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In a remarkable paper [Phys. Rev. Lett. 96, 100503 (2006)], Dimitri Gioev andIsrael Klich conjectured an explicit formula for the leading asymptotic growthof the spatially bi-partite von-Neumann entanglement entropy of non-interactingfermions in multi-dimensional Euclidean space at zero temperature. Based onrecent progress by one of us (A.V.S.) in semi-classical functional calculus forpseudo-differential operators with discontinuous symbols, we provide here acomplete proof of that formula and of its generalization to R\'enyi entropiesof all orders $\alpha>0$. The special case $\alpha=1/2$ is also known under thename logarithmic negativity and often considered to be a particularly usefulquantification of entanglement. These formulas, exhibiting a "logarithmicallyenhanced area law", have been used already in many publications.

Scaling of R\'enyi entanglement entropies of the free Fermi-gas ground state: a rigorous proof