Generalised proofs of the first law of entanglement entropy

One of the main goals of this work is to generalise first laws for holographic entanglement entropy, relaxing assumptions on boundary conditions for bulk metric perturbations. One of the tools that will be used in our analysis is renormalized entanglement entropy; this is relevant as general boundary conditions for bulk metric perturbations are associated with UV divergences in the regulated entanglement entropy. Working with quantities that are consistently renormalized allows us to work systematically with such setups.

Renormalized entanglement entropy was discussed extensively in Taylor:2016aoi , with explicit formulae for holographic renormalized entanglement entropy being derived. A convenient way to construct expressions for renormalized entanglement entropy is via the replica trick. Using the replica trick, entanglement entropy can be derived as the limit of Rényi entropy

$$S=-\alpha\partial_{\alpha}[\log\mathcal{Z}(\alpha)-\alpha\log\mathcal{Z}(1)]_{\alpha=1}$$

(3)

where ${\mathcal{Z}}(\alpha)$ is the partition function on the $\alpha$ fold cover manifold. In much of the condensed matter literature this approach is applied to UV regulated quantities, with the UV regulator being interpreted in terms of the lattice scale of the discrete condensed matter system of interest. However, from a quantum field theory perspective, it is much more natural to work directly with renormalized quantities, ie.e. ${\mathcal{Z}}$ is the renormalized partition function.

In holography the partition function is computed to leading order from the onshell bulk action i.e.

$$\mathcal{Z}_{grav}=e^{-I_{grav}},$$

(4)

where $I_{grav}$ denotes the onshell gravitational action. Applying the holographic dictionary and the replica trick to the renormalized gravity action one obtains a formal definition of the renormalized holographic entanglement entropy.

$$S=\alpha\partial_{\alpha}[I_{ren}(\alpha)-\alpha I_{ren}(1)]_{\alpha=1}$$

(5)

This approach thus directly relates the renormalization scheme for the partition function (gravitational action) to the scheme for the entanglement entropy.

To obtain a finite value for the gravitational action, one needs to use holographic renormalization. The renormalized action can then be obtained by the procedure of regularization and the introduction of appropriate covariant boundary counterterms

$$I_{ren}=I_{reg}-I_{ct}$$

(6)

For pure gravity with negative cosmological constant, the renormalized action in $(d+1)$ dimensions takes the form deHaro2001

	$\displaystyle I_{ren}=\frac{1}{16\pi G_{d+1}}$	$\displaystyle\int_{\mathcal{M}_{z\geq\epsilon}}d^{d+1}x\sqrt{g}(R+d(d-1))$		(7)
	$\displaystyle-\frac{1}{16\pi G_{d+1}}$	$\displaystyle\int_{\tilde{\mathcal{M}_{\epsilon}}}d^{d}x\sqrt{\tilde{g}}\big{[}K+2(1-d)\mathcal{R}+\frac{1}{2-d}\mathcal{R}$
		$\displaystyle-\frac{1}{(d-4)(d-2)^{2}}(\mathcal{R}_{\mu\nu}\mathcal{R}^{\mu\nu}-\frac{d}{4(d-1)}\mathcal{R}^{2})-\log\epsilon a_{(d)}+\cdots\big{]}$

In these expressions the bulk manifold is regulated using a radial coordinate $z\geq\epsilon$; $R$ denotes the curvature of the bulk manifold while $K$ and $\mathcal{R}$ refer to the extrinsic and intrinsic curvature of the boundary manifold respectively. Here the given counterterms suffice for $d\leq 5$; expressions for the additional counterterms required for $d>5$ can be found in deHaro2001 . Logarithmic counterterms associated with conformal anomalies arise for $d$ even, and explicit expressions for these can also be found in deHaro2001 .

One can then derive the renormalized entanglement entropy from the renormalized action, making use of the following expressions for the integrals of curvature invariants, expressed as series in powers of $(1-\alpha)$ Fursaev_1995 ; Fursaev_2013 :

	$\displaystyle\int_{\mathcal{M}_{\alpha}}d^{d+1}x\sqrt{g}\mathcal{R}_{\alpha}=\alpha\int_{\mathcal{M}}d^{d+1}x\sqrt{g}\mathcal{R}+4\pi(1-\alpha)\int_{\tilde{B}}d^{d-1}x\sqrt{\gamma}$		(8)
	$\displaystyle\int_{\mathcal{M}_{\alpha}}d^{d+1}x\sqrt{g}\mathcal{R}_{\alpha}^{2}=\alpha\int_{\mathcal{M}}d^{d+1}x\sqrt{g}\mathcal{R}^{2}+8\pi(1-\alpha)\int_{\tilde{B}}d^{d-1}x\sqrt{\gamma}\mathcal{R}$
	$\displaystyle\int_{\mathcal{M}_{\alpha}}d^{d+1}x\sqrt{g}{\mathcal{R}_{\alpha}}_{\mu\nu}\mathcal{R}_{\alpha}^{\mu\nu}=\alpha\int_{\mathcal{M}}d^{d+1}x\sqrt{g}\mathcal{R}_{\mu\nu}\mathcal{R}^{\mu\nu}$
	$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;+4\pi(1-\alpha)\int_{\tilde{B}}d^{d-1}x\sqrt{\gamma}\big{(}\mathcal{R}_{\mu\nu}n^{\mu}\cdot n^{\nu}-\frac{1}{2}(TrK)^{2}\big{)}$

