First Law of Entanglement Entropy in Flat-Space Holography

According to flat/Bondi-Metzner-Sachs invariant field theories (BMSFT) correspondence, asymptotically flat spacetimes in $(d+1)$-dimensions are dual to $d$-dimensional BMSFTs. In this duality, similar to the Ryu-Takayanagi proposal in the AdS/CFT correspondence, the entanglement entropy of subsystems in the field theory side is given by the area of some particular surfaces in the gravity side. In this paper we find the holographic counterpart of the first law of entanglement entropy (FLEE) in a two-dimensional BMSFT. We show that FLEE for the BMSFT perturbed states which are descried by three-dimensional flat-space cosmology, corresponds to the integral of a particular one-form on a closed curve. This curve consists of BMSFT interval and also null and spacelike geodesics in the bulk gravitational theory. Exterior derivative of this form is zero when it is calculated for the flat-space cosmology. However, for a generic perturbation of three-dimensional global Minkowski spacetime, the exterior derivative of one-form yields Einstein equation. This is the first step for constructing bulk geometry by using FLEE in the flat/BMSFT correspondence.


Introduction
Flat/BMSFT is an extension of AdS/CFT correspondence to non-AdS geometries. According to this duality quantum gravity in the asymptotically flat spacetimes in (d + 1)-dimensions can be described by a d-dimensional field theory which is BMS-invariant [1,2]. In the gravity side, BMS symmetry is the asymptotic symmetry of asymptotically flat spacetimes at null infinity [3,4]. In the field theory side , the global part of BMS algebra is given by ultra-relativistic contraction of conformal algebra . Thus one can interpret the flat-space limit (zero cosmological constant limit) in the gravity side as the ultrarelativistic limit of CFT in the boundary theory [2]. In this view, one can study flat/BMSFT by starting from AdS/CFT and taking a limit, the flat-space limit in the bulk and the ultrarelativistic limit in the boundary.
BMS symmetry as the asymptotic symmetry, is infinite-dimensional in three and four dimensions [5]- [7]. Hence one may expect to find some universal aspects for two-and three-dimensional BMSFTs. This situation is very similar to the two-dimensional conformal field theories (CFTs) which their infinite-dimensional symmetry is used to predict the structure of correlation functions as well as entanglement entropy of subsystems. Similarly, the entanglement entropy formula for some particular intervals in BMSFT 2 has been introduced in [8] by just using the infinite symmetry of two-dimensional BMSFTs and then studied more carefully in [9]- [15].
In the context of AdS/CFT correspondence, the entanglement entropy of CFT subsystems has a holographic description. According to Ryu-Takayanagi proposal, this entropy is proportional to the area of a bulk surface which has the minimum area among the surfaces connected to the boundary subsystem [16,17]. A similar proposal for the BMSFT entanglement entropy has been introduced in [12]. Accordingly, the BMSFT entanglement entropy can be given by the area of particular surfaces. These surfaces are not connected directly to the boundary of subsystem but there are null rays which connect them to null infinity where the subsystem is supposed to live.
The corresponding surface, null rays and the subsystem together construct a closed surface .
Another interesting problem which was studied in the context of AdS/CFT is the holographic description of the first law of the entanglement entropy (FLEE). It was shown in [18,19] that writing both sides of FLEE in terms of corresponding bulk parameters finally yields linearized Einstein equations. In other words, FLEE as a constraint in the boundary theory reduces to a constraint on the bulk geometry which is exactly Einstein equation. If this connection is an intrinsic property of gauge/gravity dualities, one can use entanglement entropy and its first law in an arbitrary field theory to find a dual gravitational geometry.
In this paper we study the proposal of [18,19] in the context of flat 3 /BMSFT 2 correspondence.
We start from FLEE and use flat/BMSFT correspondence to write it in terms of components of the asymptotically flat bulk metric. We focus on the BMSFT states which their gravitational dual are flat-space cosmology [20]- [23]. It is shown that both sides of the FLEE formula can be written in terms of the integral of an one-form over curves consist of BMSFT interval and the null and the spacelike geodesics introduced in [12]. These curves construct a closed curve, thus one can use Stokes's theorem to write integrals as the integral of the external derivative of the one-form over the surface bounded by the curves. For the metric of the flat-space cosmology, the exterior derivative of this form is zero. For a generic metric which satisfy BMS boundary condition (see for example [24]), the exterior derivative of one-form results in Einstein equation. Our work is not only the first step generalization of the proposal of [18,19] for the flat-space holography but also shows that the flat/BMSFT correspondence studied in several previous works (see references in [25]) is a worthwhile duality.
In section two we review the proposal of [19] in the context of AdS/CFT. In section three after briefly reviewing the flat/BMSFT correspondence and holographic description of BMSFT entanglement entropy, we write FLEE in terms of bulk metric and deduce the Einstein equation.

