A prescription for holographic Schwinger-Keldysh contour in non-equilibrium systems
MMIT-CTP/5095EFI-18-21
December 21, 2018
A prescription for holographic Schwinger-Keldysh contour innon-equilibrium systems
Paolo Glorioso
Kadanoff Center for Theoretical Physics and Enrico Fermi InstituteUniversity of Chicago, Chicago, IL 60637, USA
Michael Crossley and Hong Liu
Center for Theoretical Physics,Massachusetts Institute of Technology,Cambridge, MA 02139
Abstract
We develop a prescription for computing real-time correlation functions defined on a Schwinger-Keldysh contour for non-equilibrium systems using gravity. The prescription involves a new analyticcontinuation procedure in a black hole geometry which can be dynamical. For a system with aslowly varying horizon, the continuation enables computation of the Schwinger-Keldysh generatingfunctional using derivative expansion, drastically simplifying calculations. We illustrate the pre-scription with two-point functions for a scalar operator in both an equilibrium state and a slowlyvarying non-equilibrium state. In particular, in the non-equilibrium case, we derive a spacetime-dependent local temperature from the KMS condition satisfied by the two-point functions. We thenuse it to derive from gravity the recently proposed non-equilibrium effective action for diffusion. a r X i v : . [ h e p - t h ] D ec ONTENTS
I. Introduction 2II. A new analytic continuation procedure 4III. A scalar example 6A. Equilibrium 6B. Slowly-varying horizon 11IV. Effective field theory of diffusion from gravity 13A. Review of EFT for diffusion 13B. Obtaining generating functional W [ A , A ] 15C. Prescription for finding I EFT [ B , B ] 17D. Explicit bulk solution 20E. Near-horizon behavior to all orders 22F. The effective action 24V. Discussion and conclusions 26References 28 I. INTRODUCTION
For a quantum many-body system in a state given by a density matrix ρ , various real-timecorrelation functions can be obtained from path integrals on a Schwinger-Keldysh contour,see Fig. 1. More explicitly, it is often convenient to consider the generating functional e W [ φ i ,φ i ] = Tr (cid:104) ρ P e i (cid:82) dt ( O i ( t ) φ i ( t ) −O i ( t ) φ i ( t )) (cid:105) , (1.1)where O i denote generic operators and φ i their corresponding sources. P indicates thatthe operators are path ordered, and subscripts 1 , O i and O i denote the segments of2he contour an operator O i is inserted. The minus sign in the second term comes from thereversed time integration for the second (lower) segment. In (1.1) we have suppressed thespatial dependence of an operator. Correlation functions obtained from (1.1) correspond tothe full set of nonlinear response and fluctuating functions [1–5], and thus play key roles instudies of non-equilibrium systems. FIG. 1. A Schwinger-Keldysh contour, or often referred to as a closed time path. Operators insertedon the upper and lower segments are labeled respectively by indices 1 and 2.
For systems with a gravity dual, a prescription for computing (1.1) when ρ is a thermalequilibrium state has been developed in [6, 7] (see [8] for a recent review). More gener-ally, when state ρ can be prepared by a Euclidean path integral (which includes thermalequilibrium states), one can write (1.1) as a path integral involving some Euclidean andsome Lorentzian segments, and can obtain a corresponding gravity spacetime by patchingtogether different pieces. With the full path integration contour represented on the gravityside, the generating functional (1.1) can then be obtained using the standard procedure ofintegrating over the bulk fields with sources as boundary conditions [9, 10]. This approachis conceptually straightforward, but in practice tedious to carry out even for a thermal equi-librium computation. Also general non-equilibrium state cannot be prepared by a Euclideanpath integral. See also [11–13] for other discussions of real-time correlation functions innon-equilibrium contexts.In this paper we propose a new prescription for computing (1.1) for a non-equilibrium statewhose dual gravity solution involves a dynamical horizon which is analytic. The prescriptioninvolves a simple analytic continuation of the black hole geometry. Even for a thermal This prescription was already briefly advertised in [8]. O is given by a conserved current. The plan of the paper is as follows. In Sec. II we introduce the prescription. In Sec. IIIwe apply the prescription to obtain the large-distance and long-time behavior of two-pointfunctions of a scalar operator using derivative expansion, both in an equilibrium state and ina non-equilibrium state with a slowly varying horizon. In Sec. IV we apply the prescriptionto derive an effective action for diffusion. We conclude in Sec. V with a discussion of futuredirections.
Note:
As this work was nearing completion we became aware of [21] which also derivedthe effective action for diffusion from holography.
