / Q )( R − R )] dw − Qdwdr (cid:113) ( r − R ) − θ ( w / Q )( R − R ) − Q + dr ( r − R ) − θ ( w / Q )( R − R ) − Q + r dx , (4.6)where as before θ ( w / Q ) = θ ( w ) . Note that the radial coordinate r now runs from (cid:113) Q + R to ∞ .Letting f ( r ) ≡ (cid:40) f in ( r ) = r − Q − R for w < f out ( r ) = r − Q − R for w > , (4.7)the metric becomes ds = f ( r ) dw + (cid:18) Qdw − drf ( r ) (cid:19) + r dx , (4.8)which is manifestly positive. We now compute the length of the geodesics that connect the points ( r , τ , x ) and ( r , τ , x ) ,where r denotes the location of the regularised AdS boundary. We will assume ∆ τ = τ − τ > ∆ x = x − x = he spectral function in a strongly coupled, thermalising CFT Joris Vanhoof
The geodesics in Euclidean Vaidya consist of piecewise geodesic curves in BTZ. Above the shell( w > x ± ( r ) = x ± R arctanh (cid:32) Γ − Γ + (cid:115) r − Γ + r − Γ − (cid:33) , (4.9) w ± ( r ) = τ ± R arctan (cid:32)(cid:115) R − Γ − Γ + − R (cid:115) r − Γ + r − Γ − (cid:33) + R arctan R Q (cid:115) − ( Q + R ) r − R arctan (cid:18) R Q (cid:19) , (4.10) λ ± ( r ) = λ ± arccosh (cid:32)(cid:115) r − Γ − Γ + − Γ − (cid:33) , (4.11)and below the shell ( w < x ± ( r ) = ¯ x ± R arctanh (cid:32) ¯ Γ − ¯ Γ + (cid:115) r − ¯ Γ + r − ¯ Γ − (cid:33) , (4.12)¯ w ± ( r ) = ¯ τ ± R arctan (cid:32)(cid:115) R − ¯ Γ − ¯ Γ + − R (cid:115) r − ¯ Γ + r − ¯ Γ − (cid:33) + R arctan R Q (cid:115) − ( Q + R ) r − R arctan (cid:18) R Q (cid:19) , (4.13)¯ λ ± ( r ) = ¯ λ ± arccosh (cid:32)(cid:115) r − ¯ Γ − ¯ Γ + − ¯ Γ − (cid:33) . (4.14)Here ± denote two separate branches (appearing above and below the turning points of BTZgeodesics) and the parameters appearing in these expressions satisfy the conditions 0 (cid:54) Γ − (cid:54) R < Γ + and 0 (cid:54) ¯ Γ − (cid:54) R < ¯ Γ + . We can now distinguish three cases.1) If both endpoints are above the shell ( τ > τ > > τ > τ ), then there is a geodesic connecting themwhich is entirely below the shell. Any additional geodesics would require the existence ofgeodesics above the shell in BTZ that connect two points on the shell, and again these do notexist.The geodesic length is then given by the BTZ result, derived for example in the Appendix of [1], δ L = ln (cid:20) R (cid:18) sin (cid:18) R ∆ τ (cid:19) + sinh (cid:18) R ∆ x (cid:19)(cid:19)(cid:21) , (4.15)where R is respectively R and R in case 1) and 2).7 he spectral function in a strongly coupled, thermalising CFT Joris Vanhoof
The existence of geodesics that have two endpoints on the shell, and that are above (or below)the shell would require the existence of a local maximum (or minimum) of w ( r ) (or ¯ w ( r ) ). We notethat dw + dr ( r (cid:12) ) = d ¯ w − dr ( r (cid:12) ) = r (cid:12) = Q Γ + Γ − √ Q R − ( ¯ Γ + − R )( R − ¯ Γ − ) . Equal space geodesics aswe consider here, all have ¯ Γ − = τ > > τ ).They cross the shock wave once and hence we need to determine how they refract at the shelllocation in order to compute their length.The conditions that need to be imposed for the geodesics at the shell ( w =
