Shape dependence of mutual information in the OPE limit: linear responses

Entanglement structure has played a key role in probing deep aspects of quantum field theories (QFTs) not accessible using conventional tools and techniques. For example, it can provide the order parameter for topological phase Kitaev2005Mar ; Xiao-GangMar ; through the famous Ryu-Takayanagi formula Ryu2006 and its subsequent generalizations Hubeny2007 , it encodes information regarding bulk space-time geometry in holography Maldacena1999 ; Witten1998AntideSS ; GUBSER1998105 ; it can be used to characterize universal monotonous properties of renormalization group flow Myers2011Jan ; Casini2011May ; Casini:2015woa ; and it is also crucial for revealing connection between information theory and energy conditions HWang2016Sep ; HWang2019Sep ; Faulkner1804 , etc.

In principle, one can define various measures of the underlying entanglement structure. An important one is the entanglement entropy between subsystem $A$ and its complement $\bar{A}$. For a global pure state $|\psi\rangle$, the entanglement entropy is defined by:

This definition relies on the notion of reduced density matrix $\rho^{\psi}_{A}$, which only makes clear sense if the global Hilbert space factorizes into tensor product $\mathcal{H}=\mathcal{H}_{A}\otimes\mathcal{H}_{\bar{A}}$. This is a convenient but usually problematic assumption in quantum field theories, for example due to gauge constraints imposed at the entangling boundary $\partial A$ William2012 ; Casini2014Apr ; Lin:2018 . Another aspect associated with $\partial A$ in quantum field theories is the ubiquitous ultra-violet (UV) divergences that arise when computing entanglement entropies. Roughly speaking, they come from short-range entanglements across $\partial A$ that are only regularized by the UV cut-off of the theory. For example, in CFT the divergences take the form:

where $R,\delta$ is the scale of entangling region and UV cutoff respectively, $c_{d-2},c_{d-4},...$ are coefficients that depend on the geometry of entangling regions, and $c_{-1}$ and $c_{0}$ are universal. As a result, additional efforts are required in order to extract universal aspects of entanglement structures from the result of entanglement entropy. In order to do this, one usually construct combinations of entanglement quantities whose UV dependences cancel out. In practice, such cancellations can be arranged in a number of ways. For example, when extracting the order parameters for topological phases in 2+1 dimensional gapped systems, one computes the so-called topological entanglement entropy Kitaev2005Mar ; Xiao-GangMar by adding and subtracting vacuum entanglement entropies associated with different regions:

out of which the shape dependence of these regions cancel out completely and leaving only topological contributions. Fixing a particular entangling region $A$, one can also arrange the cancellation between states, for example when computing the so-called relative entropy between two global states $\psi$ and $\sigma$:

The relative entropy $S_{A}\left(\psi|\sigma\right)$ provides a measure of distinguishability between the global states $\psi$ and $\sigma$ based on what one can access in the subsystem $A$, i.e. between $\rho^{\psi}_{A}$ and $\rho^{\sigma}_{A}$. It satisfies several constraints such as positivity and monotonicity, and when applied to specific context they can sometimes reveal important physics such as the emergence of Einstein’s equation in AdS/CFT Faulkner2014Mar ; Faulkner2015May ; Faulkner2017Aug ; Lewkowycz2018 or the validity of the averaged null energy condition (ANEC) in QFTs HWang2016Sep , etc.

Mutual information is one of the simplest combinations among these entanglement quantities. For disjoint entangling regions $A$ and $B$, one can define the mutual information $I_{A,B}$ between them as:

where we have omitted the dependence on the state $\psi$ in these quantities, since in this paper shall focus on the vacuum state $|\psi\rangle=|\Omega\rangle$ from now on. The UV divergences associated with short-distance entanglements near $\partial\left(A\cup B\right)$ are cancelled in $I_{A,B}$, thus making it a universal quantity. Roughly speaking $I_{AB}$ measures the correlation between the subsystems $A$ and $B$. When we view $A\cup B$ together as a system in the mixed state $\rho_{AB}$, $I_{A,B}$ receives contributions from both classical and quantum correlations. In attempts to refine the distinction between classical and quantum correlations in mixed states, other measures have been proposed which include entanglement negativity Eisert1999 ; Vidal2002 ; Plenio2005 . Being a universal quantity, the mutual information behaves in ways that align more with intuitions. For example due to strong subadditivity (SSA) for entanglement entropies, the mutual information like relative entropies also satisfy monotonicity, i.e.:

for $A\subseteq\tilde{A}$, while the entanglement entropies in general do not satisfy such constraints by themselves. On the other hand, the mutual information is less constrained by symmetries and thus encode more detailed information about the underlying theories. For example, while the single-interval vacuum entanglement entropies in 2d CFT ares constrained by symmetries to only depend on the central charges $c$, the mutual information between single-intervals depend on the full operator-spectrum of the CFTs.