II. A NEW ANALYTIC CONTINUATION PROCEDURE
We will now discuss a proposal for the gravity description of the Schwinger-Keldyshcontour of Fig. 1 for a general time-dependent gravity geometry with an analytic horizon.Consider a time-dependent gravity solution in Eddington-Finkelstein coordinates, ds = − f ( r, x µ ) dv + 2 drdv + λ ij ( r, x µ ) dx i dx j , (2.1)with x µ = ( v, x i ). Throughout the paper we will use 0 to denote v -indices. We assume the See [15] for a review. See also [16] for a non-dissipative formulation, [17] for a preliminary dissipativeformulation, and [18, 19] for superspace formulations. See also [20] for a recent interesting application. r = ∞ , f ( r → ∞ ) → r , λ ij ( r → ∞ ) → δ ij r . (2.2)We also assume f and λ ij are smooth functions of r and x µ , with f ( r, x µ ) having a simplezero at some r h ( x µ ). In this section we shall not need to specify the form of f and λ ij . FIG. 2. The complexified gravity spacetime corresponding to the Schwinger-Keldysh contour inFig. 1. The two segments of gravity spacetime are identified with the two segments of the Schwinger-Keldysh contour. It should be understood that in the above plot the circle around r h is infinitesimaland the two segments along the real r -axis have infinitesimal imaginary parts. The two boundariesare denoted respectively as ∞ , with possible sources φ , . Now to describe the Schwinger-Keldysh contour in Fig. 1 on the gravity side, we treat theradial coordinate r as a complex variable, and analytically continue r around r h as indicatedin Fig. 2. The part of the contour below the real r -axis is identified with the first (upper)segment of the Schwinger-Keldysh contour of Fig. 1, while the part above the real r -axis isidentified with the second (lower) Schwinger-Keldysh segment. The two slices are connectedby a circle going around r h in a counterclockwise manner. The radius (cid:15) of the circle is takento be infinitesimal. The arrows in Fig. 2 denote the orientations of the r -direction, wherethe lower segment has the standard orientation while the upper segment has the oppositeorientation. The metric (2.1) should now be understood as that on the full r -contour. Forthe full complexified spacetime to have a single orientation, the reversal of r -orientation5ffectively reverses the orientation of the v direction in the second copy. Note that one doesno analytic continuation in Eddington-Finkelstein time v , which is already regular.For an external black hole which describes a thermal equilibrium state, the above pre-scription is equivalent to that of [7] for σ = 0. Consider for example a free bulk scalar fieldΦ. In Fourier space, with Φ = Φ( r, ω, (cid:126)k ) e − iωv + i(cid:126)k · (cid:126)x , (2.3)a general solution can be written close to the horizon r h asΦ( r, ω, (cid:126)k ) = A ( ω, (cid:126)k )( r − r h ) iβω π + B ( ω, (cid:126)k ) (2.4)where β is the inverse temperature, and A, B are some functions. Now applying the pre-scription of Fig. 2, we find that the corresponding coefficients A , , B , for Φ , on the upperand lower segments of the contour of Fig. 2 should be related by A = A e βω , B = B (2.5)which is precisely the prescription of [7]. Note that in this frequency space approach onecannot easily perform small ω expansion as in (2.4), when expanding ω , one encounterspowers of log( r − r h ) which become singular at the horizon r = r h . III. A SCALAR EXAMPLE
We will now illustrate the new prescription of the previous section using a scalar two-point function. We are interested in extracting its large-distance and long-time behaviorusing derivative expansion.
A. Equilibrium
We first consider an equilibrium state with metric (2.1) for which functions f and λ ij are x µ -independent. Consider a massless scalar field S = − (cid:90) d d +1 x √− gg MN ∂ M Φ ∂ N Φ , (3.1)6here M = 0 , , . . . , d . The equation of motion for Φ can be written as1 √ λ ∂ r (cid:16) f √ λ∂ r Φ (cid:17) + ∂ ∂ r Φ + 1 √ λ ∂ r ( √ λ∂ Φ) + λ ij ∂ i ∂ j Φ = 0 (3.2)with boundary conditionsΦ( r, x µ ) → φ ( x ) , r → ∞ , Φ( r, x µ ) → φ ( x ) , r → ∞ , (3.3)where ∞ , ∞ denote the two asymptotic infinities (see Fig. 2) and where ∂ denotes deriva-tive with respect to v . We are interested in solving (3.2) in the regime where the boundarysources φ ( x ) , φ ( x ) are slowly varying in x µ compared to the characteristic scales of thesystem, which can be taken to be the inverse temperature β = 4 π/f (cid:48) ( r h ). More explicitly,we assume β∂ µ φ φ ∼ βL (cid:28) , β∂ µ φ φ ∼ βL (cid:28) L is the typical time/length scale associated with the variations of the sources. Weassume that Φ( r, x ) satisfies the same condition for any r , i.e. β∂ µ Φ( r, x µ )Φ (cid:28) , (3.5)while the derivatives with respect to r can be arbitrary. We will see that this is a consistentassumption, thanks to the fact that the integration contour Fig. (2) goes around the horizon r h , thus avoiding the singular behavior of (2.4) at r − r h . We can then expand Φ in terms ofboundary derivatives of φ , as Φ = Φ (0) + Φ (1) + Φ (2) + · · · (3.6)and solve (3.2) order by order in the number of derivatives. The boundary conditions forvarious Φ ( n ) areΦ (0) ( r, x µ ) → φ ( x ) , r → ∞ , Φ (0) ( r, x µ ) → φ ( x ) , r → ∞ , (3.7)Φ ( n ) ( r ) → , r → ∞ , Φ ( n ) ( r ) → , r → ∞ , n ≥ . (3.8)For our discussion below it is convenient to introduce the following combinations φ r = 12 ( φ + φ ) φ a = φ − φ . (3.9)7t zeroth order we have 1 √ λ ∂ r (cid:16) f √ λ∂ r Φ (0) (cid:17) = 0 , (3.10)whose solution which satisfies the Dirichlet boundary conditions (3.7) is given byΦ (0) = − φ a Q b ( r ) + φ = φ r − φ a Q b r ( r ) (3.11)with b ( r ) = (cid:90) r ∞ dr √ λf , b ( r ) = (cid:90) r ∞ dr √ λf , b r ( r ) = 12 ( b + b ) ,Q ≡ b ( ∞ ) = − b ( ∞ ) = b − b = (cid:73) r h dr √ λf = − √ λ h iβ . (3.12)The contour of integrals in (3.12) is along that of Fig. 2. The direction that the integrationgoes around r = r h is dictated by the limits of the integrals. For example, for r lying onthe upper segment one goes around r h clockwise in the integral for b ( r ), while in b ( r ),when r lies in the lower segment, one goes around r h counterclockwise. In Q the integralgoes clockwise around r h , which determines the sign of its value − iβ . Note that it is crucialthat the integrand for b ( r ) has a logarithmic divergence at r h , which makes it possible forΦ (0) ( r, x µ ) to satisfy both the boundary conditions at ∞ , . If the integrand were regular at r = r h , then we would be forced to have φ = φ .At n -th order, we find 1 √ λ ∂ r (cid:16) f √ λ∂ r Φ ( n ) (cid:17) = s ( n ) (3.13)where s ( n ) denotes terms which were already known from lower orders, e.g. s (1) = − ∂ ∂ r Φ (0) − √ λ ∂ r (cid:16) √ λ∂ Φ (0) (cid:17) s (2) = − ∂ ∂ r Φ (1) − √ λ ∂ r (cid:16) √ λ∂ Φ (1) (cid:17) − λ ij ∂ i ∂ j Φ (0) . (3.14)Equation for Φ ( n ) can be solved in general asΦ ( n ) ( r ) = (cid:90) r ∞ dr (cid:48) √ λf (cid:34)(cid:90) r (cid:48) ∞ dr (cid:48)(cid:48) √ λs ( n ) + c n (cid:35) (3.15)8here c n is chosen so that Φ ( n ) ( r → ∞ ) →
0. For n = 1, after some manipulations we findΦ (1) = ∂ φ a Q (cid:18) a r ( r ) b r ( r ) − (cid:112) λ h Q (cid:19) + ∂ φ r ( (cid:112) λ h b r ( r ) − a r ( r )) (3.16)where we have further introduced a ( r ) = (cid:90) r ∞ drf , a ( r ) = (cid:90) r ∞ drf , a r ( r ) = 12 ( a + a ) , a − a = (cid:112) λ h Q . (3.17)One easily sees that this procedure generalizes to all derivative orders. Comparing (3.11)and (3.16) we note that in order for the derivative expansion to make sense we need to choosethe radius (cid:15) of the circle around the horizon to be not too small. Explicitly, as (cid:15) →
0, thefirst term in (3.11) diverges logarithmicallyΦ (0) ( r = r h + (cid:15) ) → − i φ a π log (cid:15) , (3.18)while the first term in (3.16) has a stronger divergenceΦ (1) ( r = r u + (cid:15) ) → iβ ∂ φ a π (log (cid:15) ) , (3.19)so, in order to have Φ (1) subleading to Φ (0) , we need φ a (cid:29) β∂ φ a | log (cid:15) | , i . e . (cid:29) (cid:15) (cid:29) e − Lβ (3.20)where we have used (3.4). In the strict derivative expansion limit L → ∞ , so (cid:15) can be takento zero at the end.We now proceed to evaluate the on-shell action for Φ, which will yield the Schwinger-Keldysh generating functional of the boundary theory to quadratic order in sources. Plugging(3.11) and (3.16) in the bulk action, we will find the on-shell action W up to second orderin derivatives. To proceed, we write the bulk action as S = − (cid:90) d d +1 x √ λ (cid:2) f ( ∂ r Φ) + 2 ∂ r Φ ∂ Φ + λ ij ∂ i Φ ∂ j Φ (cid:3) (3.21)The on-shell action at zeroth order in derivatives is W (0) = − (cid:90) d d +1 x √ λf ( ∂ r Φ (0) ) = 12 (cid:90) d d x φ a Q = iβ (cid:90) d d x (cid:112) λ h φ a . (3.22)9t first order we have W (1) = − (cid:90) d d +1 x √ λ [ f ∂ r Φ (0) ∂ r Φ (1) + ∂ r Φ (0) ∂ Φ (0) ]= − (cid:90) d d +1 x √ λ∂ r Φ (0) ∂ Φ (0) = − (cid:90) d d x (cid:112) λ h φ a ∂ φ r (3.23)where the first term in the second expression can be shown to vanish by performing integra-tion by parts, using the fact that Φ (0) is a solution of (3.10), and that Φ (1) vanishes on theboundaries ∞ , ∞ .The second order action will turn out to contain integrals which are divergent due to thebehavior of various functions as r → ∞ , which is expected from standard near-boundarybehavior of the bulk gravity theory [22]. To cure these divergences, one evaluates the integralsup to some cut-off slice r Λ < ∞ , add a local counterterm action S → S + S ct to (3.21) tocompensate for the divergence, and take r Λ → ∞ at the end. One additional ingredienthere is that we are working with two boundaries, which means that we need to add onecounterterm action per boundary: S ct = 1 d − r d − (cid:90) d d xη µν ∂ µ φ ∂ ν φ − d − r d − (cid:90) d d xη µν ∂ µ φ ∂ ν φ = 1 d − r d − (cid:90) d d xη µν ∂ µ φ a ∂ ν φ r , (3.24)where the power of r Λ and the coefficient in the prefactor are determined in order to com-pensate the divergence from bulk integrals, and η µν is the boundary Minkowski metric. Notethat there cannot be terms proportional to φ a as counterterms should always have a fac-torized structure as the first line of (3.24). Doing similar steps as above and including thecounterterm action (3.24), the second order on-shell action can be expressed as W (2) = − (cid:90) r Λ d d +1 x √ λ [ ∂ Φ (1) ∂ r Φ (0) + ∂ r Φ (1) ∂ Φ (0) + λ ij ∂ i Φ (0) ∂ j Φ (0) ]+ 12( d − r d − (cid:90) d d xη µν ∂ µ φ a ∂ ν φ r = (cid:90) d d x (cid:26) Q ra ∂ φ a ∂ φ r + Q ijra ∂ i φ a ∂ j φ r + i β Q aa ( ∂ φ a ) + i β Q ijaa ∂ i φ a ∂ j φ a (cid:27) (3.25)where in the first line, the superscript r Λ in the integration means that we integrate in goingfrom r Λ of the upper segment to r Λ of the lower segment, following the contour of Fig. 2,10nd at the end of the evaluation r Λ should be taken to ∞ and ∞ in the upper and lowersegment, respectively. The coefficients in (3.25) are Q ra = − Q (cid:90) r Λ dr √ λb r f − r d − d − , Q aa = − i βQ (cid:90) ∞ ∞ drf (cid:18) √ λb r + 14 (cid:112) λ h Q (cid:19) (3.26) Q ijra = 1 Q (cid:90) r Λ dr √ λλ ij b r + r d − d − , Q ijaa = i βQ (cid:90) ∞ ∞ dr √ λλ ij b r . (3.27)For the Schwarzschild metric with d = 4, we find Q ra = 12 r h (log 2 − , Q ijra = 12 r h δ ij . (3.28)Putting together first order and second order retarded action gives the retarded propagator G R ( ω, q ) = r h iω + 12 r h (log 2 − ω + 12 r h q , (3.29)which agrees with the retarded propagator of the tensor mode in [23].One can readily verify that W satisfies the KMS symmetry W [ ˜ φ , ˜ φ ] = W [ φ , φ ], where[15] ˜ φ ( x µ ) = φ ( − x + iθ, x i ) , ˜ φ ( x µ ) = φ ( − x − i ( β − θ ) , x i ) . (3.30)Note that, at this derivative order, KMS invariance does not impose any constraint on W (2) .One can indeed check that second derivative terms generated by W (1) [ ˜ φ , ˜ φ ] cancel witheach other.We conclude this subsection by noting that the term in W proportional to φ a arises fromthe small circle around the horizon connecting the two horizontal segments. This is consistentwith the general expectation that noises arise from fluctuations around the horizon. B. Slowly-varying horizon
Let us now consider a state which evolves with time. We consider a spacetime whosehorizon changes slowly in the boundary coordinates x µ . We shall give a proof of principlethat the above discussion can be generalized to such spacetime by performing an explicitcheck at first derivative order. 11or concreteness, we consider the AdS-Schwarzchild metric: ds = − f dv + 2 drdv + r dx i , f = r (cid:18) − ( r h ( x µ )) d r d (cid:19) , (3.31)where r h ( x µ ) is a slowly-varying function, and we treat derivatives ∂ µ r h with the same powercounting as in (3.4). To first order, we only need to account for the correction to W (1) whichis proportional to ∂ r h ( v ), which we denote by δW (1) . From (3.23) we find δW (1) = − (cid:90) d d +1 x √ λ∂ r Φ (0) δ ( ∂ Φ (0) ) , (3.32)where δ ( ∂ Φ (0) ) is the part of ∂ Φ (0) which is proportional to first derivatives of r h : δ ( ∂ Φ (0) ) = dr d − h πr d − f iφ a ∂ r h . (3.33)One then finds δW (1) = i (cid:90) d d xB ( x µ ) ∂ r h ( x µ ) φ a , (3.34)where B ( x µ ) = i d r d − h π (cid:90) ∞ ∞ dr r d − f = d − π r d − h . (3.35)Altogether, the action at first order reads W (0) + W (1) + δW (1) = (cid:90) d d xr d − h (cid:18)
12 ( φ a ∂ φ r − φ r ∂ φ a ) − i (cid:18) dr h π − d − πr h ∂ r h (cid:19) φ a (cid:19) . (3.36)Note that the KMS symmetry still holds, where the local inverse temperature is β = (cid:18) dr h π − d − πr h ∂ r h (cid:19) − , (3.37)i.e. the relation between temperature and horizon r h receives a derivative correction, whichis because we are not in equilibrium. Note that we still have KMS symmetry because weassume local equilibrium. We expect that the analytic continuation continues to hold whenthe dependence of r h on boundary spacetime coordinates is not slow.12 V. EFFECTIVE FIELD THEORY OF DIFFUSION FROM GRAVITY
The generating functional we found in Sec. III for a scalar operator is local, in the sensethat the corresponding W can be written as W = (cid:82) d d x F [ φ , φ ] where F is a functionof φ , and their derivatives, and has a well-defined derivative expansion. For a conserved U (1) current J µ or the stress tensor T µν , the corresponding generating functional will not belocal as there are gapless hydrodynamic modes associated with these conserved quantities,the integrating out of which leads to nonlocal behavior. The goal of hydrodynamic effectiveaction is to isolate the effective dynamics of these modes.In this section we shall derive from gravity the effective action for the diffusion modeassociated with a conserved U (1) current at finite temperature. The story is more complexthan the discussion of last section. In this case one cannot solve the full bulk equationsof motion, as that would amount to integrating over all modes, diffusion modes included.Instead one should integrate out the degrees of freedom except for the one corresponding todiffusion. Deriving effective actions (at non-dissipative level) of hydrodynamic modes fromgravity was initiated by [24], and subsequent work include [25, 26].Below we first present a quick review of the formulation of effective field theory for adiffusion mode given in [14], in anticipation of the structures that we will find in the gravityderivation. A. Review of EFT for diffusion
Consider a quantum system with a conserved U (1) current J µ at a finite inverse temper-ature β . The corresponding Schwinger-Keldysh generating functional is W [ A µ , A µ ], where A µ and A µ are sources for J µ along two segments of the contour, e iW [ A µ ,A µ ] ≡ Tr (cid:104) ρ P e i (cid:82) d d x ( A µ J µ − A µ J µ ) (cid:105) . (4.1)Due to current conservation, the generating functional satisfies W [ A µ , A µ ] = W [ A µ + ∂ µ λ , A µ + ∂ µ λ ] , (4.2)13here λ , λ are independent functions. The generating functional W [ A µ , A µ ] can be ob-tained from the path integrals over a pair of diffusion fields ϕ , e iW [ A µ ,A µ ] = (cid:90) Dϕ Dϕ e iI EFT [ B µ ,B µ ] , (4.3)where B µ = A µ + ∂ µ ϕ , B µ = A µ + ∂ µ ϕ , (4.4)and I EFT [ B µ , B µ ] is a local action of B , B . The combinations (4.4) ensure that (4.2)is satisfied after integrating out ϕ , ϕ , and furthermore, equations of motion for ϕ , areequivalent to imposing the current conservation along two segments of the