Using these replica curvature integrals the explicit expression for the holographic renormalized entanglement entropy becomes Taylor:2016aoi

	$\displaystyle S_{ren}=\frac{1}{4G_{d+1}}\int_{\tilde{B}}d^{d-1}x\sqrt{\gamma}-\frac{1}{4(d-2)G_{d+1}}\int_{\partial\tilde{B}}d^{d-2}x\sqrt{\tilde{\gamma}}$		(9)
	$\displaystyle-\frac{1}{4(d-2)(d-4)G_{d+1}}\int_{\partial\tilde{B}}d^{d-2}x\sqrt{\tilde{\gamma}}\bigg{(}\mathcal{R}_{\mu\nu}n^{\mu}\cdot n^{\nu}-\frac{1}{2}(TrK)^{2}-\frac{d}{2(d-1)}\mathcal{R}\bigg{)}$

where $\mathcal{R}$ is the Ricci scalar of the metric $g_{\mu\nu}$, $\mathcal{R}_{aa}=\sum_{a}(-1)^{a}\mathcal{R}_{\mu\nu}n^{\mu}_{a}n^{\nu}_{a}$ is the projection of the Ricci tensor on the subspace orthogonal to $\partial\tilde{B}$ with temporal and spatial normals $n_{a}^{\mu}$, $a=1,2$, $\widetilde{\gamma}$ is the determinant of the induced metric on $\partial\tilde{B}$ and $k^{2}=\sum_{a}K_{a}K_{a}$ with $K_{a}$ trace of the extrinsic curvature corresponding to the two normals $n_{a}^{\mu}$. Here $\tilde{B}$ denotes the entangling surface with boundary $\partial\tilde{B}$. The counterterms given here are sufficient for $d<6$, but can straightforwardly be computed for $d\geq 6$. For $d$ even there are logarithmic counterterms related to conformal anomalies, see Taylor:2016aoi for details.

When the CFT dimension $d$ is odd, the renormalized entanglement entropy can be written in terms of the Euler characteristic and other renormalized curvature invariants of the bulk entangling surface Taylor:2020uwf ,

$\displaystyle S_{ren}(\tilde{B})\sim(-1)^{n+1}{\cal F}_{n}\;\chi(\tilde{B})-\sum_{r}{\cal W}_{r}(\tilde{B})-\sum_{p}{\cal H}_{p}(\tilde{B})-\sum_{q}{\cal I}_{q}(\tilde{B}).$

(10)

where $\mathcal{W}_{r}$ are renormalized integral of projections of the Weyl curvature, $\mathcal{H}_{p}$ are renormalized integral of even powers of the extrinsic curvature and for $d>5$ there are ${\cal I}_{q}$ renormalized integrals containing products of Weyl and extrinsic curvature. More explicitly the renormalized entanglement entropy proportional to the renormalized area of the bulk entangling surface $\tilde{B}$

$\displaystyle S_{ren}(\tilde{B})=\frac{\mathcal{A}(\tilde{B})}{4G_{d+1}}$

(11)

and renormalized area integral in $d=3$ is

$\displaystyle{\cal A}(\tilde{B})=-2\pi\chi(\tilde{B})-\frac{1}{2}\int_{\tilde{B}}d^{2}x\sqrt{g}|{K}|^{2}-\int_{\tilde{B}}d^{2}x\sqrt{g}{W}_{1212},$

(12)

and in $d=5$ is

	$\displaystyle{\cal A}(\tilde{B})$	$\displaystyle=$	$\displaystyle\frac{4\pi^{2}}{3}\chi(\tilde{B})-\frac{1}{6}{\cal H}(\tilde{B})-\frac{1}{3}{\cal W}(\tilde{B})$
			$\displaystyle-\frac{1}{24}\int_{\tilde{B}}d^{4}x\sqrt{g}\left(H^{2}-4H_{\mu\nu}H^{\mu\nu}+H_{\mu\nu\rho\sigma}H^{\mu\nu\rho\sigma}\right.$
			$\displaystyle\left.+4{W}_{1212}^{2}-4{W}_{\mu n\nu n}\widetilde{W}^{\mu n\nu n}+{W}_{\mu\nu\rho\sigma}{W}^{\mu\nu\rho\sigma}\right).$

In what follows we will find these geometric expressions for renormalized entanglement entropy useful. Note in particular that these will simplify considerably in the context of first variations around AdS backgrounds.

