A Special Case Of A Conjecture By Widom With Implications To Fermionic Entanglement Entropy

TL;DR

This paper proves a special case of Widom's conjecture, deriving asymptotic behavior of certain integral operators, and shows that the entanglement entropy of a free Fermi gas grows at least proportionally to surface area times a logarithm.

Contribution

It establishes the leading and next-to-leading terms in a semi-classical expansion related to Widom's conjecture, with implications for fermionic entanglement entropy.

Findings

Proves a specific case of Widom's conjecture in asymptotic analysis.

Derives asymptotic expansion terms for trace of integral operators.

Shows entanglement entropy grows at least as fast as surface area times a logarithm.

Abstract

We prove a special case of a conjecture in asymptotic analysis by Harold Widom. More precisely, we establish the leading and next-to-leading term of a semi-classical expansion of the trace of the square of certain integral operators on the Hilbert space $L^2(\R^d)$. As already observed by Gioev and Klich, this implies that the bi-partite entanglement entropy of the free Fermi gas in its ground state grows at least as fast as the surface area of the spatially bounded part times a logarithmic enhancement.