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arXiv:2611.03630

Mutual information to all orders from modular flow: a replica-free approach

Hewei Frederic Jia

hep-thcond-mat.stat-mechquant-phSubmitted Nov 2, 2026• Announced Nov 3, 2026
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Abstract

We derive an all-orders long-distance expansion of vacuum mutual information between two disjoint balls in generic $d$-dimensional CFTs. The derivation uses a novel radial sewing expansion of the joint vacuum density matrix, together with a modular response formula for relative entropy and the geometric modular flow of a single ball, without analytic continuation in replica number. The expansion coefficients are modular integrals of modular-flowed vacuum correlators. Our formula provides a systematic treatment of arbitrary multi-point sectors, and recovers known results on two-point sector derived from replica methods. The full two-point sector resums into a modular integral of the thermal partition function on $S^{d-1}$, with an effective temperature depending on ball configuration and modular time. We compute the expansion explicitly in CFTs. In two-dimensional Ising CFT, using analytic reductions and multidimensional numerical integration, we obtain the first ten radial orders in the long-distance expansion, which involve sphere correlators with up to twenty spin insertions. The truncated expansion agrees well with lattice data at finite separation, and approximately satisfies crossing symmetry. Analytic calculations also reproduce known expansion coefficients for 2d free compact boson at self-dual radius and its current sector, as well as for free scalars in higher dimensions.

Comments
36 pages, 4 figures, 4 tables

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We derive an all-orders long-distance expansion of vacuum mutual information between two disjoint balls in generic ddd-dimensional CFTs. The derivation uses a novel radial sewing expansion of the joint vacuum density matrix, together with a modular response formula for relative entropy and the geometric modular flow of a single ball, without analytic continuation in replica number. The expansion coefficients are modular integrals of modular-flowed vacuum correlators. Our formula provides a systematic treatment of arbitrary multi-point sectors, and recovers known results on two-point sector derived from replica methods. The full two-point sector resums into a modular integral of the thermal partition function on Sd−1S^{d-1}Sd−1, with an effective temperature depending on ball configuration and modular time. We compute the expansion explicitly in CFTs. In two-dimensional Ising CFT, using analytic reductions and multidimensional numerical integration, we obtain the first ten radial orders in the long-distance expansion, which involve sphere correlators with up to twenty spin insertions. The truncated expansion agrees well with lattice data at finite separation, and approximately satisfies crossing symmetry. Analytic calculations also reproduce known expansion coefficients for 2d free compact boson at self-dual radius and its current sector, as well as for free scalars in higher dimensions.
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.
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.
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‾ \overline{\psi} 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 a novel replica trick that computes the relative entropy of two arbitrary states in conformal field theory. Our replica trick is based on the analytic continuation of partition functions that break the Z_{n} replica symmetry. It provides a method for computing arbitrary matrix elements of the modular Hamiltonian corresponding to excited states in terms of correlation functions. We show that the quantum Fisher information in vacuum can be expressed in terms of two-point functions on the replica geometry. We perform sample calculations in two-dimensional conformal field theories.
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.
Abstract We study the two point correlation function of a local operator on an n-sheeted replica manifold corresponding to the half-space in the vacuum state of a conformal field theory. In analogy with the inverse Laplace transform, we define the Renyi transform of this correlation function, which is a function of one complex variable w, dual to the Renyi parameter n. Inspired by the inversion formula of Caron-Huot, we argue that if the Renyi transform f (w) has bounded behavior at infinity in the complex w plane, the discontinuity of the Renyi transform disc f (w) provides the unique analytic continuation in n of the original replica correlation function. We check our formula by explicitly calculating the Renyi transform of a particular replica correlator in a large N holographic CFTd in dimensions d > 2. We also discover that the discontinuity of the Renyi transform is related to the matrix element of local operators between two distinct eigenstates of the modular Hamiltonian. We calculate the Renyi transform in 2d conformal field theories, and use it to extract the off-diagonal elements of (modular) ETH. We argue that in 2d, this is equivalent to the off-diagonal OPE coefficients of a CFT and show that our technique exactly reproduces recent results in the literature.