In this section we review the description of Wald Hamiltonians Lee:1990nz ; Wald:1993nt ; Iyer:1994ys ; Iyer:1995kg ; Wald:1999wa and charges in anti-de Sitter spacetimes. Our review follows closely the work of Papadimitriou:2004ap ; Papadimitriou:2005ii , and more details may be found in these references. The Wald approach assumes that the gravitational theory is described by a diffeomorphism covariant Lagrangian d-form ${\bf{L}}(\psi)$, where ${\bf{L}}(\psi)$ will depend both on the metric and other fields, denoted collectively as $\psi$. In the anti-de Sitter context we work with a renormalized Lagrangian

$\displaystyle\boldsymbol{L}^{ren}=\boldsymbol{L}-{d}\boldsymbol{B}$

(14)

where $\boldsymbol{L}$ is the bulk Lagrangian form and $\boldsymbol{B}$ may be viewed as the combination of the Gibbon-Hawking term and boundary counterterms. The onshell regular Lagrangian is exact i.e.

$\displaystyle\boldsymbol{L}^{onshell}=-\frac{{d}(\boldsymbol{\varepsilon}_{a}n^{a}\lambda)}{16\pi G_{d+1}}.$

(15)

where $n^{a}$ is the outward normal in the asymptotic radial direction,

$\displaystyle n=-\frac{dz}{z},$

(16)

$\boldsymbol{\varepsilon}$ is the volume form and $\boldsymbol{\varepsilon}_{a_{1}\dots a_{n}}$ is a $(d-n+1)$ form

$\displaystyle\boldsymbol{\varepsilon}_{a_{1}\dots a_{n}}=\frac{1}{(d-n+1)!}\varepsilon_{a_{1}\dots a_{n}b_{1}\dots b_{d-n+1}}dx^{b_{1}}\wedge\cdots\wedge dx^{b_{d-n+1}},$

(17)

with the orientation

$\displaystyle\varepsilon_{ztx^{1}\cdots}=+\sqrt{-g}.$

(18)

In the Hamiltonian formalism, fields can be expanded asymptotically near the conformal boundary in series of dilatation eigenfunctions with ascending weight, see appendix B.1 for more detailed explanation. The general structure of the boundary term is then

	$\displaystyle\boldsymbol{B}$	$\displaystyle=\boldsymbol{B}^{GH}-\boldsymbol{B}^{ct}$
		$\displaystyle=-\frac{\boldsymbol{\varepsilon}_{a}n^{a}}{16\pi G_{d+1}}\left(K-(K_{ct}-\lambda_{ct})\right)$
		$\displaystyle=\frac{-\boldsymbol{\varepsilon}_{a}n^{a}}{16\pi G_{d+1}}\left(K_{(d)}+\lambda_{ct}\right).$		(19)

where typically counterterms contribute up to terms at the $d^{th}$ order i.e.

$\displaystyle O_{ct}\sim\sum_{i<d}O_{(}i).$

(20)

Variations can then be expressed as

	$\displaystyle\delta\boldsymbol{L}^{ren}$	$\displaystyle=\delta\boldsymbol{L}-d\delta\boldsymbol{B}$		(21)
		$\displaystyle=\boldsymbol{E}^{\psi}\delta\psi+d\boldsymbol{\Theta}[\delta\psi]-d\delta\boldsymbol{B}$		(22)

where $\boldsymbol{E}^{\psi}$ denotes the equations of motion and $\boldsymbol{\Theta}$ is the symplectic potential. This expression can be rewritten as

$\displaystyle\delta\boldsymbol{L}^{ren}=\boldsymbol{E}^{\psi}\delta\psi+d\boldsymbol{\Theta}^{ren}[\delta\psi]$

(23)

where the renormalized symplectic potential form can be expressed as

$\displaystyle\boldsymbol{\Theta}^{ren}[\delta\psi]=\boldsymbol{\varepsilon}_{a}n^{a}\pi_{(d)}^{\mu\nu}\delta\gamma_{\mu\nu}$

(24)

The canonical momentum $\pi_{\mu\nu}$ can be expressed in terms of the extrinsic curvature as

$\displaystyle\pi^{\mu\nu}=-\frac{1}{16\pi G_{d+1}}\sqrt{\gamma}\left(K^{\mu\nu}-K\gamma^{\mu\nu}\right).$

(25)

If we expand both sides of the equality in the dilatation eigenfunction expansion, we can match the dilatation weights and obtain,

$\displaystyle\pi_{(n)}^{\mu\nu}=-\frac{1}{16\pi G_{d+1}}\left(K^{\mu\nu}_{(n)}-K_{(n)}\gamma^{\mu\nu}\right).$

(26)

In (24) $\pi_{(d)}^{\mu\nu}$ is the $d^{th}$-term in the dilatation eigenfunction series of the conjugate momentum with respect to the metric and this is in turn related to the renormalized CFT stress tensor as

