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 $$ \overline{\psi} $$ 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.
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 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.
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 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.
In this work, we study the universal behaviors in the mutual information of two disjoint spheres in a conformal field theory (CFT). By using the operator product expansion of the spherical twist operator in terms of the conformal family, we show that the large distance expansion of the mutual information can be cast in terms of the conformal blocks. We develop the 1/n prescription to compute the coefficients before the conformal blocks. For a single conformal family, the leading nonvanishing contribution to the mutual information comes from the bilinear operators. We show that the coefficients of these operators take universal forms and such universal behavior persists in the bilinear operators with derivatives as well. Consequently the first few leading order contributions to the mutual information in CFT take universal forms. To illustrate our framework, we discuss the free scalars and free fermions in various dimensions. For the free scalars, we compute the mutual information to the next-to-leading order and find good agreement with the improved numerical lattice result. For the free fermion, we compute the leading order result, which is of universal form, and find the good match with the numerical study. Our formalism could be applied to any CFT potentially.
A bstract The operator product expansion (OPE) of twist operators in the replica trick framework enables a long-distance expansion of the mutual information (MI) in conformal field theories (CFTs). In this expansion, the terms are labeled by primary operators, as contributions from descendant operators can be resummed into conformal blocks. However, for the MI, the expansion involves primaries from the multi-replica theory, which includes far more operators than those in the original theory. In this work, we develop a method to resum this series, yielding an expansion in terms of the primaries of the original theory, specifically restricted to the two-copy sector. This is achieved by expressing the twist operators in a non-local manner across different replicas and using a modular flow representation to obtain the n → 1 limit of the Rényi index. We explicitly compute the resulting “enhanced conformal blocks,” which, surprisingly, provide excellent approximations to the MI of generalized free fields across the full range of cross ratios. Remarkably, this approximation appears to be exact in the limit of large spacetime dimensions.