contour. I EFT canbe obtained as the most general local action of B , B which are: (i) translationally invariant;(ii) rotationally invariant; (iii) invariant under the following diagonal shift symmetry ϕ ( x , x i ) → ϕ ( x , x i ) + λ ( x i ) , ϕ ( x , x i ) → ϕ ( x , x i ) + λ ( x i ) , (4.5)(iv) invariant under dynamical KMS symmetry. With I EFT = (cid:82) d d x L , then to second orderin derivatives and quadratic in B , , L can be written as as L = i a B a + i b B ai + g B a B r + if B a ∂ i B ai + g ∂ B a B r + u ∂ i B ai B r + v B ai ∂ B ri + i a ( ∂ B a ) + i a ( ∂ i B a ) + if ∂ B a ∂ i B ai + i b ( ∂ B ai ) + i b ( ∂ i B aj ) + i c ( ∂ i B ai ) + g ∂ B a ∂ B r + g ∂ i B a ∂ i B r + h B a ∂ i ∂ B ri + v ∂ B ai ∂ B ri − u ∂ i B ai ∂ B r + w F aij F rij , (4.6) The origin of (4.5) is due to that the U (1) symmetry associated with current conservation is not spon-taneously broken. We then have the freedom of independently relabeling the phase of each chargedconstituent of the system on one given time slice. As we are considering a global symmetry, the phaseredefinition cannot depend on time x , but since the constituents of the system are independent of oneanother, they should have the freedom of making independent phase rotations, i.e. we should allow phaserotations of the form e iλ ( x i ) . When the U (1) symmetry is spontaneously broken, i.e. the system is in asuperfluid phase, the phases of all the constituents are locked together, and (4.5) should be dropped. a , b , · · · are constants and we have introduced B rµ = 12 ( B µ + B µ ) , B aµ = B µ − B µ F rij = ∂ i B rj − ∂ j B ri , F aij = ∂ i B aj − ∂ j B ai , (4.7)The explicit expressions of the currents can be found by varying I EFT with respect to B rµ and B aµ . Introducing J µr = 12 ( J µ + J µ ) , J µa = J µ − J µ , (4.8)one finds J µr = δI EFT δB aµ , J µa = δI EFT δB rµ . (4.9)More details, elaborations and generalizations of the above action can be found in [14]. B. Obtaining generating functional W [ A , A ] Before discussing how to obtain the effective action I EFT [ B , B ] for diffusion fields ϕ , ,we first discuss how to obtain the full generating functional W [ A , A ] of (4.1), in a mannerwhich helps motivating the prescription for obtaining I EFT [ B , B ].Suppose J µ is dual to a bulk gauge field C M with an action S = − (cid:90) d d +1 x √− gF MN F MN , (4.10)where F MN = ∂ M C N − ∂ N C M . The background spacetime is again (2.1) with f, λ ij being x µ independent. The boundary conditions are C µ ( r → ∞ ) = A µ , C µ ( r → ∞ ) = A µ (4.11)with A µ , A µ as sources for J µ , J µ . Equations of motion of (4.10) are E M ≡ √− g ∂ N ( √− gF MN ) = 0 . (4.12)It is also convenient to introduce Π µ = √− gF rµ (4.13)15hich is the conjugate momentum to C µ , with ratial direction r treated as “time.” Recallthat when evaluated on the solutions to equations of motion, the boundary values of Π µ simply give the expectation values of boundary currents J µ and J µ . The radial componentof the bulk equations (4.12) can be written in terms of (4.13) as E r = 1 √− g ∂ µ Π µ = 0 , (4.14)which are then equivalent to the conservation of J µ , when evaluated at the boundary. Fromthe Bianchi identity, which can be written as ∂ r ( √− gE r ) + ∂ ( √− gE ) + ∂ i ( √− gE i ) = 0 , (4.15)it is enough to impose E r = 0 at a single slice of r .Now consider performing a gauge transformation C M ( r, x µ ) → C M ( r, x µ ) + ∂ M Λ( r, x µ ) , Λ( r, x µ ) = (cid:90) r c r dr (cid:48) C r ( r (cid:48) , x µ ) (4.16)to set C r to zero. Here r c is an arbitrary reference point, which for later convenience we willtake to lie on the small circle around r h . The boundary conditions (4.11) are modified bysuch a gauge transformation to C µ ( r → ∞ ) = B µ = A µ + ∂ µ ϕ , C µ ( r → ∞ ) = B µ = A µ + ∂ µ ϕ (4.17)where ϕ = (cid:90) r c ∞ dr C r , ϕ = (cid:90) r c ∞ dr C r . (4.18)As in previous sections all the r -integrals should be viewed as contour integrals along thecomplex r -contour of Fig. 2 with directions of integrations specified by the integration limits.Note that ϕ , should be considered as dynamical variables: they are the Wilson linedegrees of freedom associated with C r . They are not independent, as in the gauge C r = 0there is still a residual gauge transformation C µ → C µ + ∂ µ λ ( x µ ) with λ ( x µ ) an arbitraryfunction of boundary coordinates. Under this transformation, ϕ → ϕ + λ and ϕ → ϕ + λ ,thus we could shift one of them away, and the combination ϕ − ϕ remains.The generating functional W [ A , A ] of (4.1) can be obtained from a two-step procedure:16. Find the gauge fixed on-shell action ˜ S on − shell [ B , B ] by plugging solution C µ into equa-tions of motion E µ = 0 with boundary conditions (4.17). Note that one does not imposeequation of motion E r = 0 for C r . From the above discussion, ˜ S is a functional of A , A and ϕ − ϕ .2. Extremizing ˜ S on − shell with respect to one of ϕ ’s, say ϕ , δ ˜ S on − shell δϕ = 0 , (4.19)which corresponds to imposing E r = 0 at slice r = ∞ . Then W [ A , A ] is equal to˜ S on − shell evaluated on the solution ϕ . C. Prescription for finding I EFT [ B , B ] Comparing (4.17) with (4.4) we would like to identify (4.18) as the hydrodynamic fields fordiffusion. Clearly to derive the effective action I EFT [ B , B ], we need to skip the step (4.19),as ϕ , equations of motion correspond to imposing ∂ µ J µ , = 0 at the boundary. But thisis not enough, as discussed above the gauge