Scaling of Rényi entanglement entropies of the free Fermi-gas ground state: a rigorous proof

Abstract

In a remarkable paper [Phys. Rev. Lett. 96, 100503 (2006)], Dimitri Gioev and Israel Klich conjectured an explicit formula for the leading asymptotic growth of the spatially bi-partite von-Neumann entanglement entropy of non-interacting fermions in multi-dimensional Euclidean space at zero temperature. Based on recent progress by one of us (A.V.S.) in semi-classical functional calculus for pseudo-differential operators with discontinuous symbols, we provide here a complete proof of that formula and of its generalization to Rényi entropies of all orders $\alpha >0$. The special case $\alpha =1/2$ is also known under the name logarithmic negativity and often considered to be a particularly useful quantification of entanglement. These formulas, exhibiting a "logarithmically enhanced area law", have been used already in many publications.

5 pages, added a discussion following (in)equalities (9) and concluding remarks

References in corpus (5)

• Entanglement entropy of fermions in any dimension and the Widom conjecture
• Entanglement scaling in critical two-dimensional fermionic and bosonic systems
• Entropy and Entanglement in Quantum Ground States
• The entanglement entropy of one-dimensional gases
• Wiener-Hopf operators in higher dimensions: the Widom conjecture for piece-wise smooth domains

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