0) were derived in [1].They are given by the following refraction law1 f dxdr (cid:18) Q f dwdr − (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) in = f dxdr (cid:18) Q f dwdr − (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) out , (cid:0) f dxdr (cid:1) (cid:34) f (cid:18) dwdr (cid:19) + (cid:18) Q f dwdr − (cid:19) (cid:35)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) in = (cid:0) f dxdr (cid:1) (cid:34) f (cid:18) dwdr (cid:19) + (cid:18) Q f dwdr − (cid:19) (cid:35)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) out , (4.16)which is supplemented by the continuity conditions x in ( r ∗ ) = x out ( r ∗ ) and w in ( r ∗ ) = w out ( r ∗ ) = r ∗ is the value of the radial coordinate at which the geodesic reaches the shell.The boundary conditions at the AdS boundary are x + ( r → ∞ ) = x w + ( r → ∞ ) = τ ¯ x − ( r → ∞ ) = x ¯ w − ( r → ∞ ) = τ ⇒ x = x + R arctanh (cid:16) Γ − Γ + (cid:17) τ = τ + R arctan (cid:18)(cid:114) ( R − Γ − )( Γ + − R ) (cid:19) x = ¯ x − R arctanh (cid:16) ¯ Γ − ¯ Γ + (cid:17) τ = ¯ τ − R arctan (cid:18)(cid:114) ( R − ¯ Γ − )( ¯ Γ + − ¯ R ) (cid:19) (4.17)and the continuity conditions x + R arctanh (cid:32) Γ − Γ + (cid:115) r ∗ − Γ + r ∗ − Γ − (cid:33) = ¯ x + R arctanh (cid:32) ¯ Γ − ¯ Γ + (cid:115) r ∗ − ¯ Γ + r ∗ − ¯ Γ − (cid:33) , (4.18)0 = τ + R arctan (cid:32)(cid:115) R − Γ − Γ + − R (cid:115) r ∗ − Γ + r ∗ − Γ − (cid:33) + R arctan R Q (cid:115) − ( Q + R ) r ∗ − R arctan (cid:18) R Q (cid:19) , (4.19)0 = ¯ τ + R arctan (cid:32)(cid:115) R − ¯ Γ − ¯ Γ + − R (cid:115) r ∗ − ¯ Γ + r ∗ − ¯ Γ − (cid:33) + R arctan R Q (cid:115) − ( Q + R ) r ∗ − R arctan (cid:18) R Q (cid:19) . (4.20)8 he spectral function in a strongly coupled, thermalising CFT Joris Vanhoof
In terms of the solutions (4.9)-(4.14), the refraction conditions read Γ − Γ + R = ¯ Γ − ¯ Γ + R , (4.21)1 ( r ∗ − R ) R R (cid:113) ( r ∗ − Γ − )( r ∗ − Γ + ) − Qr ∗ (cid:113) ( R − Γ − )( Γ + − R ) (cid:113) r ∗ − Q − R = ( r ∗ − R ) R R (cid:113) ( r ∗ − ¯ Γ − )( r ∗ − ¯ Γ + ) − Qr ∗ (cid:113) ( R − ¯ Γ − )( ¯ Γ + − R ) (cid:113) r ∗ − Q − R . (4.22)Altogether, these are nine conditions, for nine unknowns ( x , τ , ¯ x , ¯ τ , Γ + , Γ + , ¯ Γ + , ¯ Γ − and r ∗ ).The renormalised length of the geodesic is given by δ L = lim r → ∞ (cid:0) λ + ( r ) − ¯ λ − ( r ) − ( r ) (cid:1) = λ − ¯ λ +
12 ln (cid:18) Γ + − Γ − (cid:19) +
12 ln (cid:18) Γ + − ¯ Γ − (cid:19) , (4.23)which, using the continuity condition λ + ( r ∗ ) = ¯ λ + ( r ∗ ) , can be written as δ L = ln ( ¯ Γ + − ¯ Γ − ) (cid:113) r ∗ − ¯ Γ − + (cid:113) r ∗ − ¯ Γ + (cid:113) r ∗ − Γ − + (cid:113) r ∗ − Γ + . (4.24)We will focus on equal space correlators, i.e. ∆ x = x − x =