fixed on-shell action ˜ S on − shell [ B , B ] dependsonly on the combination ϕ − ϕ , while we want independent ϕ , . For this purpose we willimpose an additional boundary condition C ( r c ) = 0 (4.20)where recall that r c is the reference point used to define ϕ , in (4.18), see Fig. 3. r c may beviewed as a “stretched horizon” and its precise location on the small circle is not important.Note that due to the residual gauge symmetry from fixing C r = 0, one could equivalentlychoose any fixed function of x µ to be on the right hand side of (4.20).After imposing (4.20) there is still a residual gauge freedom C i → C i + ∂ i λ ( x i ) where thegauge transformation λ ( x i ) depends only on the boundary spatial coordinates. Such a gaugetransformation also shifts the boundary values of C µ leading to the shifts ϕ ( x , x i ) → ϕ ( x , x i ) + λ ( x i ) , ϕ ( x , x i ) → ϕ ( x , x i ) + λ ( x i ) . (4.21)17his precisely corresponds to the diagonal shift symmetry (4.5) for the EFT. Here we seethe “dual” statement: on the gravity side it is a consequence of a boundary condition at thehorizon. FIG. 3. A boundary condition (4.20) is imposed at the “stretched horizon” r = r c . As willbe discussed below, such a boundary condition implies presence of a nonzero surface charge ρ c at r = r c , with a discontinuity in the radial component of the electric field at r c . To summarize, our prescription for computing the effective action in (4.3) for diffusion is I EFT [ B µ , B µ ] = ˜ S on − sell | C ( r c )=0 (4.22)where the right hand side means that the solution to E µ = 0 should satisfy both boundaryconditions (4.17) and (4.20). In fact, as we will see in next subsection when analyzing theexplicit equations, the independent boundary conditions for C at ∞ and ∞ are onlypossible with the imposing of (4.20).Now let us elaborate a bit further on the physical meaning and implications of (4.20).Imposing (4.20) means that the equation of motion for C at r c is not imposed as we are notfree to vary the value C at that point. More explicitly, the 0-th component of (4.12) canbe written as − ∂ r Π + ∂ i (cid:0) √− gF i (cid:1) = 0 . (4.23)For any r , integrating the above equation from r − c = r c − δ to r + c = r + δ with δ → ( r + c ) − Π ( r − c ) = 0 . (4.24)18hus not imposing E = 0 at r = r c means that there can be discontinuity in Π at r c , i.e.in general ρ c ≡ Π ( r + c ) − Π ( r − c ) (cid:54) = 0 . (4.25)Since Π = √− gF r is the radial component of the electric field, the above equation canbe interpreted that there is a surface charge density proportional to ρ c at the hypersurface r = r c , see Fig. 3.Equation (4.22) can be written more explicitly as I EFT [ B µ , B µ ] = ˜ S [ C µ ( r c ) , B µ ] + ˜ S [ C µ ( r c ) , B µ ] (4.26)where ˜ S denotes the part of the on-shell action evaluated for the region ( r + c , ∞ ) while ˜ S isthat for the region ( r − c , ∞ ). Now consider the variation of ˜ S under a gauge transformation C µ → C µ + ∂ µ λ ( x µ ), we have0 = (cid:90) d d x (cid:20) δS δB µ ∂ µ λ + δS δC µ ( r c ) ∂ µ λ (cid:21) (4.27)which leads to ∂ µ Π µ ( ∞ ) = ∂ µ Π µ ( r + c ) . (4.28)Similarly from ˜ S we have ∂ µ Π µ ( ∞ ) = ∂ µ Π µ ( r − c ) . (4.29)Taking the sum and difference of (4.28)–(4.29) we find that ∂ µ J µa = ∂ ρ c + ∂ i (Π i ( r + c ) − Π i ( r − c )) , ∂ µ J µr = 12 ∂ µ (cid:0) Π µ ( r + c ) + Π µ ( r − c ) (cid:1) . (4.30)Finally, let us note that one possible concern with the boundary condition (4.20) iswhether it can be consistently imposed as C does couple to other components of the gaugefield. Discontinuity of Π can potentially lead to discontinuities in Π i and make the othercomponents of equations of motion not self-consistent. We will pay particular attention tothese aspects in our discussion below and show it does not happen (more details in Sec. IV E).19 . Explicit bulk solution In this subsection we solve explicitly the bulk equations of motion E µ = 0 with boundaryconditions (4.17) and (4.20) order by order in derivatives, assuming the sources are slowlyvarying. We discuss the general structure of the derivative expansion of the equations ofmotion and solve them up to first derivative order. For simplicity from now on we will take λ ij in (2.1) to be diagonal.The equations of motion E µ = 0 can be written explicitly as ∂ r Π = − ∂ i (cid:16) √ λλ ii ∂ r C i (cid:17) (4.31) ∂ r Π i = ∂ (cid:16) √ λλ ii ∂ r C i (cid:17) + √ λ ( λ ii ) (cid:88) j ∂ j F ji (4.32)with Π = √ λ∂ r C , Π i = −√ λλ ii ( f ∂ r C i + ∂ C i − ∂ i C ) . (4.33)Like in the scalar case, we solve (4.31)–(4.32) by doing derivative expansion of C µ , C µ = C (0) µ + C (1) µ + C (2) µ + · · · . (4.34)The boundary conditions at infinities are C (0) µ ( r, x µ ) → B µ ( x ) , r → ∞ , C (0) µ ( r, x µ ) → B µ ( x ) , r → ∞ , (4.35) C ( n ) µ ( r ) → , r → ∞ , C ( n ) µ ( r ) → , r → ∞ , n ≥ . (4.36)At zeroth order, the equations for C (0)0 and C ( i ) i read ∂ r (cid:16) √ λ∂ r C (0)0 (cid:17) = 0 , ∂ r (cid:16) √ λλ ii f ∂ r C (0) i (cid:17) = 0 . (4.37)The solution for C (0)0 can be written as C (0)0 = k b ( r ) + h r c ≤ r ≤ ∞ k b ( r ) + h r c ≤ r ≤ ∞ , (4.38)where k , k , h , h are arbitrary functions of v , and b ( r ) = (cid:90) rr c dr √ λ , Q = b ( ∞ ) . (4.39)20mposing the UV boundary conditions (4.35), together with the boundary condition at thehorizon (4.20), we find the unique zeroth order solution C (0)0 = B Q b ( r ) r c ≤ r ≤ ∞ B Q b ( r ) r c ≤ r ≤ ∞ . (4.40)The equation for C i