Entanglement entropy and its first law
For a quantum field theory state |ψ , the density matrix is ρ = |ψ ψ|. (2.1) If we decompose a spatial (time constant) slice Σ to two subsystems B andB (Σ = B ∪B), then the density matrix associated to B can be obtained from ρ by tracing out the degrees of freedom of the complement subsystemB as The Entanglement entropy of subsystems B is the von Neumann entropy associated to the density matrix ρ B , For a small perturbation |ψ(ε) to the initial state |ψ(0) of the whole system, the first law of entanglement entropy (FLEE) is where H B is modular Hamiltonian which is independent of perturbation and defined through Formula (2.4) is a quantum generalization of the first law of thermodynamics. This formula holds for any arbitrary small perturbation of quantum state and for any subsystem B.
Mostly, it is difficult to compute the modular Hamiltonian H B and its associated density matrix ρ B . However, for the cases that H B is a local operator, one may find a unitary transformation (and hence reversible which acts also on the corrdinates) which maps ρ B to a thermal density matrix. Hence the resulatant entropy is a thermal one (see [26]). If we denote the unitary transformation by U and the final thermal density matrix by ρ H , then It is not difficult to check that the thermal entropy given by is the same as the entanglement entropy (2.3). Since ρ H is thermal, it can be written as 1 .
We consider a spacial time slice Σ of d−dimensional Minkowski space and divide it to two regions B andB (Σ = B ∪B). Let B be a (d − 1)− dimensional ball with radius R.
In order to find δE B in (2.4), we need to calculate the vacuum expectation value of the modular Hamiltonian. The modular Hamiltonian for this ball shaped region is calculated in [26] as follows where x i 0 are the coordinates of the center of ball B and T µν is the stress tensor of CFT. We use the convention x µ = (t, x i ). Hence FLEE (2.4) can be written as Now we use holography to calculate δS B . When the CFT vacuum state |Ψ(0) is perturbed to the state |Ψ(ε) , in the dual gravitational theory, the metric of the dual AdS spacetime will be perturbed as where h µν are infinitesimal. By means of the Ryu-Takayanagi formula [16,17] we can write where AB is the minimal area of the co-dimension two surfaceB in the bulk AdS space which is homologous to B and given by Here γ AB is the induced metric onB.
Let us illustrate the holographic counterpart of δS B and δE B ,respectively, as δS grav.
B and δE grav.

B
. It was shown in [18,19] that they are given as follows in terms of bulk perturbed metric h ij : Thus the FLEE formula (2.4) is written as This is a non-local equation which is correct for any ball shaped region with arbitrary radius R and center coordinate {x i 0 }. Thus one may think about a local equation which is equivalent to (2.17). In order to find this local constraint, we look for a form χ such that (2.18) If such a form χ exists, using (2.4) we can write where Π is the hypersurface bounded by B andB (B ∪B = ∂Π) and located at t = t 0 . For the asymptotically AdS spacetimes, χ is given by [19] where ξ a is the bulk modular flow For this form, the exterior derivative is given by where δG ab are linearized Einstein equations around AdS spacetimes, and ǫ b is related to volume form as follows Moreover, the exterior derivative is zero on the boundary.
From (2.19) and (2.22) it is obvious that the holographic interpretation of the first law of entanglement entropy leads to Using the fact that only the t component of ξ a is non-vanishing on Π and also FLEE is valid for all of the ball shaped regions with arbitrary R, from (2.25) one can deduce that [27] δG tt = 0. that δG zµ and δG zz are zero everywhere [28].
We see that the gravitational interpretation of FLEE in CFTs leads to the linearized equations of motion of the dual AdS gravity. In the next section we will apply the above procedure for asymptotically flat spacetimes in the context of flat/BMSFT correspondence. with well-defined flat space limit [29,30]. A relevant question is finding a counterpart for the flat space limit of the gravity theory in the field theory side. To answer this question one needs to study the asymptotic symmetry of the asymptotically flat spacetimes. This study has been done in [3] for the four dimensional and in [4] for the three dimensional spacetimes. More recent studies show that for the four dimensional cases the asymptotic symmetry algebra at null infinity is the semi-direct sum of infinite dimensional local conformal symmetry algebra on a two-sphere and the abelian ideal algebra of supertranslations [6]. This algebra is known as bms 4 . Such an infinite dimensional locally well-defined symmetry algebra also exists at null infinity of three dimensional asymptotically flat spacetimes [5] . This algebra is called bms 3 .
The observation of [2] is that the bms 3 is isomorphic to an infinite-dimensional algebra in two dimensions which is given by ultra-relativistic contraction of conformal algebra. Thus it was proposed in [2] that the holographic dual of asymptotically flat spacetimes in (d + 1) dimensions are field theories in d dimensions which have BMS symmetry. We call these BMS invariant field theories BMSFT and the correspondence between them and asymptotically flat spacetimes flat/BMSFT.
To be more precise, let us consider Einstein-Hilbert action with negative cosmological constant in three dimensions An appropriate coordinate with well-defined flat space limit is BMS gauge [29] where M and N are functions of u and φ and are constrained by using the equations of motion as  The algebra of conserved charges is centrally extended with central charges c =c = 3ℓ/2G.
Taking the flat space limit from metric (3.2) yields asymptotically flat spacetimes with metric where M and N are functions of u and φ and they satisfy (3.7) The algebra of conserved charges is also centrally extended.
The generators of bms 3 can be obtained by taking flat space limit from the generators of conformal algebra [29], It was argued in [2] that the limit (3.8) which is taken in the gravity side corresponds to the ultra relativistic limit in the field theory side. In the rest of this paper by BMSFT 2 we mean a field theory which has the symmetry algebra (3.7).
From BMSFT 3 we mean a field theory with the following symmetry algebra

