Abstract We present several new results for the N-partite information, IN, of spatial regions in the ground state of d-dimensional conformal field theories. First, we show that IN can be written in terms of a single N-point function of twist operators. Using this, we argue that in the limit in which all mutual separations are much greater than the regions sizes, the N-partite information scales as IN ~ r−2N∆, where r is the typical distance between pairs of regions and ∆ is the lowest primary scaling dimension. In the case of spherical entangling surfaces, we obtain a completely explicit formula for the I4 in terms of 2-, 3- and 4-point functions of the lowest-dimensional primary. Then, we consider a three- dimensional scalar field in the lattice. We verify the predicted long-distance scaling and provide strong evidence that IN is always positive for general regions and arbitrary N for that theory. For the I4, we find excellent numerical agreement between our general formula and the lattice result for disk regions. We also perform lattice calculations of the mutual information for more general regions and general separations both for a free scalar and a free fermion, and conjecture that, normalized by the corresponding disk entanglement entropy coefficients, the scalar result is always greater than the fermion one. Finally, we verify explicitly the equality between the N-partite information of bulk and boundary fields in holographic theories for spherical entangling surfaces in general dimensions.
We present several new results for the
N-partite information,
IN, of spatial regions in the ground state of
d-dimensional conformal field theories. First, we show that
IN can be written in terms of a single
N-point function of twist operators. Using this, we argue that in the limit in which all mutual separations are much greater than the regions sizes, the
N-partite information scales as
IN∼r−2NΔ, where
r is the typical distance between pairs of regions and
Δ is the lowest primary scaling dimension. In the case of spherical entangling surfaces, we obtain a completely explicit formula for the
I4 in terms of 2-, 3- and 4-point functions of the lowest-dimensional primary. Then, we consider a three-dimensional scalar field in the lattice. We verify the predicted long-distance scaling and provide strong evidence that
IN is always positive for general regions and arbitrary
N for that theory. For the
I4, we find excellent numerical agreement between our general formula and the lattice result for disk regions. We also perform lattice calculations of the mutual information for more general regions and general separations both for a free scalar and a free fermion, and conjecture that, normalized by the corresponding disk entanglement entropy coefficients, the scalar result is always greater than the fermion one. Finally, we verify explicitly the equality between the
N-partite information of bulk and boundary fields in holographic theories for spherical entangling surfaces in general dimensions.
A bstract The mutual information, I 2 , of general spacetime regions is expected to capture the full data of any conformal field theory (CFT). For spherical regions, this data can be accessed from long-distance expansions of the mutual information of pairs of regions as well as of suitably chosen linear combinations of mutual informations involving more than two regions and their unions — namely, the N -partite information, I N . In particular, the leading term in the I 2 long-distance expansion is fully determined by the spin and conformal dimension of the lowest-dimensional primary of the theory. When the operator is a scalar, an analogous formula for the tripartite information I 3 contains information about the OPE coefficient controlling the fusion of such operator into its conformal family. When it is a fermionic field, the coefficient of the leading term in I 3 vanishes instead. In this paper we present an explicit general formula for the long-distance four-partite information I 4 of general CFTs whose lowest-dimensional operator is a fermion ψ . The result involves a combination of four-point and two-point functions of ψ and
ψ ψ ¯ evaluated at the locations of the regions. We perform explicit checks of the formula for a (2 + 1)-dimensional free fermion in the lattice finding perfect agreement. The generalization of our result to the N -partite information (for arbitrary N ) is also discussed. Similarly to I 3 , we argue that I 5 vanishes identically at leading order for general fermionic theories, while the I N with N = 7 , 9 , … only vanish when the theory is free.
