Large-scale behaviour of local and entanglement entropy of the free Fermi gas at any temperature In memory of Enrico Fermi (1901-1954) on the occasion of the 90 th birthday of the ideal Fermi gas

• Published in: Journal of physics. A, Mathematical and theoretical Vol. 49; no. 30
• Main Authors: Leschke, Hajo, Sobolev, Alexander V, Spitzer, Wolfgang
• Format: Journal Article
• Language: English
• Published: IOP Publishing 16.06.2016
• Subjects: entanglement entropy; free Fermi gas in equilibrium; local entropy; semiclassical trace formulas
• ISSN: 1751-8113; 1751-8121
• DOI: 10.1088/1751-8113/49/30/30LT04

The leading asymptotic large-scale behaviour of the spatially bipartite entanglement entropy (EE) of the free Fermi gas infinitely extended in multidimensional Euclidean space at zero absolute temperature, T = 0, is by now well understood. Here, we present and discuss the first rigorous results for the corresponding EE of thermal equilibrium states at T > 0 . The leading large-scale term of this thermal EE turns out to be twice the first-order finite-size correction to the infinite-volume thermal entropy (density). Not surprisingly, this correction is just the thermal entropy on the interface of the bipartition. However, it is given by a rather complicated integral derived from a semiclassical trace formula for a certain operator on the underlying one-particle Hilbert space. But in the zero-temperature limit T ↓ 0 , the leading large-scale term of the thermal EE considerably simplifies and displays a ln ( 1 / T ) -singularity which one may identify with the known logarithmic enhancement at T = 0 of the so-called area-law scaling.