Entanglement Entropy in Flat Holography
BMS symmetry, which is the asymptotic symmetry at null infinity of flat spacetime, is an important input for flat holography. In this paper, we give a holographic calculation of entanglement entropy and Rényi entropy in three dimensional Einstein gravity and Topologically Massive Gravity. The geometric picture for the entanglement entropy is the length of a spacelike geodesic which is connected to the interval at null infinity by two null geodesics. The spacelike geodesic is the fixed points of replica symmetry, and the null geodesics are along the modular flow. Our strategy is to first reformulate the Rindler method for calculating entanglement entropy in a general setup, and apply it for BMS invariant field theories, and finally extend the calculation to the bulk.
Holography tHooft:1993dmi (); Susskind:1994vu (), which relates a theory containing gravity in higher spacetime dimensions to a quantum field theory in lower dimensions, is believed to be a promising way to understand quantum gravity. In particular, holography for asymptotically locally AdS (AlAdS) spacetimes, the so-called AdS/CFT correspondence Maldacena:1997re (); Gubser:1998bc (); Witten:1998qj () is one of the most active research fields. A priori, it is not clear whether the rich conceptual achievements from AdS/CFT are contingent to AlAdS. To understand the generality, it is important to extend the great success of holography beyond the context of AdS/CFT. Progresses for non-AlAdS holography includes dS/CFT correspondence Strominger:2001pn (); Anninos:2011ui (), the Shrdinger or Lifshitz spacetime/non-relativistic field theory duality Son:2008ye (); Balasubramanian:2008dm (); Kachru:2008yh (); Taylor:2015glc (), the Kerr/CFT correspondence Guica:2008mu (); Bredberg:2009pv (); Castro:2010fd (); Bredberg:2011hp (); Compere:2012jk (), the WAdS/CFT Anninos:2008fx () or WAdS/WCFT Detournay:2012pc () correspondence, and illuminating results toward flat holography in four dimensions Bondi:1962px (); Sachs:1962wk (); Strominger:2013jfa (); Hawking:2016msc () and three dimensions Barnich:2010eb (); Bagchi:2010eg (); Bagchi:2012cy (); Bagchi:2014iea ().
A general item in the dictionary of holography is that the asymptotic symmetry for the gravitational theory in the bulk agrees with the symmetry of the dual field theory, if the later exists. For four dimensional flat spacetime, the asymptotic symmetry group at null future (past) is the BMS group, first studied by Bondi, van der Burg, Metzner and Sachs Bondi:1962px (); Sachs:1962wk (). In a recent resurgence, Strominger Strominger:2013jfa () pointed out that the diagonal elements of BMS BMS BMS is the symmetry of the matrix. BMS group is connected to infra properties of scattering amplitude He:2014laa (); Cachazo:2014fwa (), and memory effects Zeldovich (), in a triangle Strominger:2014pwa (). See the lecture notes Strominger:2017zoo () for a review. As a simple toy model, the three dimensional version of BMS group has also generated lots of interested. BMS on was discussed in Ashtekar:1996cd (); Barnich:2006av (); Barnich:2010eb (); Bagchi:2012yk (); Detournay:2014fva (). Interesting developments include connections with Virasoro algebra Barnich:2006av (), isomorphism between BMS algebra and Galileo conformal algebra Bagchi:2010eg (), representations and bootstrap Barnich:2012aw (); Barnich:2012rz (); Barnich:2014kra (); Barnich:2015uva (); Campoleoni:2016vsh (); Oblak:2016eij (); Carlip:2016lnw (); Bagchi:2016geg (); Batlle:2017llu (). Flat holography based on BMS symmetry was proposed in Bagchi:2010eg (); Bagchi:2012cy () and supporting evidence can be found in Bagchi:2012xr (); Barnich:2012xq (); Bagchi:2016bcd (). The antipodal identification in three dimensions was discussed in Prohazka:2017equ (); Compere:2017knf ().
