We provide a rigorous and asymptotically exact expression of the mutual information of translationally invariant free fermionic lattice systems in a Gibbs state. In order to arrive at this result, we introduce a novel framework for computing determinants of Töplitz operators with smooth symbols, and for treating Töplitz matrices with system size dependent entries. The asymptotically exact mutual information for a partition of the 1D lattice satisfies an area law, with a prefactor which we compute explicitly. As examples, we discuss the fermionic XX model in one dimension and free fermionic models on the torus in higher dimensions in detail. Special emphasis is put on the discussion of the temperature dependence of the mutual information, scaling like the logarithm of the inverse temperature, hence confirming an expression suggested by conformal field theory. We also comment on the applicability of the formalism to treat open systems driven by quantum noise. In the appendix, we derive useful bounds to the mutual information in terms of purities. Finally, we provide a detailed error analysis for finite system sizes. This analysis is valuable in its own right for the abstract theory of Töplitz determinants.