is almost identical to that of the scalar, eq. (3.10), and can be solvedall the way from ∞ to ∞ C (0) i = B ri − B ai Q i b ri ( r ) , (4.41)where we have already imposed the boundary conditions (4.35), Q was defined in (3.12),and b ri ( r ) = (cid:90) r ∞ dr λ ii √ λf − Q i , Q i = (cid:90) ∞ ∞ dr λ ii √ λf = − K h iβ , K h = √ λλ ii (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) r h . (4.42)Notice from (4.37) that the equation for C is regular across the horizon, while the equa-tion for C i has a similar structure to (3.10) of the scalar. The difference is reflected in thatthe integrand of b in (4.39) is completely regular at r = r h . Without the boundary condi-tion (4.20), then we would not be able to satisfy independent boundary conditions at ∞ and ∞ .At n -th order the equations for C ( n )0 and C ( n ) i read1 √ λ ∂ r (cid:16) √ λ∂ r C ( n )0 (cid:17) = s ( n )0 , √ λ ∂ r (cid:16) √ λλ ii f ∂ r C ( n ) i (cid:17) = s ( n ) i (4.43)where the right hand sides are expressed solely in terms of ( n − s ( n )0 = − λ ii ∂ i ∂ r C ( n − i s ( n ) i = − λ ii ∂ ∂ r C ( n − i − √ λ ∂ r (cid:16) √ λλ ii ( ∂ C ( n − i − ∂ i C ( n − ) (cid:17) − λ ii λ jj ∂ j (cid:16) ∂ j C ( n − i − ∂ i C ( n − j (cid:17) . (4.44)We now solve (4.43) to find C (1)0 , C (1) i . Using (4.40) and (4.41), at first order we have s (1)0 = ∂ i B ia Q i √ λf , s (1) i = ∂ B ia Q i √ λf − √ λ ∂ r (cid:16) √ λλ ii ( ∂ C (0) i − ∂ i C (0)0 ) (cid:17) . (4.45)21e find the temporal component C (1)0 repeating the steps done to find (4.40), yielding C (1)0 = ∂ i B ia Q i (cid:90) rr h dr (cid:48) √ λ (cid:34)(cid:90) r (cid:48) ∞ dr (cid:48)(cid:48) f − ˜ Q Q (cid:35) , ˜ Q = (cid:90) ∞ r h dr (cid:48) √ λ (cid:90) r (cid:48) ∞ dr (cid:48)(cid:48) f (4.46)for both region I and region II. The spatial components C (1) i give C (1) i = − a ( r ) ∂ C (0) i ( r ) + (cid:90) r ∞ drf ∂ i C (0)0 + c b i ( r ) , (4.47)where for compactness we wrote the right hand side in terms of derivatives of C (0)0 , C (0) i , a ( r ) is defined in eq. (3.17), and c is an integration constant to ensure C (1) i ( ∞ ) = 0, c = K h ∂ B i + ∂ i B a Q Q i (cid:90) ∞ r h drf b ( r ) . (4.48)This concludes the evaluation of the bulk solution up to first order.As anticipated, the conjugate momentum Π of C is discontinous at the horizon. Wefind that at the horizon Π ( r + c ) − Π ( r − c ) = B a Q + K h ∂ i B ia . (4.49)Additionally, the non-analiticity in C induces a non-analiticity in C i . More explicitly, wecan write C (1) i = (cid:16) C (1) i (cid:17) ana + (cid:16) C (1) i (cid:17) non (4.50)where (cid:16) C (1) i (cid:17) ana = − a ( r ) ∂ C (0) i ( r ) + c b i ( r ) , (cid:16) C (1) i (cid:17) non = (cid:90) r ∞ drf ∂ i C (0)0 (4.51)and the subscript ana and non denote the analytic and non-analytic part respectively.As in the discussion of (3.18)–(3.20), for the perturbation series to be valid near r h weneed (cid:15) to lie in the range of (3.20). E. Near-horizon behavior to all orders
Below we shall analyze the near-horizon behavior of C and C i to all derivative orders,which is a crucial test of consistency of our prescription.22o proceed, we first divide C i into analytic and non-analytic part C i = ( C i ) ana + ( C i ) non . (4.52)The analytic part ( C i ) ana does not have a well-defined value at the horizon and is definedvia analytic continuation around the horizon through an infinitesimal contour. C and thenon-analytic part ( C i ) non are finite at the horizon and are continuous, but their derivativesare not. Similarly we can write Π i asΠ i = (cid:0) Π i (cid:1) ana + (cid:0) Π i (cid:1) non . (4.53)Now let us look at the qualitative behavior to all orders for various quantities. From (4.43)we find that the equations of motion for the analytic part of C ( n ) i can be written schematicallyas (we will suppress the subscript “ana” for notational simplicity) ∂ r ( f ∂ r C ( n ) i ) = ∂ r ∂ C ( n − i + ∂ C ( n − i + ∂ i ∂ j C ( n − i (4.54)we thus find that near the horizon (cid:16) C ( n ) i (cid:17) ana ∼ (log( r − r h )) n +1 , (cid:0) Π i ( n ) (cid:1) ana ∼ (log( r − r h )) n . (4.55)Now let us look at the non-analytic part. From (4.43) we find that C ( n )0 ∼ ( r − r h )(log( r − r h )) n , (cid:16) C ( n ) i (cid:17) non ∼ ( r − r h )(log( r − r h )) n − . (4.56)For Π µ we find, using (4.33),Π n ) ∼ (log( r − r h )) n , (cid:0) Π i ( n ) (cid:1) non ∼ ( r − r h ) (log( r − r h )) n − . (4.57)Note that the exponent for log( r − r h ) is one higher in (4.56) for C i than that for Π i in (4.57) asin Π i the leading non-analytic behavior from C i cancels with that from C . The non-analyticpart for Π i should appear at second order and it goes to zero on the horizon r = r h .The first equation of (4.30) implies that when we impose conservation of J µa , the diver-gence of the discontinuous part of the momentum at the horizon vanishes, i.e. ∂ µ Π µ ( r + c ) − µ Π µ ( r − c ) = 0. Eq. (4.57) shows that Π i ( r + c ) − Π i ( r − c ) → (cid:15) →
0. In this limit, the firsteq. in (4.30) then becomes ∂ µ J µa = ∂ ρ c . (4.58)After imposing conservation of J µa , the surface charge ρ c located on the horizon becomesthen time-independent. In particular, with the initial condition at v = −∞ that ρ c = 0, wefind that ρ c = 0 for all times, recovering the equation of motion for C at r c . F. The effective action
Plugging the solution of section IV D into the bulk action S = − (cid:90) d d +1 x √− g (cid:20) f g ii F ri + 2 g ii F i F ri − F r + 12 F ij F ij (cid:21) , (4.59)evaluating the radial integral and taking (cid:15) →