Apart from the choice of entanglement measures, shape dependence on the entangling region can also reveal important information regarding entanglement structure. For example, in the presence of corner or cusp the entanglement entropy receives log divergent contributions whose coefficients on the other hand contain universal information Fradkin2006 ; Casini2007Dec ; Hirata2007 . For regions with smooth boundaries, more generally one can study shape dependence by computing perturbation theory under shape deformation Rosenhaus2014Dec ; Rosenhaus2015Feb , usually about symmetric shapes i.e. spheres, half-planes etc. For spheres in CFTs, computing the first-order response indicates that its entanglement entropy is extremal under symmetry-breaking deformations Mezei2015Feb ; Mezei2015Feb02 ; while it being perturbatively minimal was verified by the second-order response which also revealed universal non-local contributions known as the entanglement density Faulkner2016Apr ; the minimization was later proven beyond perturbation theory Bueno2021 . Sometimes by probing shape dependence of carefully chosen entanglement quantities, one discover surprising connections between entanglement behaviors and other aspects of QFTs, such as the irreversibility of RG flows and the validity of energy conditions, manifested through analyzing shape dependence of entanglement entropies (appropriately regularized) and relative entropies.

It is therefore natural to expect that the shape dependence of mutual information can also reflect important physics. For example, it has been shown that the shape dependence of $I_{A,B}$ satisfies constraints on the null cone of $A$ which in the large separation limit (see 8 below) between $A$ and $B$ can give rise to the unitarity bound Casini2021Sep . However, explicit results for shape dependence of mutual information using the general perturbative approach have been missing so far. One reason for this is that we do not have configurations with non-trivial mutual information that are sufficiently symmetric. In the case of a single sphere $A$ the vacuum entanglement structure in CFT is fixed by symmetry. In particular the modular Hamiltonian which encodes the complete entanglement data is known explicitly Casini2011May :

Therefore it makes sense to do perturbation theory about it as in Mezei2015Feb ; Mezei2015Feb02 ; Faulkner2016Apr . On the other hand, the mutual information $I_{A,B}$ between two spheres $A$ and $B$ in CFT vacuum is not known in general form, despite being the most symmetric configuration with non-trivial mutual information. As mentioned before it is sensitive to the details (e.g. operator spectrum) of the theory. Explicit results can be obtained by restricting to special theories, e.g. Dirac fermions in two-dimensions Casini2005Jul ; Casini2009Mar ; Casini2009Sep , in which the shape dependence on entangling region is degenerate; or by restricting to special limits in general CFTs, e.g. of large separation $L$ between $A$ and $B$ (see Fig 1), where $I_{AB}$ admits an OPE-like expansion Cardy2013 ; Faulkner2016Aug ; ChenBin2017 ; Long2016 :

where $\Delta$ denotes the conformal dimension of an internal scalar primary operator that “carries” the correlation between $A$ and $B$ in some loose sense. The OPE result (8) is valid in any dimensional CFTs, and therefore provides a good starting point to develop perturbative expansion in shape deformations.

In this paper, we study the shape dependence of mutual information perturbatively about spheres. We will focus on the leading order contribution in the OPE (8). More specifically, we will compute the first-order linear response under shape deformation about one of the spheres $A$. The OPE expansion (8) is conformally invariant, i.e. depending only on the conformal-ratio:

where $R_{A,B}$ are the radius of the spheres $A,B$ and $L$ is the distance between them. Therefore, we have the freedom to work in a conformal frame where $R_{A}\sim L\gg R_{B}$, see Fig. 1, so that the fine-details in the shape-response of $I_{A,B}$ on $A$ are optimally pronounced. A key input into our calculation is the modular Hamiltonian version of the OPE expansion (8) derived in Faulkner2021Aug . This is because at the first order, the shape perturbation theory can be implemented in the context of the entanglement first law, which requires knowledge about the unperturbed modular Hamiltonian.

