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arXiv:2608.23692

Exact mutual information for free massless fermions in any dimension

Nicolás Abate, Leandro Martinek

Published Aug 24, 2026• 0 citations
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Abstract

We obtain an exact expression for the Rényi mutual information (RMI) between two spherical regions for a free, massless Dirac field in $d$ spacetime dimensions and for a massless Rarita-Schwinger field in $d=4$. We identify each angular momentum mode in the spherical dimensional reduction of the parent theory with a massive Dirac fermion in two dimensional anti-de Sitter space (AdS). The RMI for each mode can be computed analytically for any Rényi index $n$ in terms of solutions of a Painlevé differential equation. We explore the long and short distance asymptotics of the result, where the sum over modes can be computed explicitly. As a result we are able to find subleading contributions to the long distance expansion in $d$ dimensions for arbitrary $n$, as well as to extract the area and logarithmic coefficients for the case $n=1$ in the short distance regime.

DOI
10.48550/arxiv.2608.23692

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We compute the R\'enyi mutual information of two disjoint spheres in free massless scalar theory in even dimensions higher than 2. The spherical twist operator in a conformal field theory can be expanded into the sum of local primary operators and their descendants. We analyze the primary operators in the replicated scalar theory and find the ones of the fewest dimensions and spins. We study the one-point function of these operators in the conical geometry and obtain their expansion coefficients in the operator product expansion of spherical twist operators. We show that the R\'enyi mutual information can be expressed in terms of the conformal partial waves. We compute explicitly the R\'enyi mutual information up to order zd{z}^{d}zd, where zzz is the cross ratio and ddd is the spacetime dimension.
We study the universal logarithmic coefficient of the entanglement entropy in a sphere for free fermionic field theories in a d=4d=4d=4 Minkowski spacetime. As a warm-up, we revisit the free massless spin-1/21/21/2 field case by employing a dimensional reduction to the d=2d=2d=2 half-line and a subsequent numerical real-time computation on a lattice. Surprisingly, the area coefficient diverges for a radial discretization but is finite for a geometric regularization induced by the mutual information. The resultant universal logarithmic coefficient $\ensuremath{-}11/90$ is consistent with the literature. For the free massless spin-3/23/23/2 field, the Rarita-Schwinger field, we also perform a dimensional reduction to the half-line. The reduced Hamiltonian coincides with the spin-1/21/21/2 one, except for the omission of the lowest total angular momentum modes. This gives a universal logarithmic coefficient of $\ensuremath{-}71/90$. We discuss the physical interpretation of the universal logarithmic coefficient for free higher-spin field theories without a stress-energy tensor.
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.
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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.