0, one obtains the effective action for hydro-dynamic modes ϕ , . We find that (4.6) is precisely recovered. The zeroth and first ordercoefficients are easily evaluated: b = 1 iQ i , g = 1 Q , f = iQ i Q (cid:90) ∞ r h dr b f , v = − K h , a = g = u = 0 . (4.60)The second order coefficients are a = 0 , g = 0 , b = − βK h − iQ i (cid:90) ∞ ∞ √− gb ri λ ii f , (4.61) v = 1 Q i (cid:90) ∞ ∞ a f − βK h i − Q i (cid:90) ∞ ∞ √− gb ri λ ii f , (4.62) f = − iQ i Q (cid:90) r h ∞ √− gb ri b λ ii f − iQ i Q (cid:90) r h ∞ b f (cid:90) r ∞ f − i Q (cid:90) r h ∞ √− gb λ ii f , (4.63) h = − K h Q (cid:90) r h ∞ b f + 1 Q (cid:90) r h ∞ √− gb λ ii f , a = 0 , g = 1 Q (cid:90) r h ∞ √− gb λ ii f , (4.64) u = β iQ i Q (cid:90) r h ∞ b f + 1 Q (cid:90) r h ∞ √− gb λ ii f , (4.65)24 = − iQ i (cid:90) ∞ ∞ f (cid:90) rr h dr (cid:48) √− g (cid:32)(cid:90) r (cid:48) ∞ dr (cid:48)(cid:48) f − ˜ Q Q (cid:33) − iQ i (cid:90) ∞ ∞ √− g (cid:32)(cid:90) r ∞ dr (cid:48) f − ˜ Q Q (cid:33) , (4.66) b = iQ i (cid:90) ∞ ∞ √− g ( λ ii ) b ri , w = 1 Q i (cid:90) ∞ ∞ √− g ( λ ii ) b ri . (4.67)Due to the asymptotic behavior of metric and gauge field at infinity, some of the integralswill be divergent for the same reason explained above (3.24). We again add a counterterm S → S + S ct , where S ct = C d (cid:90) d d x ( F µν F µν − F µν F µν ) , (4.68)with C d = d − r d − for d (cid:54) = 4, and C = log r Λ √ r h + log 2 − , and F µν = ∂ µ A ν − ∂ ν A µ ,and similarly for F µν .Let us now specialize to the AdS Schwarzschild metric with d = 4. We find that thecoefficients evaluate to a = 0 , b = 2 πβ , g = 2 π β , f = − πβ , g = u = 0 , v = − πβ ,a = 0 , a = 0 , f = − G + 7 π π , b = π , b = − π , c = π ,g = 0 , g = 1 − , h = 1 − log 22 − π , v = 1 − log 22 ,u = 1 − log 22 + π , w = − , (4.69)where G is the Catalan’s constant. One can verify that the above satisfies dynamical KMSinvariance I EFT [ ˜ B µ , ˜ B µ ] = I EFT [ B µ , B µ ] , (4.70)where [15]˜ B µ ( x µ ) = B µ ( − x + iθ, − x i ) , ˜ B µ ( x µ ) = B µ ( − x − i ( β − θ ) , − x i ) . (4.71)The expressions of the currents are obtained from I EFT by varying with respect to B aµ and B rµ as in (4.9). The conservation equation of J µa to first order reads, explicitly This particular expression of C has been chosen for the sake of comparison with literature discussedbelow. µ J µa = 1 Q ∂ ˆ B a + K h ∂ ∂ i ˆ B ia = 0 , (4.72)which, using (4.49), can also be written as ∂ ρ c = 0 (4.73)thus confirming the statement that conservation of the horizon charge is equivalent to con-servation of the noise current.Using (4.43) of [14] one finds, for the transverse sector, G Sαα = 2 πT + π ω − π q + O ( ω , q , ω q ) G Rαα = iπT ω + 1 − log 22 ω − q − iπβ ωq + iπβ ω + O ( ω , q ω , q ) . (4.74)From (4.48) of [14] we have, for longitudinal sectorΠ L = − πT + i − log 22 ω − − πT q − πβ ω + · · · iω − q πT − i log 24 π T ωq + O ( q , ω , ω q ) · · · (4.75)and G L = 2 πT − ( π + log 2 π ) q + π ω + O ( q , ω , q ω ) ω + q π T − log 22 π T ω q + O ( ω , · · · ) . (4.76)As a non-trivial check of the construction, one can verify that (4.75),(4.76) agree preciselywith [27], in the low-energy limit ω, q (cid:28) T . V. DISCUSSION AND CONCLUSIONS
In this paper we proposed a simple prescription for real-time correlation functions definedon a Schwinger-Keldysh contour for gravity geometries with a dynamical horizon. We firstapplied the prescription to a scalar field in an external black brane geometry, showing thatit dramatically simplifies the extraction of the long-time and large-distance limit of real-timecorrelation functions. We also showed that the same procedure can be straightforwardlygeneralized to a black hole geometry with a slowly-varying horizon. Finally, we applied26he formalism to give a gravity derivation of the non-equilibrium effective field theory ofdiffusion proposed in [14] to quadratic order in deviations from equilibrium and second orderin derivatives. The derivation contains a few new elements compared with calculation ofreal-time correlation functions, as one needs to keep the hydrodynamic fields for diffusionoff-shell. We showed that this can be achieved by not imposing the Gauss Law and imposinga horizon boundary condition.There are a number of immediate future directions. Firstly, one should study how theprescription works in the calculation of higher-point real-time correlation functions. We sawthat the bulk fields exhibit logarithmic behavior near the horizon. It is important to under-stand whether such behavior leads to complications in the presence of nonlinearity. Secondly,it would be interesting to perform a more systematic study of correlation functions for non-equilibrium geometries with a dynamical horizon. This includes generalizing the discussionof Sec. III B to higher derivative orders and studying more general time-dependent geome-tries. Such investigations may lead to a more geometric way to identify local temperatureof a slowly-varying black hole horizon which we initiated in Sec. III B. Finally, it shouldbe possible to generalize the derivation of the effective action for diffusion to the effectiveaction for the full dissipative hydrodynamics (see [14, 15, 18, 19, 28–32] for discussions ofdissipative hydrodynamic actions). This requires dealing with full nonlinear bulk gravity.
Acknowledgements
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