0. Because of the refraction condition, dxdr does not change sign at the shell. Also the geodesics in BTZ have no local extremum of x ( r ) .Therefore, we must set Γ − = ¯ Γ − =
0. It then follows that x = ¯ x = x = x . We are thus leftwith five conditions for five unknowns. After solving this system, we find that the renormalisedgeodesic length is given by δ L = ln ( r ∗ − cos ( ¯ γ ) (cid:113) r ∗ − R )( r ∗ + cos ( γ ) (cid:113) r ∗ − R ) , (4.25)where the unknown r ∗ is implicitly determined by the relation (cid:113) r ∗ − R cos ( γ ) − Q sin ( γ ) (cid:113) r ∗ − Q − R = (cid:113) r ∗ − R cos ( ¯ γ ) − Q sin ( ¯ γ ) (cid:113) r ∗ − Q − R , (4.26)and we have used the expressionssin ( γ ( r ∗ )) = R sin ( R Ω ( r ∗ )) r ∗ − cos ( R Ω ( r ∗ )) (cid:113) r ∗ − R , (4.27) The expressions (4.26)-(4.28) were obtained assuming 0 (cid:54) γ ( r ∗ ) (cid:54) π / (cid:54) ¯ γ ( r ∗ ) (cid:54) π /
2. Outside this range,similar expressions can be derived from (4.22). he spectral function in a strongly coupled, thermalising CFT Joris Vanhoof and sin ( ¯ γ ( r ∗ )) = − R sin ( R ¯ Ω ( r ∗ )) r ∗ − cos ( R ¯ Ω ( r ∗ )) (cid:113) r ∗ − R , (4.28)with Ω ( r ∗ ) = τ + R arctan R Q (cid:115) − ( Q + R ) r ∗ − R arctan (cid:18) R Q (cid:19) , (4.29)and ¯ Ω ( r ∗ ) = τ + R arctan R Q (cid:115) − ( Q + R ) r ∗ − R arctan (cid:18) R Q (cid:19) . (4.30)The parameters Γ + and ¯ Γ + in the geodesics (4.9)-(4.14) are given by the formulas Γ + ( r ∗ ) = R +( r ∗ − R ) sin ( γ ( r ∗ )) and ¯ Γ + ( r ∗ ) = R + ( r ∗ − R ) sin ( ¯ γ ( r ∗ )) . In Fig. 2, we plot the profile of asample of equal space geodesics in the Euclidean AdS-Vaidya geometry. r (cid:45) (cid:45) w Figure 2:
Equal space geodesics in the Euclidean shell background for R = . , R = , Q = . , τ = − and, increasing from the bottom up, τ = − . , . , . , . , , . . If we perform the double Wick rotation τ = it , τ = it and Q = iE and take the limit E → ∞ ,in order to recover the ‘lightlike’ Vaidya result, we finally find δ L = ln (cid:34)(cid:18) R cosh (cid:18) R t (cid:19) sinh (cid:18) R t (cid:19) − R sinh (cid:18) R t (cid:19) cosh (cid:18) R t (cid:19)(cid:19) (cid:35) , (4.31)which reduces to the expression obtained in [1] in the R = The Euclidean two-point function associated to a CFT in thermal equilibrium at a tempera-ture θ = R / ( π ) can be computed in the geodesic approximation from the renormalised geodesiclength in Euclidean BTZ (4.15). Remember that, in the geodesic approximation, the renormalisedEuclidean two-point function in the dual CFT equals (cid:104) O ( τ , x ) O ( τ , x ) (cid:105) ren ∼ e − ∆ δ L . (4.32)10 he spectral function in a strongly coupled, thermalising CFT Joris Vanhoof