$\displaystyle 2\pi_{(d)}^{\mu\nu}=-\frac{1}{16\pi G_{d+1}}T^{\mu\nu}_{ren},$

(27)

i.e. the first variation of the renormalized action is

$\displaystyle\delta I^{onshell}_{ren}=\frac{-1}{32\pi G_{d+1}}\int_{\partial\mathcal{M}}d^{d}x\sqrt{-\gamma}T^{\mu\nu}_{ren}\delta\gamma_{\mu\nu}.$

(28)

Now let us consider the asymptotic behaviour of metric perturbations. Expressing the AdS_d+1 metric as

$\displaystyle ds^{2}=\frac{dz^{2}}{z^{2}}+\frac{1}{z^{2}}\eta_{\mu\nu}dx^{\mu\nu}$

(29)

where $\eta$ is the Minkowski metric, only the normalizable mode is allowed to vary under a Dirichlet condition and

$\displaystyle\delta\gamma_{\mu\nu}=z^{d-2}\delta\gamma_{(d)\;\mu\nu}+O(z^{d-1}),\quad\quad z\rightarrow 0.$

(30)

For $d$ odd, using the tracelessness of the stress tensor and absence of trace anomaly, the Dirichlet boundary condition can be automatically generalized to a conformal Dirichlet boundary condition which fixes the conformal class only. In the present of a conformal anomaly, for the onshell action to be stationary under perturbations a representative of the conformal class has to be fixed.

Now let us turn to Noether charges. If the field variation is induced by a vector field $\xi$, we can define the Noether current form as

$\displaystyle\boldsymbol{J}[\xi]=\boldsymbol{\Theta}[\delta_{\xi}\psi]-\iota_{\xi}\boldsymbol{L}$

(31)

where $\iota_{\xi}$ contracts $\xi$ with the first index of $\boldsymbol{L}$. The exterior derivative of the Noether current is proportional to the equation of motion

$\displaystyle d\boldsymbol{J}[\xi]=-\boldsymbol{E}^{\psi}\delta_{\xi}\psi,$

(32)

and thus vanishes onshell. Hence we can define the Noether charge form $\boldsymbol{Q}[\xi]$ as the exact term in the Noether current

$\displaystyle\boldsymbol{J}[\xi]=d\boldsymbol{Q}[\xi]-\boldsymbol{N}[\xi]$

(33)

where

$\displaystyle d\boldsymbol{N}[\xi]=\boldsymbol{E}^{\psi}\delta_{\xi}\psi.$

(34)

There is another conserved charge in $AlAdS$ induced by $\xi$ called the holographic charge. Using $(\ref{eq:pidTren})$ and the fact that the renormalized CFT stress tensor is conserved, we can construct the total relativistic momentum of the boundary system. The holographic charge form $\boldsymbol{\mathcal{Q}}[\xi]$ is defined by

$\displaystyle\boldsymbol{\mathcal{Q}}[\xi]=-\boldsymbol{\varepsilon}_{ab}n^{a}2\pi_{(d)}^{bc}\xi_{c}.$

(35)

and this form is integrated over a timeslice at the boundary to obtain the holographic charge. This can be interpreted as the renormalized relativistic momentum along the $\xi$ direction.

In Lemma 4.1 in Papadimitriou:2005ii it was proved that for any asymptotically locally anti-de Sitter space ${\mathcal{M}}$ the two definitions of charges corresponding to asymptotic conformal Killing vector, $\xi$, on a spatial slice on the conformal boundary, $\partial\mathcal{M}\cap C$, are equivalent i.e.

$\displaystyle-\int_{\partial\mathcal{M}\cap C}\boldsymbol{\mathcal{Q}}[\xi]=\int_{\partial\mathcal{M}\cap C}\textbf{Q}^{full}[\xi].$

(36)

where

$\displaystyle\textbf{Q}^{full}[\xi]=\textbf{Q}[\xi]-\iota_{\xi}\textbf{B}.$

(37)

Note that this equivalence is defined up to exact terms since $\partial\mathcal{M}\cap C$ is a cycle and the asymptotic conformal Killing vector $\xi$ has the following fall off condition:

$\displaystyle\xi^{z}=O(z^{d}),$

$\displaystyle\xi^{\mu}=\zeta^{\mu}(1+O(z^{d+2}))$

(38)

where $\zeta$ is a boundary conformal Killing vector. We will later need to generalise this equivalence to less restrictive fall off conditions on the vector field.

In this section we briefly review the first law of entanglement entropy. Given a reduced density matrix $\rho_{B}$ the modular Hamiltonian $H_{B}$ is given by

$$\rho_{B}=e^{-H_{B}}$$

(39)

Under a small variation of the entanglement entropy, we obtain the relation

$\displaystyle\delta S_{B}=\delta\expectationvalue{H_{B}}$

(40)

and the equivalence between $\delta\expectationvalue{H_{B}}$ and the change in energy $\delta E$ gives the first law of entanglement entropy.