One useful probe of holography is the entanglement entropy, which describes the correlation structure of a quantum system. In the context of AdS/CFT correspondence, Ryu and Takayanagi Ryu:2006bv (); Ryu:2006ef () (RT) proposed that the entanglement entropy is given by the area of a codimension-two minimal surface in the bulk, which anchored on the entangling surface of the subsystem on the boundary. A covariant version was proposed by Hubeny, Rangamani and Takayanagi (HRT) Hubeny:2007xt (). Using AdS/CFT for Einstein gravity, RT and HRT proposal have been proved by Casini:2011kv (); Hartman:2013mia (); Faulkner:2013yia (); Lewkowycz:2013nqa (); Dong:2016hjy (). It is interesting to ask if the connection between spacetime structure in the bulk and entanglement in the boundary still exist beyond the context of AdS/CFT, and if so, how it works for non-AlAdS spacetimes. So far, in the literature, there are three approaches. The first approach is to start with the RT or HRT proposal, and study the implications in the holographic dual, see Li:2010dr (); Anninos:2013nja (); Basanisi:2016hsh (); Gentle:2015cfp (); Sun:2016dch (). The second approach is to directly propose a prescription in the bulk, and check its consistency Sanches:2016sxy (); Bakhmatov:2017ihw (). The third approach, which we will advocated in the current paper, is to derive an analog of RT proposal using the dictionary of holography, along the lines of Casini:2011kv (); Hartman:2013mia (); Faulkner:2013yia (); Lewkowycz:2013nqa (); Dong:2016hjy (). In Song:2016pwx (); Song:2016gtd (), holographic entanglement entropy in Warped AdS spacetime was derived by generalizing the gravitational entropy Lewkowycz:2013nqa () and Rindler method Casini:2011kv (); Castro:2015csg (), respectively. Interestingly, it was found that the HRT proposal indeed need to be modified, and moreover the modification depends on different choices of the boundary conditions which determines the asymptotic symmetry group. Another important lesson is that the Rindler method Casini:2011kv (), which maps entanglement entropy to thermal entropy by symmetry transformations, can be generalized to non-AlAdS dualities.
In the context of flat holography, entanglement entropy for field theory with BMS symmetry was considered in Bagchi:2014iea () using twist operators. Using the Chern-Simons formalism of 3D gravity, Basu:2015evh () took the Wilson line approach Ammon:2013hba (); deBoer:2013vca (); Castro:2014tta (), and found agreement with Bagchi:2014iea (). However, a direct calculation in metric formalism is still missing. No geometric picture has been proposed and it is not clear whether RT (HRT) proposal is applicable for asymptotic flat spacetime. In this paper, we will address this question along the lines of Song:2016pwx (); Song:2016gtd ().
In this paper, the Rindler method is formulated in general terms, without referring to any particular example of holographic duality. We argue that the entanglement entropy for a subregion is given by the thermal entropy on , if there exists a Rindler transformation from the causal development of to . Moreover, under such circumstances, the modular Hamiltonian implements a geometric flow generated by the boost vector . Then we apply this generalized Rindler method to holographic dualities governed by BMS symmetry, and provide a holographic calculation of entanglement entropy and Rényi entropy in Einstein gravity and Topologically massive gravity. On the field theory side, our result of entanglement entropy agrees with that of Bagchi:2014iea () obtained using twist operators Calabrese:2004eu (). On the gravity side, by extending the Rinlder method to the bulk, we provide a holographic calculation of the entanglement entropy in metric formalism and provide a geometric picture (see figure 1). We also expect a generalization in higher dimensions.