The paper is organized as follows. In section 2, we recall some basic ingredients regarding shape perturbation theory and its application through the entanglement first law for computing linear responses. In section 3, we summarize previous results for mutual information and the corresponding modular Hamiltonians in the OPE limit, which our calculations will be based upon. In section 4, we carry out the main computation and obtain results for linear response of mutual information under shape deformation on one of the spheres. During this we identify some subtleties associated with the integral expression of modular Hamiltonian in terms of local operators. In section 5, we explore the implications of our results from a symmetry point of view, and discuss the extremality property under shape deformations for mutual information on spheres. In section 6, we conclude the paper with some discussions and suggestions for future directions.

In this section, we quickly recall the necessary ingredients needed for our computations. More details can be found in Rosenhaus2014Dec ; Rosenhaus2015Feb ; Mezei2015Feb ; Mezei2015Feb02 ; Faulkner2016Apr .

Our goal is to study the shape dependence of mutual information perturbatively about two spheres in the CFT vacuum. Similar to the computation of $\delta S_{A}$ in Sec (2.3), this can be done via the entanglement first law. The pre-requisite for such an approach is explicit knowledge of the unperturbed result for both the mutual information and its modular Hamiltonians. However, mutual information in general is highly theory dependent, even for the case of our interest, i.e. two spheres in the vacuum. Explicit results are only accessible in the so-called OPE limit, in which the distance between two spheres are much larger than their radii. As a result, in this paper we shall also focus on the shape dependence of mutual information in the same (or conformally equivalent) limit (see Fig 1). In this section, we review the results in Faulkner2016Aug ; Faulkner2021Aug calculations of for the OPE limit of mutual information and the corresponding modular Hamiltonian using the replica technique.

We consider two disjoint spherical subregions $A$ and $B$ with radius $R_{A},R_{B}$ separated by a distance $L$ (see Fig 1). The mutual information $I_{A,B}$ can be obtained by the replica trick as: $I_{A,B}\equiv\lim_{n\rightarrow 1}I_{n}(A,B)$, where:

is the Rényi mutual information. For QFTs defined on Euclidean spacetime $\mathcal{M}$, we can rewrite these traces as partition functions on appropriately constructed conifolds, i.e.

where $\mathcal{C}_{n}^{X}$ is constructed from $n$-copies of $\mathcal{M}$, glued across subregion $X$. Equivalently, we can write the partition functions on conifold as expectation values of twist operators in the corresponding orbifold theory: $CFT^{\otimes n}/\mathds{Z}_{n}$, defined on $\mathcal{M}^{n}$:

Analogous to ordinary correlation functions, the twist operators in $\left\langle\Sigma_{n}^{A}\Sigma_{n}^{B}\right\rangle_{\mathcal{M}^{n}}$ also admit operator product expansions into local operators in $CFT^{\otimes n}/\mathds{Z}_{n}$ :

i.e. expanding $\Sigma_{n}^{A,B}$ into a complete set of local operators at $r_{A,B}$, where in (27) we have written the leading identity operator separately. Assuming a $\mathds{Z}_{2}$ symmetry for the underlying CFT, the coefficients for single copy operators vanish. As a result, the first few orders in the OPE take the form :

where $\mathcal{O}$ is a primary operator with the lowest scaling dimension $\Delta$, and it enters the OPE by appearing twice in different replica $j\neq k$. We call such form of OPE the bi-local channel. Due to replica symmetry, the OPE coefficients only depend on $j-k$. The subsequent $\cdots$ denotes contributions from other operators in the bi-local channel or from multi-local channels. For future convenience, we have rescaled the OPE coefficients with a factor $(2R_{A,B})^{-2\Delta}$ relative to (27), i.e. $c_{j-k}=(2R_{A,B})^{-2\Delta}C_{j-k}$. In this paper, we focus on the leading-order contribution to $I_{A,B}$. This corresponds to $\mathcal{O}^{(j)}\mathcal{O}^{(k)}$ composed by the lowest-dimension primary operator $\mathcal{O}$. For simplicity, we begin by assuming $\mathcal{O}$ as a scalar, and will discuss the cases for spinning operators later.

Analogous to the usual OPE, we can extract the coefficients $c_{j-k}$ by inserting corresponding operators at $r\to\infty$ and inverting (27):

The correlation function in the presence of $\Sigma^{A,B}_{n}$ has been written in terms of thermal correlator $G_{n}$ in hyperbolic space $\mathds{H}_{d-1}$ at inverse temperature $T^{-1}=2\pi n$. This is achieved via the coordinate transformation into hyperbolic space coordinates $Z=(\tau,Z^{i})\in\mathds{S}_{(2\pi n)}\times\mathds{H}_{d-1}$:

From this form, one can easily perform the analytic continuation in $n$ for the replica trick and obtain the contribution to the mutual information:

The prefactor is just the conformal ratio $\rho$ in (9) in the OPE limit. More details of the computations can be found in Cardy2013 ; Faulkner2016Aug ; Faulkner2021Aug .