Following Casini:2011kv , we now review relevant properties of the modular Hamiltonian and modular flow for CFTs on Minkowski space. There is a symmetry group associated with the modular Hamiltonian: the modular group, a group of one-parameter transformations of the form

$\displaystyle U_{B}(s)=e^{-isH_{B}}$

(41)

where $\partial_{s}$ is called the modular flow. For QFTs on Minkowski space, the modular flow generates a boost. In null coordinates $X^{\pm}$ this is given by

$\displaystyle X^{\pm}(s)=X^{\pm}e^{\pm 2\pi s}.$

(42)

For an accelerated observer in Rindler coordinates, the state is thermal in $\tau$ where the longitudinal part of the metric is given by

$\displaystyle dX^{+}dX^{-}=-\frac{\rho^{2}}{R^{2}}d\tau^{2}+d\rho^{2},$

(43)

where $R$ relates to the imaginary time periodicity i.e. $\beta=2\pi R$. The thermal density matrix of the state is

$\displaystyle\rho_{\mathcal{R}}=\frac{e^{\beta H_{\tau}}}{Tr(e^{\beta H_{\tau}})}.$

(44)

The modular flow generator is $2\pi R\partial_{\tau}$ and the modular Hamiltonian is given by $2\pi RH_{\tau}+\log Tr(e^{\beta H_{\tau}})$.

For a spatial ball $B$ of radius $R$ centred at $x^{i}=0,\,t=0$ on $d$-dimensional Minkowski space, we can conformally map the causal development of the spatial ball $D(B)$ to the Rindler wedge. This conformal map $X^{\mu}\rightarrow x^{\mu}$ can also map the modular flow $(\ref{eq:modflowR})$ to

$\displaystyle x^{\pm}(s)=R\frac{(R+x^{\pm})-e^{\mp 2\pi s}(R-x^{\pm})}{(R+x^{\pm})+e^{\mp 2\pi s}(R-x^{\pm})}$

(45)

and hence the modular flow generator $\partial_{s}$ as $\zeta_{B}$

	$\displaystyle\partial_{s}$	$\displaystyle=\frac{\pi}{R}\left((R^{2}-t^{2}-\vec{x}^{2})\partial_{t}-2tx^{i}\partial_{i}\right)$		(46)
	$\displaystyle\zeta_{B}$	$\displaystyle=\frac{i\pi}{R}(R^{2}P_{t}+K_{t})$		(47)

where $P_{t}$ and $K_{t}$ are the time translation and special conformal transformation generators, respectively.

Since the modular Hamiltonian is the translation operator in $s$, on $B$ we get

$\displaystyle H_{B}=\int_{B}d^{d-1}xT^{ts}.$

(48)

We can write $(\ref{eq:HBs})$ in covariant form as

$\displaystyle H_{B}=\int_{B}d\sigma^{\mu}T_{\mu\nu}\zeta_{B}^{\nu}$

(49)

and the modular energy as

$\displaystyle E_{B}=\int_{B}d\sigma^{\mu}\expectationvalue{T_{\mu\nu}}\zeta_{B}^{\nu}.$

(50)

The entanglement entropy of region $B$ can be calculated holographically by the area of the corresponding bulk entangling surface $\tilde{B}$.

A CFT in the vacuum state on the causal wedge $D(B)$ can also be mapped conformally to a CFT in a thermal state on the hyperbolic cylinder. This can be easily seen from writing the Rindler metric as:

$\displaystyle ds^{2}=\frac{\rho^{2}}{R^{2}}\left(-d\tau^{2}+\frac{R^{2}}{\rho^{2}}(d\rho^{2}+dX^{i}dX^{i})\right)=\frac{\rho^{2}}{R^{2}}ds^{2}_{\mathbb{R}\times\mathbb{H}^{d-1}}.$

(51)

As discussed in Faulkner:2013ica , the first law of entanglement entropy of the CFT thermal state on hyperbolic cylinder can be related to the first law of black hole dynamics via holography. Essentially, the CFT on hyperbolic cylinder $\mathbb{R}\times\mathbb{H}^{d-1}$ is dual to the Rindler $AdS_{d+1}$ black hole exterior and the bulk entangling surface $\tilde{B}$ can be viewed as the black hole horizon. The perturbation of entanglement entropy $\delta S_{B}$ is equal to the perturbation of black hole entropy calculated from the Wald functional $\delta S_{Wald}$ where

$\displaystyle S_{Wald}=-2\pi\int_{\mathcal{H}}d\sigma\frac{\delta\mathcal{L}}{\delta R^{ab}_{\;\;cd}}n^{ab}n_{cd}.$

(52)