The geometric picture (figure 1) for holographic entanglement entropy in three dimensional flat spacetime involves three special curves, a spacelike geodesic , and two null geodesics . is the set of fixed points of the bulk extended modular flow (and also fixed points of bulk extended replica symmetry), while are the orbits of the boundaey end points under . The end points of is connected to the boundary end points by . The holographic entanglement entropy for the boundary interval is given by . The main difference between our picture and the RT (HRT) proposal is that the spacelike geodesic is not directly connected to the boundary end points . The reason is that points on are the fixed points of the boundary modular flow but not the fixed points of the bulk modular flow . Our results are consistent with all previous field theory Bagchi:2014iea () and as well as a Chern-Simons calculation Basu:2015evh (). In this paper, explicit calculations are done in Bondi gauge at future null infinity. Similar results follows for the past null infinity. We also expect the techniques and results here can be reinterpreted in the hyperbolic slicing deBoer:2003vf (); Ashtekar (); Beig (); deHaro:2000wj (); Compere:2017knf ().
The paper is organized as follows. In section 2 we explain the generalized Rinder method and provide a formal justification for its validity. In section 3, BMS and asymptotically flat spacetimes is reviewed. In section 4, we calculate the EE in the BMS invariant field theory (BMSFT) by Rindler method. Then we calculate the entanglement entropy holographically for Einstein gravity in section 5. In section 6, we give geometric picture of holographic entanglement entropy. In section 7, we calculate the HEE by taking the flat limit of AdS. In section 8 we calculate the holographic entanglement entropy in topological massive gravity. In section 9, we calculate the Rényi entropy both on the field theory side and the gravity side. In appendix A, we rederive the “Cardy formula” for BMSFT using the BMS symmetries. In appdneix B, we present the Killing vectors of 3D bulk flat spacetime. In appendix C we give the details of Rindler coordinate transformations for BMSFT in finite temperature and on cylinder.
2 Generalized Rindler method
The Rindler method was developed in the context of AdS/CFT Casini:2011kv () with the attempt to derive the Ryu-Takayanagi formula. For spherical entangling surfaces on the CFT vacuum, the entanglement entropy can be calculated as follows. In the CFT side, certain conformal transformations map the entanglement entropy in the vacuum to the thermodynamic entropy on a Rindler or hyperbolic spacetime. In the bulk, certain coordinate transformations map vacuum AdS to black holes with a hyperbolic horizon. Using the AdS/CFT dictionary, the Bekenstein-Hawking entropy calculates the thermal entropy on the hyperbolic spacetime, and hence provides a holographic calculation of the entanglement entropy. Going back to vacuum AdS, the image of the hyperbolic horizon then becomes an extremal surface ending on the entangling surface at the boundary. Recently, the Rindler method has been generalized to holographic dualities beyond AdS/CFT. The field theory story was generalized to Warped Conformal Field Theories (WCFT) in Castro:2015csg (), while the gravity story was generalized to Warped Anti-de Sitter spacetimes (WAdS) in Song:2016gtd (). The results are consistent with the WAdS/WCFT correspondence Anninos:2008fx ().
In this section, we summarized the Rindler method for holographic entanglement entropy, without referring to the details of the holographic pair. The goal is to provide a general prescription which could be potentially used in a broader context. Schematic prescriptions in the field theory side and the gravity side are as follows.
2.1 Field theory calculation of entanglement entropy
2.1.1 Generalized Rindler method
In the field theory, the key step is to find a Rindler transformation, a symmetry transformation which maps the calculation of entanglement entropy to thermal entropy. Consider a QFT on a manifold with a symmetry group , which act both on the coordinates and on the fields. The vacuum preserve the maximal subset of the symmetry, whose generators are denoted by . Consider the entanglement entropy for a subregion with a co-dimension two boundary . Acting on positions, a Rindler transformation is a symmetry transformation. The image of is a manifold 111 Throughout this paper, we always use tilded variables to describe the spacetimes and their bulk extensions after the Rindler transformation. , and the domain is , with , and . For theories with Lorentz invariance, is just the causal development of . The image of should also be the boundary of . A Rindler transformation is supposed to have the following features:
The transformation should be in the form of a symmetry transformation.
The coordinate transformation should be invariant under some imaginary identification of the new coordinates . Such an identification will be referred to as a “thermal” identification hereafter.
The vectors annihilate the vacuum. i.e.
where are arbitrary constants.