Holographic entanglement entropy in flat/BMSFT
Similar to other field theories, it is possible to define entanglement entropy for the subsystems of BMSFT. The infinite dimensional symmetry of BMSFTs admits to find universal formulas for the entanglement entropy of sub-regions [8]. Moreover, using the flat/BMSFT correspondence one can find a holographic description for the BMSFT entanglement entropy. Recently, a prescription (similar to the Ryu-Takayanagi's proposal for the CFT entanglement entropy [16,17]) has been proposed for the BMSFT entanglememnt entropy [12] that relates it to the area of some particular curves into the bulk flat spacetimes. According to [12], the entanglement entropy of sub-region B of BMSFT 2 is given by where γ is a spacelike geodesic and γ + and γ − are null rays from ∂γ to ∂B.
The most generic solution of Einstein gravity with zero cosmological constant in three dimensions is given by (3.5). In the rest of this paper we will consider an interval B in the BMSFT which is determined by − lu 2 < u < lu 2 and − (3.12) In this case the bulk modular flow is Here γ is given by (3.14) By using the coordinate transformations we can change the metric of null-orbifold to the Cartesian coordinate In this coordinates the bulk modular flow is given by (3.17) and geodesics are γ : 2. Global Minkowski with metric (M = −1 and N = 0 in (3.5)) The bulk modular flow is where γ is given by 2 Using coordinate transformation [31] t = (r + u) csc l φ 2 − r cos φ cot l φ 2 , we have In this Cartesian coordinates the bulk modular flow is the same as (3.17) and geodesics are (3.28) 3. Flat-space cosmology (FSC) with metric (M = m and N = j ) ds 2 = mdu 2 − 2dudr + 2jdudφ + r 2 dφ 2 , (3.29) where m and j are constants. It has a cosmological horizon at radius r c = j √ m . FSC is a shift-boost orbifold of Minkowski spacetime [21] and can be brought into the Cartesian coordinate locally by using the following transformation: The holographic entanglement entropy of interval B is given by (3.32)