Changing back to the interpretation in terms of a Minkowski boundary, we label $\Sigma$ as the bulk region enclosed by $B$ and $\tilde{B}$ and the bulk causal wedge of $\Sigma$ as $D(\Sigma)$. We extend the boundary modular flow to the bulk as the Killing vector

$\displaystyle\xi_{B}=-\frac{2\pi}{R}(t-t_{0})[z\partial_{z}+(x^{i}-x^{i}_{0})\partial_{i}]+\frac{\pi}{R}[R^{2}-z^{2}-(t-t_{0})^{2}-(\vec{x}-\vec{x}_{0})^{2}]\partial_{t}.$

(53)

Note that this Killing vector does not satisfy $(\ref{eq:falloff})$ but instead has the weaker fall-off behaviour

$\displaystyle\xi^{z}_{B}=O(z),$

$\displaystyle\xi^{\mu}_{B}=\zeta^{\mu}_{B}(1+O(z^{2})).$

(54)

One can check $\xi_{B}$ vanishes on $\tilde{B}$.

It was shown in Faulkner:2013ica that for metric perturbations limited to normalizable modes, $\delta g_{\mu\nu}=z^{d-2}h^{(d)}_{\mu\nu}$, one get the infinitesimal first law of entanglement entropy as

	$\displaystyle\delta\expectationvalue{T_{tt}\ }$	$\displaystyle=\frac{d^{2}-1}{2\pi\Omega_{d-2}}\lim_{R\rightarrow 0}\left(\frac{1}{R^{d}}\delta S_{B}\right)$
	$\displaystyle\delta\expectationvalue{T_{tt}}$	$\displaystyle=\frac{d}{16\pi G_{d+1}}h^{(d)}_{tt}$		(55)
	$\displaystyle\delta\expectationvalue{T_{\mu\nu}}$	$\displaystyle=\frac{d}{16\pi G_{d+1}}h^{(d)}_{\mu\nu}$

where the tracelessness condition of $h^{(d)}_{\mu\nu}$ is used to go from the second line to the final covariant expression. The final expression matches the holographic dictionary between stress tensor and normalizable metric coefficient found in deHaro2001 .

The covariant first law of entanglement entropy utilises the charges associated with with energy and entropy corresponding to the bulk Killing vector $\xi_{B}$ introduced in section 2.2. The entanglement entropy is

$\displaystyle S_{B}^{grav}=\int_{\tilde{B}}\textbf{Q}^{full}[\xi_{B}]$

(56)

and the modular energy is

$\displaystyle E_{B}^{grav}=-\int_{B}\boldsymbol{\mathcal{Q}}[\xi_{B}].$

(57)

Limiting to the variation involving only the normalizable mode with no boundary variation on $\partial B=\partial\tilde{B}$ and using $(\ref{eq:Lemma4.1})$,

$\displaystyle-\int_{B}\delta\boldsymbol{\mathcal{Q}}[\xi_{B}]=\int_{B}\delta\textbf{Q}^{full}[\xi_{B}].$

(58)

The off-shell difference is expressed in terms of the Einstein equations

	$\displaystyle\delta E_{B}^{grav}-\delta S_{B}^{grav}$	$\displaystyle=\int_{{B}}\delta\boldsymbol{Q}^{full}[\xi_{B}]-\int_{\tilde{B}}\delta\boldsymbol{Q}^{full}[\xi_{B}]$		(59)
		$\displaystyle=\int_{\Sigma}d\delta\textbf{Q}^{full}[\xi_{B}]=\int_{\Sigma}\delta\textbf{J}^{full}[\xi_{B}]=\int_{\Sigma}-2\boldsymbol{\varepsilon}^{a}\delta E_{ab}\xi^{a}_{B},$

hence recovering the first law of entanglement entropy onshell. We also obtain the version of $(\ref{eq:Lemma4.1})$

$\displaystyle-\int_{B}\delta\boldsymbol{\mathcal{Q}}[\xi_{B}]=\int_{\tilde{B}}\delta\boldsymbol{Q}^{full}[\xi_{B}].$

(60)

Note there are many caveats regarding boundary terms and fall-off condition when we allow the variation of the non-normalizable modes. We shall address them in the following sections.

The metric of $AdS_{d+1}$ on the Poincare patch may be parameterized as

$$ds^{2}=\frac{dz^{2}}{z^{2}}+\frac{1}{z^{2}}g_{\mu\nu}dx^{\mu}dx^{\nu},$$

(61)

where $g_{\mu\nu}=\eta_{\mu\nu}$ is flat Minkowski metric with signature ($-$,$+$,$\cdots$,$+$). In the case of spherical entangling regions, the $(d-1)$-dimensional bulk extending entangling surface $\tilde{B}$ with boundary $\partial\tilde{B}$ as the entangling surface of the boundary CFT can be described by

$$r^{2}+z^{2}=R^{2}$$

(62)

where $r$ is the radial coordinate on the boundary and $R$ is the radius of the spherical entangling region. The induced metric on the entangling surface in $AdS_{d+1}$ is then

$$ds^{2}=\frac{R^{2}}{z^{2}r^{2}}dz^{2}+\frac{r^{2}}{z^{2}}d\Omega_{d-2}^{2}.$$

(63)

where $d\Omega_{d-2}^{2}$ is the standard unit sphere metric. Another convenient choice of coordinates $(w,\,u)$ are defined by