Let , then generates a translation along the thermal circle, and induce the flow . A thermal identification can be expressed as . The boundary of the causal domain should be left invariant under the flow. In particular, can only become degenerate222For Warped conformal field theory Song:2016gtd (), keeps invariant, but will not degenerate anywhere for . at the entangling surface ,
Now we argue that the vacuum entanglement entropy on is given by the thermal entropy on , if such a Rindler transformation with the above properties can be found. Property 2 defines a thermal equilibrium on . Property 3 implies that the vacuum state on is mapped to a state invariant under translations of , which is just the thermal equilibrium on . The Modular Hamiltonian on , denoted by then implements the geometric flow along . With property 1, the symmetry transformation acts on the operators by an unitary transformation . By reversing the Rindler map, a local operator on can be defined by , where . Note that generates the geometric flow, and is just the conserved charge up to an additive constant. Property 4 indicates that implements a symmetry transformation which keeps invariant. Then we can always decompose the Hilbert space of in terms of eigenvalues of . Therefore, the modular Hamiltonian on can indeed be written as , which again generates the geometric flow along . In particular, the density matrix are related by a unitary transformation
Since unitary transformations does not change entropy, the entanglement entropy on is given by the thermal entropy on .
More explicitly, at the thermal equilibrium, the partition function and density of matrix on can be written as 333If there are internal symmetries, the partition function should be modified accordingly.
where are the conserved charges associated with the translation symmetries. The modular flow is now local, and is generated by . The action of the flow on positions and fields are given by
The entanglement entropy which is equivalent to thermal entropy is then given by
Réyni entropy and the Modular entropy Dong:2016fnf () can be calculated in a similar fashion. In fact, parameterizes the Rindler time. The thermal identification is just . The replica trick can be performed by making multiple copies of and impose the periodicity boundary condition
To actually find the Rindler map, the strategy is to follow the steps below,
Take an arbitrary symmetry transformation, and impose the condition (1). This will give a system of differential equations, whose solution will depends on the constants .
The temperatures can be read off from the transformations, and will be determined by certain combinations of .
Further solving condition (2) will relate the to position and size of the entangling surface .
2.1.2 “Cardy” formula
In conformal field theory, -transformation is used to estimate the resulting thermal entropy, leading to an analog of the Cardy formula Cardy:1986ie (); Hartman:2014oaa (). A successful generalization has been applied to WCFT in Detournay:2012pc (); Castro:2015csg (), and BMSFT in Bagchi:2012xr (); Barnich:2012xq (). More generally, -transformation can be realized as a coordinate transformation compatible with the symmetry, which effectively switches the spatial circle and thermal circle. In some region of parameters, the partition function is dominated by the vacuum contribution, and the entropy can hence be estimated.
For our purpose in BMSFT, we revisit the Cary-like formula in appendix A and obtain the approximated entropy formula for BMSFT on arbitrary torus. Our derivation is based on the BMS symmetries only, without resorting to flat limit of CFT.
2.2 Holographic entanglement entropy
The gravity story is the extension of the field theory story using holography. There are two possible routes.
The first route is to find the classical solution in the bulk which is dual to thermal states on . This can be obtained by extending the boundary coordinate transformation to the bulk, by performing a quotient. More detailed discussion can be found in Song:2016gtd ().
The second route is to extend replica symmetry to the bulk along the lines of Lewkowycz:2013nqa (); Dong:2016hjy (). The field theory generator has a bulk extension via the holographic dictionary. Since is the fixed point of , we expect a special bulk surface satisfying
Such a bulk surface will be the analog of RT( HRT ) surface. However, if the homologous condition can not be imposed directly. As we will see later, is connected to by two null geodesics , which are along the bulk modular flow . We will discuss a local version of this approach in a future work JSW ().