For the application of the entanglement first law, we also need the explicit form of the modular Hamiltonians in the combination $\Delta H_{A,B}\equiv H_{A,B}-H_{A}-H_{B}$. In the OPE limit, it admits an expansion similar to that of the mutual information. In this case the OPE is only taken for one of the twist operators $\Sigma_{n}^{B}$. After analytic continuation, the local operators from OPE are modular flown by $H_{A}$. The details can be found in Faulkner2021Aug . In the end, one obtains the OPE for $\Delta H_{A,B}$ in the bi-local channel of primary scalar operator $\mathcal{O}$ as:

where $c_{1}(is)$ is the analytic continuation of $c_{n}$ at $n=1$:

The operator $\mathcal{O}_{A}(-is)$ is obtained from $\mathcal{O}_{B}=\mathcal{O}(r_{B})$ by first applying a modular reflection in $H_{A}$, followed with a modular flow by real $s$ (see Fig 4):

As a check, one can compute the expectation value of the modular Hamiltonian (3) in the vacuum state:

where $\langle\Delta H_{A,B}^{(1)}\rangle$ and $\langle\Delta H_{A,B}^{(1)}\rangle$ represents the contribution from the single integral and double integral of Eq (3) respectively. It can be found that:

which once combined in (34) agrees with the result for mutual information (31).

In this section, we give the main calculations of this paper. The task is to compute the linear response of mutual information $I_{A,B}$ between two spheres $A$ and $B$ under shape deformation on one of the spheres $A\to\tilde{A}$. The spheres have radius $R_{A}$ and $R_{B}$ whose centers are separated by a distance $L$. We focus on a conformally equivalent version of the OPE limit $L\sim R_{A}\gg R_{B}$ (see Figure 5).

More specifically, the deformed sphere $\tilde{A}$ can be parametrized by an angular-dependent modification of the radius: $R(\Omega)=R_{A}+\zeta(\Omega)$. In this parametrization, we can work with the ansatz in which the deformation vector field $\zeta^{\mu}$ that enters the entanglement first law (15) only has a radial component in polar coordinates about the center of $A:\zeta^{\mu}\propto\delta^{\mu}_{r}$.

Now, let us calculate the shape deformation of mutual information by using the ingredients that we have prepared in previous. Similar to the warm-up exercise in 2.3, application of the entanglement first law yields that:

where $\partial\mathcal{M}\sim S^{1}\times S^{d-2}$ is the UV cut-off surface in the form of codimension one tube of radius $\epsilon$ enclosing the spherical boundary $\partial A$, and $(\theta,\Omega_{d-2})$ are coordinates on $\partial\mathcal{M}$. Similar to (18), here we can also drop the contact term in the second line. The reason is that the OPE of $\Delta H_{A,B}$ consists of bi-local operators whose location we can avoid by restricting the support of $\zeta^{\mu}$ in the bulk.

From Eq. (3), we see that the OPE limit of modular Hamiltonian $\Delta H_{A,B}$ is written as the integral of local operators which are flowed by the modular Hamiltonian of spherical subregion $A$. In terms of actual calculations, this is not convenient because the expressions for the modular trajectories are very complicated. We can simplify the calculations by exploiting the conformal symmetry. More specifically, we observe that (36) involves integrals of CFT correlators of the form

where $Y_{B}$ is the center of a small sphere $B$. For CFTs, there is a conformal transformation $\mathcal{U}$ that maps from $A$ to the Rindler half plane. The modular evolution operator can be related to Rindler boost $K$ by:

Plugging (38) into (37), and using the invariance of the conformal vacuum under $\mathcal{U}$, we can rewrite the correlator in the Rindler frame:

where $\tilde{T}_{\mu\nu}=\mathcal{U}^{-1}\;T_{\mu\nu}\;\mathcal{U}$, and $\tilde{\mathcal{O}}(-is)=\Lambda(Y_{B})^{-\Delta}\;e^{iKs}\mathcal{O}\Big{(}\mathcal{U}(Y_{B})\Big{)}e^{-iKs}$ are boosted operators in Rindler frame rescaled by the conformal factor $\Lambda(Y_{B})$ of $\mathcal{U}$.