A theoretical study presents an exact way to calculate Rényi mutual information—the shared quantum information between two regions—for spherical regions in a free massless Dirac field in any spacetime dimension. It also treats a massless Rarita–Schwinger field in four dimensions.
The document identifies itself as arXiv:2608.23692v1 [hep-th], dated 24 August 2026.
Breaking the calculation into angular modes
To build the calculation, the authors use spherical dimensional reduction: the higher-dimensional theory is recast as an infinite tower of two-dimensional models on a half line. Each mode has a finite-boundary solution and a position-dependent mass.
Each reduced mode is equivalent to a standard-quantized Dirac fermion in AdS2, and its Rényi mutual information can be computed exactly in terms of Painlevé transcendents.
Putting the modes together gives the full d-dimensional nth Rényi mutual information as a sum weighted by angular degeneracy. In the n → 1 limit, ordinary mutual information is represented by a corresponding integral over κ.
The separation changes the answer
At long distances, the Dirac expansion in η begins with the ℓ = 0 sector. For arbitrary d, its first terms are proportional to η^(d−1) and η^d, followed by O(η^(d+1)).
For ordinary mutual information at n = 1, the reported terms are (1/35)η^3 + (4/105)η^4 + O(η^5) in d = 4, and (1/15)η^2 + (2/35)η^3 + O(η^4) in d = 3.
For the massless Rarita–Schwinger field in d = 4, the n = 1 expansion begins at the fifth power of η: (2/693)η^5 + (20/3003)η^6 + O(η^7). The first two terms come from the ℓ = 1 sector.
Near-touching spheres bring a different pattern
As η approaches 1, the short-distance limit for each half-line mode contains a universal term, −(1/6) log(1−η), plus a mass-dependent integral contribution.
For large μℓ, this short-distance result has the asymptotic form −(1/3) log μℓ + c0, followed by inverse-power corrections including −1/(30μℓ^2) and 1/(84μℓ^4).
Near complementary spheres, the leading d-dimensional term is 2^[d/2] × Area(S^(d−2)) × κ_d × (a/ϵ)^(d−2), with subleading terms of order O((a/ϵ)^(d−4)). The coefficient κ_d is given by an integral over t^(d−3)c_flat(t).
For the Dirac field, the logarithmic coefficient is twice the type-A trace anomaly in even dimensions and vanishes in odd dimensions. The reported values are c_log^(4) = −11/45 and c_log^(6) = 191/1890.
A sharp four-dimensional difference
In d = 4, the Rarita–Schwinger mutual information differs from the Dirac result by exactly the omitted ℓ = 0 Dirac contribution, with μℓ = 1.
A numerical check with explicit limits
The numerical check used N = 8000 half-line lattice sites. The lattice results were divided by 2 because of fermion doubling.
The analytic and lattice approaches were reported to agree well, although small deviations were attributed to numerical limitations in the lattice implementation or in solving the Painlevé VI equation. No quantitative error metrics were reported.
The n → 1 continuation relies on assumptions about the mode functions that are not fully proved. The long-distance, short-distance and large-μℓ formulas are asymptotic and regime-dependent, while the Rarita–Schwinger result is limited to the stated d = 4 model.
Paper data and sources
Original title: Exact mutual information for free massless fermions in any dimension
Authors: Nicolás Abate, Leandro Martinek
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-24
DOI: Not available
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