3 Review of BMS group and asymptotically flat spacetime
3.1 BMS invariant field theory
In this subsection, we review a few properties of two dimensional field theory with BMS symmetry( BMSFT). On the plane Bagchi:2009pe (), BMS symmetries are generated by the following vectors
The finite BMS transformations can be written as Barnich:2012xq ()
Let denote the current associated to the reparameterization of , and let denotes the current associated to the -dependent shift of . We can define charges 444We believe the analytic continuations of to complex numbers are inessential. Barnich:2012xq ()
The conserved charges satisfy the central extended algebra
where are the central charges. Under the transformation (13), the currents transform as
where the Schwarzian derivative is
In particular, the transformation below maps a plane to a cylinder,
The currents transform as
where the generators on the cylinder are defined as
3.2 BMS as asymptotic symmetry group
Minkowski spacetime is the playground of modern quantum field where the Poincaré symmetry gives very stringent constraint on the properties of particles. The Poincaré algebra includes translation and Lorentz transformation, while the Lorentz transformations further consist of spatial rotation and boost. At the null infinity of flat spacetime, the finite dimensional Poincaré isometry group is enhanced to infinitely dimensional asymptotic symmetry group, called BMS group Bondi:1962px (); Sachs:1962wk ().
In three spacetime dimensions, the topology of the boundary at null infinity is where is the null direction. Under proper boundary conditions Barnich:2006av (); Barnich:2010eb (), the general solution to Einstein equation in the Bondi gauge is
where the null infinity is at . The asymptotic symmetry group is the three dimensional BMS group, whose algebra is given in the previous subsection.
BMS group is generated by the super-translation and super-rotation. The super-translation can be thought as the translation along the null direction which may vary from one point to another in , while the super-rotation is the diffeomorphism of . Furthermore, the corresponding conserved charges generates BMS group on the phase space. The infinitely dimensional algebra now has central extensions, which also coincide with (16). For Einstein gravity, Barnich:2006av ().
3.3 Global Minkowski, null-orbifold and FSC
The zero mode solutions in (25), describing some classical background of spacetime, are of particular interest. With the standard parameterization of the
general classical solutions of Einstein gravity without cosmology constant takes the following form Barnich:2010eb ()
where in the second line we have used the convention which will be adopted throughout the paper. We will spell it out whenever it is necessary to restore .
: Null-orbifold. It was first constructed in string theory Horowitz:1990ap (). They are supposed to play the role of zero temperature BTZ, being the holographic dual of BMSFT on a torus with zero-temperature and a fixed spatial circle.
The flat-space metric (27) can also be written in the ADM form555The coordinates are related by (28)
The thermal circle of (27) is given by
The thermal entropy of the horizon is given by the Bekenstein-Hawking formula. Meanwhile, this thermal entropy also can be obtained from the dual field theory by considering a Cary-like counting of states. The equality of thermal entropy between bulk gravity and boundary field was shown in Bagchi:2012xr (); Barnich:2012xq ()
3.4 Poincaré coordinates
If we decompactify the angular direction , the boundary theory will be put on the plane instead of cylinder. In particular, the resulting spacetime with is more like a flat version of AdS in Poincaré patch. The coordinate transformation from the Poincaré coordinate and the Cartesian coordinate is
It is easy to check that
the patch with covers the region
The inverse transformation from Cartesian coordinate to the Poincaré to is
As both and will diverge except when .
3.5 Solutions with general spatial circle
When we study the bulk extension of the Rindler transformations in section 5, we will encounter the bulk extensions of , which are usually this kind of spacetime. Hereafter we will refer to (41) as . The proper length on Cauchy horizon is . Integration along this proper length will give the length of the horizon, and the Bekenstein-Hawking entropy
where and represent the extension of the spacetime along the and direction.
4 Entanglement entropy in field theory side
In this section we apply the generalized Rindler method to a general BMS field theory with arbitrary and . As we will show below, our results in this section agree with the previous calculation using twist operators Bagchi:2014iea ().