$\displaystyle r=w\cos u,$

$\displaystyle z=w\sin u$

(64)

so the $AdS_{d+1}$ metric can be written as

$\displaystyle ds^{2}=\frac{1}{w^{2}\sin^{2}u}(dw^{2}-dt^{2})+\frac{1}{\sin^{2}u}dz^{2}+\frac{\cos^{2}u}{\sin^{2}u}d\Omega_{d-2}^{2}$

(65)

and the induced metric on $\tilde{B}$ becomes

$\displaystyle ds^{2}=\frac{1}{\sin^{2}u}dz^{2}+\frac{\cos^{2}u}{\sin^{2}u}d\Omega_{d-2}^{2}.$

(66)

The regularised bulk contribution to the entanglement entropy for such an entangling surface is then

	$\displaystyle S^{reg}_{B}$	$\displaystyle=\frac{1}{4G_{d+1}}\int_{\tilde{B}_{\epsilon}}d^{d-1}x\sqrt{\gamma}$
		$\displaystyle=\frac{\Omega_{d-2}}{4G_{d+1}}\int_{\epsilon}^{R}dz\,\frac{Rr^{d-3}}{z^{d-1}}$		(67)
		$\displaystyle=\frac{\Omega_{d-2}}{4G_{d+1}}\int_{\xi}^{\frac{\pi}{2}}du\,\frac{\cos^{d-2}u}{\sin^{d-1}u}$		(68)

where $\Omega_{d-2}$ is the area of ($d-2$)-dimensional unit sphere and $R\sin\xi\equiv\epsilon$.

The divergent contributions are of the form $\epsilon^{-n}$ except in even $d$ where there are extra logarithmic terms. Focussing first on odd $d$, from $(\ref{eq:renhee})$ we know the counterterms for $d<6$ are

	$\displaystyle S^{ct}_{B}$	$\displaystyle=\frac{1}{4(d-2)G_{d+1}}\int_{\partial\tilde{B}_{\epsilon}}d^{d-2}x\sqrt{\widetilde{\gamma}}\left[1+\frac{1}{(d-2)(d-4)}(\mathcal{R}_{aa}-\frac{1}{2}k^{2}-\frac{d}{2(d-1)}\mathcal{R})\right]$		(69)
		$\displaystyle=\frac{\Omega_{d-2}}{4(d-2)G_{d+1}}\frac{r^{d-2}}{\epsilon^{d-2}}\left[1-\frac{(d-2)\epsilon^{2}}{2(d-4)r^{2}}\right].$

Note that the intrinsic curvature terms do not contribute here since the boundary metric is flat, but they will contribute to the variation of the entanglement entropy under metric perturbations later. For odd $d$, using the definition of the entangling surface one obtains counterterm contributions

$\displaystyle S^{ct}_{B}=\frac{\Omega_{d-2}}{4G_{d+1}}\left[\frac{R^{d-2}}{(d-2)\epsilon^{d-2}}-\frac{R^{d-4}}{(d-4)\epsilon^{d-4}}+\cdots\right].$

(70)

Combing $(\ref{eq:Sctoddd})$ with $(\ref{eq:IntzSreg})$ we get

	$\displaystyle d=3:\quad\quad\quad\quad S_{ren}=-\frac{\pi}{2G_{4}}$		(71)
	$\displaystyle d=5:\quad\quad\quad\quad S_{ren}=\frac{\pi^{2}}{3G_{6}}.$

For $d=4$, the regularized entanglement entropy from $(\ref{eq:IntzSreg})$ is

$\displaystyle S^{reg}_{B}=\frac{\Omega_{2}}{4G_{5}}\left[-\frac{\ln R}{2}+\frac{\ln(R^{2})}{2}+\frac{R(R^{2}-\epsilon^{2})^{\frac{1}{2}}}{2\epsilon^{2}}+\frac{\ln\epsilon}{2}-\frac{\ln(R^{2}+R(R^{2}-\epsilon^{2})^{\frac{1}{2}})}{2}\right]$

(72)

and the corresponding the full set of counterterms, including the logarithmic counterterm, gives

	$\displaystyle S^{ct}_{B}$	$\displaystyle=\frac{1}{8G_{5}}\int_{\partial\tilde{B}}d^{2}x\sqrt{\widetilde{\gamma}}-\frac{\ln\epsilon}{16G_{5}}\int_{\partial\tilde{B}}d^{2}x\sqrt{\widetilde{\gamma}}(\mathcal{R}_{aa}-\frac{1}{2}k^{2}-\frac{2}{3}\mathcal{R})$
		$\displaystyle=\frac{\Omega_{2}}{8G_{5}}\left[\frac{r^{2}}{\epsilon^{2}}+{\ln\epsilon}\right].$		(73)

Combining these we obtain the renormalized entanglement entropy in $d=4$

$\displaystyle S^{ren}_{B}=\frac{\Omega_{2}}{4G_{5}}\left[-\frac{\ln 2R}{2}+\frac{1}{4R}\right].$

(74)

Since the action in this case has logarithmic counterterms there is an intrinsic scheme dependence in the renormalised entanglement entropy, which is completely determined by the scheme chosen for the renormalization of the action.