Holographic FLEE
In this section we will consider the BMSFT dual to the global Minkowski. The starting point is FLEE formula (2.4) which is written in the field theory side. We want to use Flat 3 /BMSFT 2 to write both sides of this formula in the gravity side. BMSFT lives on a cylinder with coordinates (u, φ) and interval B is given by − lu where l u , l φ , u 0 and φ 0 are constants.
Let us start from the right hand side of (2.4). In order to calculate the expectation value of modular Hamiltonian, we use the fact that up to an additive constant, the modular Hamiltonian H B is the same as conserved charge of the modular flow ξ. If we show the stress tensor of BMSFT by T ab , the corresponding charge of ξ can be calculated on a spacelike surface Σ with metric σ ab as [32] where σ is the coordinate on the surface Σ and n a is the unit timelike vector normal to Σ.
The most challenging problem in the flat-space holography is definition of Σ. In the AdS/CFT correspondence, Σ is a spacelike ( surface on the conformal boundary of the asymptotically AdS spacetimes. However, such a definition for conformal infinity of asymptotically flat spacetimes is not appropriate in the flat-space holography . In the previous works [30], [33]- [38], in the flatspace holography, Σ has been defined by using the corresponding surface of asymptotically AdS spacetimes which their flat-space limit yields the asymptotically flat metric. To be precise, let us consider AdS 3 metric written in the BMS coordinate, where ℓ is the radius of AdS space. At fixed but large r we can write, Thus we can write the metric of conformal boundary as In the AdS/CFT correspondence, the metric of Σ in (3.33) is given by using (3.36). The new point in all of papers [30], [33]- [38] is that (3.36) is also appropriate for writing metric of Σ in the ℓ → ∞ limit. The proposal of [30] for the definition of Σ is that we use a metric similar to (3.36) but replace ℓ with three dimensional Newton constant G. In this paper we employ this definition of Σ. Since we want to study FLEE in a BMSFT which is holographic dual of global Minkowski, the metric of bulk spacetime is given by (3.21) which is the ℓ → ∞ limit of (3.34). Thus we choose Σ as a spacelike subspace of a space which is determined by metric It will prove convinient to first make a coordinate transformation as In this coordinate, our interval will be on the φ axe between − Moreover, by taking r → ∞ limit from (3.22), we can find the BMSFT modular flow on the interwal (w = 0) as If we determine Σ as w = 0, − l φ 2 < φ − φ 0 < l φ 2 then using (3.37) and (3.39) we find Since h uu and h uφ are infinitesimal constants, we can use (3.31) to calculate δS. We find Using (3.40) and (3.42), we can write the FLEE as This formula is valid for all of intervals determined by l φ , l u and (u 0 , φ 0 ). For a very small interval which is given by l φ → 0, l u → 0 but lu l φ =fixed, the expectation value of stress tensor can be considered as a function of center of the interval. Since center of interval is an arbitrary point, using (3.43) we find, (3.44) Putting (3.44) into (3.40), we find δE B as The interesting point is that both of δS B and δE B given by (3.42) and (3.45) are written as the integral of a specific one-form χ. Precisely, we can write 3 ξ is the bulk modular flow (3.22), h = h ν µ and ǫ µνα is the completely antisymmetric tensor with component ǫ 012 = |g 0 | where g 0 is the determinant of global Minkowski (3.21

Summary and Conclusion
In this paper we studied another aspect of flat/BMSFT which was previously introduced in the context of AdS/CFT. We wrote FLEE of BMSFT 2 in terms of three-dimensional asymptotically flat metrics. The steps are analogue to those that are used in the context of AdS/CFT correspondence. We rewrite both sides of FLEE (2.4) by using corresponding bulk parameters. δS B in (2.4) is the variation of entanglement entropy with respect to the state by which the system is described. Using the proposal of [12] one can write this variation as the variation of length of some spatial curves in the bulk geometry. δE B in the right hand side of FLEE (2.4) is variation of the expectation value of the modular Hamiltonian. For calculating this quantity, we used the fact that the modular Hamiltonian is the conserved charge of modular flow upto an additive constant which can be ignored in the variation. BMSFT conserved charges are given by using stress tensor.
Using flat/BMSFT dictionary we relate the calculation of the conserved charges to a bulk calculation similar to the Brown-York proposal [32]. The keypoint in this calculation is the definition of the spatial surface over which the integration is performed. In the AdS/CFT correspondence this surface is given by using the conformal boundary of asymptotically AdS spacetimes. In this case we do not use the standard definition of conformal boundary. Our proposal is that this surface for the flat spacetimes is the same as that one for the asymptotically AdS case whose flat-space limit yields the asymptotically flat spacetimes [30]. This proposal works again in this problem similar to all previous works [33]- [38], however, a thorough investigation is necessary that we hope to do in our future studies.
In this paper we assumed that the perturbed state in the field theory side corresponds to a metric similar to the flat-space cosmology [20]- [23] in the bulk theory. Hence, the gravitational counterpart of FLEE was the exterior derivative of a one-form which is zero for the flat-space cosmology. The exterior derivative of this form for a generic metric which satisfy BMS boundary condition results in Einstein equations for undermined components of the metric. This is a good hint that holographic FLEE is Einstein equation in the flat/BMSFT correspondence.
Note added: While we were ready to submit this work, ref. [39] was posted on the arXiv whose results overlap with ours.