From (4.15), we then find the (Euclidean) thermal two-point function G thermal E ( τ , x ; τ , x ) = (cid:2) R (cid:0) sin (cid:0) R ∆ τ (cid:1) + sinh (cid:0) R ∆ x (cid:1)(cid:1)(cid:3) ∆ . (4.33)Despite the fact that we have used a geodesic approximation, the result for the thermal two-pointfunction is exact, being fully constrained by conformal invariance (apart from an overall scaling).Note that we can obtain the vacuum result by taking R → G vacuum E ( τ , x ; τ , x ) = lim R → G thermal E ( τ , x ; τ , x ) = ( ∆ τ + ∆ x ) ∆ . (4.34)It is well known that by Wick rotating the Euclidean two-point function, one finds the time-ordered(Feynman) two-point function. Therefore let τ = lim ε → e i ( π − ε ) t (cid:39) it , such that τ = − t + i ε and iG F ( t , x ; t , x ) = G E ( it , x ; it , x ) . (4.35)The retarded two-point function can be obtained from the time-ordered one by applying the relation(2.3). For non-integer scaling dimension ∆ , this leads to iG thermal F ( t , x ; t , x ) = θ (cid:2) ( ∆ x ) − ( ∆ t ) (cid:3)(cid:2) R (cid:0) sinh (cid:0) R ∆ x (cid:1) − sinh (cid:0) R ∆ t (cid:1)(cid:1)(cid:3) ∆ + θ (cid:2) ( ∆ t ) − ( ∆ x ) (cid:3) e − i π ∆ (cid:2) R (cid:0) sinh (cid:0) R ∆ t (cid:1) − sinh (cid:0) R ∆ x (cid:1)(cid:1)(cid:3) ∆ , (4.36)and G thermal R ( t , x ; t , x ) = − ( π ∆ ) θ ( ∆ t ) θ (cid:2) ( ∆ t ) − ( ∆ x ) (cid:3)(cid:2) R (cid:0) sinh (cid:0) R ∆ t (cid:1) − sinh (cid:0) R ∆ x (cid:1)(cid:1)(cid:3) ∆ . (4.37) In an analogous way, from (4.31) we can then find (the geodesic approximation of) the Feyn-man and retarded two-point functions in the thermalizing CFT dual to three-dimensional Vaidya: iG F ( t , x ; t , x ) = e − i π ∆ θ ( − t ) θ ( − t ) (cid:12)(cid:12)(cid:12) R sinh (cid:0) R ( t − t ) (cid:1)(cid:12)(cid:12)(cid:12) ∆ + e − i π ∆ θ ( t ) θ ( t ) (cid:12)(cid:12)(cid:12) R sinh (cid:0) R ( t − t ) (cid:1)(cid:12)(cid:12)(cid:12) ∆ + e − i π ∆ θ ( t ) θ ( − t ) (cid:12)(cid:12)(cid:12) R cosh (cid:0) R t (cid:1) sinh (cid:0) R t (cid:1) − R sinh (cid:0) R t (cid:1) cosh (cid:0) R t (cid:1)(cid:12)(cid:12)(cid:12) ∆ + e − i π ∆ θ ( − t ) θ ( t ) (cid:12)(cid:12)(cid:12) R cosh (cid:0) R t (cid:1) sinh (cid:0) R t (cid:1) − R sinh (cid:0) R t (cid:1) cosh (cid:0) R t (cid:1)(cid:12)(cid:12)(cid:12) ∆ , (4.38) The case of integer ∆ is discussed in detail in [1]. he spectral function in a strongly coupled, thermalising CFT Joris Vanhoof and G R ( t , x ; t , x ) = − ( π ∆ ) θ ( t − t ) θ ( − t ) θ ( − t ) (cid:12)(cid:12)(cid:12) R sinh (cid:0) R ( t − t ) (cid:1)(cid:12)(cid:12)(cid:12) ∆ + θ ( t ) θ ( t ) (cid:12)(cid:12)(cid:12) R sinh (cid:0) R ( t − t ) (cid:1)(cid:12)(cid:12)(cid:12) ∆ + θ ( − t ) θ ( t ) (cid:12)(cid:12)(cid:12) R cosh (cid:0) R t (cid:1) sinh (cid:0) R t (cid:1) − R sinh (cid:0) R t (cid:1) cosh (cid:0) R t (cid:1)(cid:12)(cid:12)(cid:12) ∆ . (4.39)In the limit R → R of equal initial and final temperatures, these expressions reduce to the equi-librium ones, (4.36) and (4.37). Also, in the limit of zero initial temperature, R →
0, we recoverthe expressions for a thermalising CFT that were presented in [1].