We now consider the variation of the entanglement entropy under a linear perturbation of the bulk metric. We will express the perturbed metric in radial gauge so that

$$ds^{2}=\frac{dz^{2}}{z^{2}}+\frac{1}{z^{2}}(\eta_{\mu\nu}+h_{\mu\nu})dx^{\mu}dx^{\nu},$$

(75)

A general perturbation $h_{\mu\nu}$ can be expanded near the conformal boundary as

$$h_{\mu\nu}=h^{(0)}_{\mu\nu}+z^{2}h^{(2)}_{\mu\nu}+\cdots+z^{d}h^{(d)}_{\mu\nu}+z^{d}\log z\tilde{h}^{(d)}_{\mu\nu}+\cdots$$

(76)

where the logarithmic terms arise in even $d$ and all coefficients in the expansion can be expressed in terms of the pair of data $(h^{(0)}_{\mu\nu},h^{(d)}_{\mu\nu})$ using the Einstein equations.

The goal of this section is to show that the change in the renormalized entropy $\delta S_{B}^{ren}$ under such metric perturbations is equal to the change in modular energy i.e.

$\displaystyle\delta E_{B}=\delta S_{B}^{ren}.$

(77)

In the previous literature Faulkner:2013ica , the first law was derived by restricting the variation of metric to only normalizable modes i.e. imposing $h_{\mu\nu}^{(0)}=0$ with $h_{\mu\nu}^{(d)}\neq 0$. Accordingly, the change in the entanglement entropy $\delta S_{B}$ is finite even without including the counterterms. Here we will derive the first law for general perturbations for which $h^{(0)}_{\mu\nu}$ is not necessarily zero; from a QFT perspective a general bulk metric perturbation corresponds to changing the background for the dual QFT as well as the state in the theory.

We will first demonstrate the renormalized first law in the infinitesimal limit where the radius of the boundary entangling region $B$ tends to zero $R\rightarrow 0$. The modular energy may be approximated by

	$\displaystyle\delta E_{B}$	$\displaystyle=\int_{B}d\sigma^{\mu}\delta T_{\mu\nu}^{ren}\xi_{B}^{\nu}$		(78)
	$\displaystyle\delta E_{B}$	$\displaystyle\stackrel{{\scriptstyle\scalebox{0.5}{$R\rightarrow 0$}}}{{\scalebox{1.3}{$\longrightarrow$}}}\frac{2\pi R^{d}\Omega_{d-2}}{d^{2}-1}\delta T_{tt}^{ren}.$

From holographic renormalization deHaro2001 , the variation of the renormalized energy momentum tensor for odd $d$ is

$\displaystyle\delta T_{\mu\nu}^{ren}=\frac{d}{16\pi G_{d+1}}h^{(d)}_{\mu\nu}.$

(79)

In even dimensions the relation between the renormalized stress tensor and the coefficients of the asymptotic expansion is more complicated, capturing the conformal anomalies. For example, in $d=4$

$\displaystyle\delta T_{\mu\nu}^{ren}=\frac{1}{16\pi G_{5}}\left(4h^{(4)}_{\mu\nu}+6\tilde{h}^{(4)}_{\mu\nu}\right),$

(80)

i.e. there is an additional contribution associated with the coefficient of the logarithmic term $\tilde{h}^{(4)}$. At linearized order we can express $\tilde{h}^{(4)}$ in terms of the curvature $R^{(0)}$ of the perturbation of the QFT metric $h^{(0)}$ as

$$\tilde{h}^{(4)}_{\mu\nu}=-\frac{1}{48}\partial_{\mu}\partial_{\nu}R^{(0)}+\frac{1}{16}\partial^{\rho}\partial_{\rho}R^{(0)}_{\mu\nu}-\frac{1}{96}(\partial^{\rho}\partial_{\rho}R^{(0)})\eta_{\mu\nu}.$$

(81)

The infinitesimal first law of entanglement entropy for general variation is thus equivalent to showing that the variation of renormalized entanglement entropy can be expressed in terms of the renormalized stress tensor as

$\displaystyle\delta S^{ren}_{B}=\frac{2\pi R^{d}\Omega_{d-2}}{d^{2}-1}\delta T_{tt}^{ren}$

(82)