4.1 Rindler transformations and the modular flow in BMSFT
In this section, following the guidelines we give in section 2.1.1, we derive the most general Rindler transformations in BMSFT. According to property 1 in subsection 2.1.1, the Rindler transformations should be a symmetry of the field theory, thus for BMSFT it should be the following BMS transformations Barnich:2012xq ()
where we use to denote derivative with respect to . Furthermore, this indicates the theory after Rindler transformation is also a BMS invariant field theory, which we call . The inverse transformation can be written as
and denotes derivative with respect to . The property 4 of the Rindler transformation indicates that the vectors and have to be linear combinations of the global BMS generators. The conditions for is
which implies that the condition for is automatically satisfied, as
then we can get two differential equations
It is interesting to note
Furthermore by noting the relation in (48), the previous differential equations can be simplified as
Rindler transformation on the plane
Now we consider the BMSFT on the plane with zero temperature. In this case the symmetry generators of BMS are
where the part form a subalgebra, and generate the global symmetries. By matching the general form of Killing vectors in (52), one can easily see that
Substituting them into (56), one gets
Note that and shift the origin of or , and therefore we can set without losing generality. Taking
Plugging and into (57) , we get
The two parameters and can be absorbed by a shift of and . Finally, up to some trivial shifts, we get the most general coordinate transformations
The above Rindler transformation satisfies the property 2 as it induces a thermal circle
Note that the subregion of which maps to is a strip bounded by and .
Rewritten in the original coordinate system, we get
Thus, the generator of modular flow is
It is easy to verify that at the points , . Following our prescription (2), this implies that is that endpoints of the interval. Thus, we can naturally interpret as the extension of the interval along and direction.
The global BMS generators have bulk extensions, which are just the Killing vectors (B.2) of the flat-space (27) in Poincaré patch. Substitute (B.2) into (50) and (51), the modular generator (71) can also be extended to the bulk and is given by
Following the similar analysis, we give the construction of Rindler transformations in BMSFT with a thermal or spatial circle in Appendix C. We also calculate the modular flow and its bulk extension in these two cases.
4.2 Entanglement entropy for BMSFT
In this subsection we consider zero temperature BMSFT on the plane, finite temperature BMSFT, and BMSFT on a cylinder respectively, and calculate the entanglement entropy of the following interval in these BMSFTs,
where the arrow means a line connecting the two endpoints. After the Rindler transformation, the entanglement entropy equals to the thermal entropy of the on , which can be calculated via a Cardy-like formula. Since the extension of is essentially infinite, we need to introduce the cutoffs to regulate the interval
In Bagchi:2012xr (); Barnich:2012xq (), the Cardy-like formula for BMSFT is derived from the modular invariance of the theory, which is inherited from the modular invariance of CFT under flat limit. In appendix A we re-derive the Cardy-like formula using the BMS symmetry only.
The manifold can be considered as a torus with the following identifications
4.2.1 Zero temperature BMSFT on the plane
After performing the Rindler transformation (67), the image of this regularized interval is
where we have neglected irrelevant terms and terms , but keep terms of order . The endpoint effects are expected to be negligible for this large interval, thus we can identify the endpoints to form a spatial circle. The same story happens in the later two cases. Together with the thermal circle induced by the Rindler transformation (67), can be considered as a torus parametrized by
The canonical torus parameters defined by (242) are
Substituting these quantities into the entropy formula (77) derived before, we can obtain the entropy
This agrees with the result in literature Bagchi:2014iea () when we set . In general, we would like to keep the cutoff related terms. It is interesting to note that term is the same as that in CFT. This is not surprising since the generators satisfy the chiral part of the Virasoro algebra.
4.2.2 Finite temperature BMSFT
The temperature of the quantum field theory is dictated by the periodicity along the imaginary axis of time, namely . More generally, we can have the following thermal identification
The Rindler transformation for thermal BMSFT is given in (272), which in particular respects the above thermal circle. Following the similar steps, we find the torus is parametrized by
4.2.3 Zero temperature BMSFT on the cylinder
Consider the cylinder with periodicity , we should apply the Rindler transformation (273). Similarly we find can be considered as a torus parametrized by