5. Spectral function and effective temperature in thermalising CFT
In a two-dimensional CFT in equilibrium at a temperature θ = R / ( π ) , we know the spectralfunction analytically [1, 14]. It is obtained from the Fourier transform of the retarded two-pointfunction (4.37) using equation (2.6): ρ ( ω , k ) = R ∆ − ( Γ ( ∆ )) (cid:12)(cid:12)(cid:12)(cid:12) Γ (cid:18) ∆ + i ( ω + k ) R (cid:19) Γ (cid:18) ∆ + i ( ω − k ) R (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) sinh (cid:16) πω R (cid:17) . (5.1)When averaged over momenta, it becomes ρ ( ω ) = (cid:90) d k π ρ ( ω , k ) = R ∆ − Γ ( ∆ ) (cid:12)(cid:12)(cid:12)(cid:12) Γ (cid:18) ∆ + i ω R (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) sinh (cid:16) πω R (cid:17) . (5.2)To compute the momentum average of the time-dependent spectral function (2.13) in a thermalis-ing CFT, we need to perform a numerical Fourier transform of the equal space retarded two-pointfunction (4.39). As explained in section 2, one can also define a notion of time-dependent tem-perature (2.14) from the Fourier transforms of the thermalising Feynman and retarded equal spacetwo-point functions. The results of this analysis are shown in Figures 3 and 4. Ω Ρ (cid:72) Ω ,T (cid:76) T (cid:61)(cid:165) T (cid:61) (cid:61) (cid:61) (cid:61)(cid:45) (cid:61)(cid:45) (cid:61)(cid:45)(cid:165) Ω Ρ (cid:72) Ω ,T (cid:76) T (cid:61)(cid:165) T (cid:61) (cid:61) (cid:61) (cid:61)(cid:45) (cid:61)(cid:45) (cid:61)(cid:45)(cid:165) (A) (B) Figure 3:
The spectral function as a function of the frequency ω for different values of the average time T .In both plots, we have taken ∆ = . and R = . The plot (A) has R = and plot (B) has R = . . In the distant past ( T → − ∞ ) the spectral function is given exactly by the equilibrium expres-sion (5.2) for a temperature θ = R / ( π ) , while in the distant future ( T → ∞ ) it coincides with the12 he spectral function in a strongly coupled, thermalising CFT Joris Vanhoof equilibrium result for a temperature θ = R / ( π ) . The notion of time-dependent spectral functionthat we have defined interpolates smoothly between these curves. However, for a definite frequency ω , ρ ( ω , T ) does not increase monotonically from the past limiting value to the future one, but isan oscillating function. (cid:45) (cid:45) (cid:45) T Θ (cid:72) T (cid:76) (cid:68)(cid:61) (cid:68)(cid:61) (cid:68)(cid:61) (cid:45) (cid:45) (cid:45) T Θ (cid:72) T (cid:76) (A) (B) Figure 4:
The temperature as a function of average time for different values of the conformal dimension ∆ . In both plots, we have taken R = and R = . . In plot (A) we have taken different values for ∆ asindicated in the legend. Plot (B) has ∆ = . . For sufficiently small values of ∆ , the time-dependent notion of temperature that we havedefined gives a smooth transition from the initial temperature to the final temperature. However,for larger values of ∆ there are values of the average time for which the spectral function will havezero slope at ω =
0. This results in a divergence in the temperature just before the time T = Acknowledgments
JV would like to thank the organizers of the Corfu Summer Institute and the XVIIIth Euro-pean Workshop on String Theory for the nice workshop and the opportunity to present this work.We would also like to thank V. Balasubramanian, V. Keränen, E. Keski-Vakkuri, B. Müller and L.Thorlacius for enjoyable collaboration. This work is supported by the FWO-Vlaanderen, ProjectsNo. G.0651.11 and G.0114.10N, by the Belgian Federal Science Policy Office through the Interuni-versity Attraction Pole P7/37, by the European Science Foundation Holograv Network and by theVrije Universiteit Brussel through the Strategic Research Program “High-Energy Physics”. AB isa Postdoctoral Researcher FWO-Vlaanderen. JV is an Aspirant FWO-Vlaanderen.
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