We consider the mutual Rényi information of disjoint compact spatial regions A and B in the ground state of a d + 1-dimensional conformal field theory (CFT), in the limit when the separation r between A and B is much greater than their sizes R A , B . We show that in general , where α is the smallest sum of the scaling dimensions of operators whose product has the quantum numbers of the vacuum, and the constants depend only on the shape of the regions and universal data of the CFT. For a free massless scalar field, where α = d − 1, we show that is proportional to the capacitance of a thin conducting slab in the shape of A in d + 1-dimensional electrostatics, and give explicit formulae for this when A is the interior of a sphere S d − 1 or an ellipsoid. For spherical regions in d = 2 and 3 we obtain explicit results for C ( n ) for all n and hence for the leading term in the mutual information by taking n → 1. We also compute a universal logarithmic correction to the area law for the Rényi entropies of a single spherical region for a scalar field theory with a small mass.
A bstract The entanglement entropy of spacetime regions A in odd-dimensional conformal field theories (CFTs) contains a universal constant term,
(−1)2d−1 F(A) − 1 d − 1 2 F A . This quantity can be robustly defined by considering the mutual information of pairs of slightly deformed versions of A . In the case of general three-dimensional CFTs, F ( A ) is positive definite and bounded below by the round disk result, F ( A ) ≥ F 0 ≡ F ( ∂A = 𝕊 1 ). Additionally, strong evidence has been provided that for every region A , F ( A ) /F 0 is maximized, within the space of CFT 3 ’s, by the free scalar field result. In this paper we show that while F ( A ) remains a local minimum around F 0 ≡ F ( ∂A = 𝕊 3 ) for small deformations of the spherical entangling surface, it can take values of arbitrarily large magnitude with either sign for more general regions, and hence it is neither upper- nor lower-bounded in general CFT 5 ’s. We argue that an analogous conjecture regarding the extremization of F ( A ) /F 0 for general regions within the space of theories fails in d = 5. We instead analyze the viability of the weaker bound, F ϵ /F 0 ≤ F ϵ /F 0 | free scalar , ∀CFT 5 for general small geometric deformations of the spherical entangling surface. This is equivalent to a general constraint involving the stress-tensor two-point function C T and the Euclidean partition function on the sphere, namely, C T /F 0 ≤ C T /F 0 | free scalar ≈ 0 . 314, which we show to hold for all known CFT 5 ’s. We also comment on possible extensions of this result to higher dimensions.
We compute the leading term of the tripartite information at long distances for three spheres in a CFT. This falls as r^{-6\Delta} r − 6 Δ , where r r is the typical distance between the spheres, and \Delta Δ the lowest primary field dimension. The coefficient turns out to be a combination of terms coming from the two- and three-point functions and depends on the OPE coefficient of the field. We check the result with three-dimensional free scalars in the lattice finding excellent agreement. When the lowest-dimensional field is a scalar, we find that the mutual information can be monogamous only for quite large OPE coefficients, far away from a perturbative regime. When the lowest-dimensional primary is a fermion, we argue that the scaling must always be faster than r^{-6\Delta_f} r − 6 Δ f . In particular, lattice calculations suggest a leading scaling r^(6\Delta_f+1) r ( 6 Δ f + 1 ) . For free fermions in three dimensions, we show that mutual information is also non-monogamous in the long-distance regime.
The vacuum mutual information (MI) of subregion algebras provides a universal window into the data of general conformal field theories (CFTs). Exploiting the geometric nature of the modular flow associated to ball-shaped regions and the operator product expansion of twist operators implementing the replica symmetry in an n -fold version of a CFT, it is possible to construct a hierarchy of increasingly refined approximations to the full MI. In this letter, we use the two-point functions of primaries of arbitrary spin in the replicated theory to constrain the twist operators, and find their contribution to the MI of arbitrarily boosted balls in any d -dimensional CFT. When the two-point functions involve the primary with the lowest scaling dimension, our result provides the most precise approximation for the long-distance behavior of the MI, superseding all previous expansions. Building upon this result and certain universal properties of the short- and long-distance regimes, we put forward a new high-precision analytic approximation to the MI for arbitrary separations. The accuracy of our approach is validated against exact d = 2 and lattice d = 3 results. We further apply it to characterize the MI of a d = 4 Maxwell field, a case for which no prior results are available.