How Much Delocalisation is Needed for an Enhanced Area Law of the Entanglement Entropy?

Abstract

We consider a multi-dimensional continuum Schrödinger operator which is given by a perturbation of the negative Laplacian by a compactly supported potential. We establish both an upper and a lower bound on the bipartite entanglement entropy of the ground state of the corresponding quasi-free Fermi gas. The bounds prove that the scaling behaviour of the entanglement entropy remains a logarithmically enhanced area law as in the unperturbed case of the free Fermi gas. The central idea for the upper bound is to use a limiting absorption principle for such kinds of Schrödinger operators.

Stability of the Enhanced Area Law of the Entanglement Entropy

“…It should be pointed out that spectral localisation alone is not sufficient for the validity of an area law. This has been recently demonstrated [24] for the random dimer model if the Fermi energy coincides with one of the critical energies where the localisation length diverges and dynamical delocalisation takes over.…”

“…Entanglement properties of the ground state of quasi-free Fermi gases have received considerable attention over the last two decades, see, for example, [1,2,[8][9][10]15,16,18,[20][21][22]24,25,27,37]. Here, entanglement is understood with respect to a spatial bipartition of the system into a subsystem of linear size proportional to L and the complement.…”

Entanglement Entropy Bounds in the Higher Spin XXZ Chain

“…While an area law is generally considered as an indicator of MBL, many delocalized systems seem to exhibit a logarithmic correction, which means that the EE scales like the logarithm of the subchain's length. See the results in [4,9,12,13,15,16,18,23], where such logarithmic corrections to an area for the EE have been obtained.…”

A numerical study of entanglement measures of clusters joined by a point contact

“…That is, there is an anomalous ln L, which is known to occur in gapless systems in 1-d and non interacting fermions even in higher dimensions [18]. These logarithms can also occur even when disorder is present, for an itinerant model see [19], [20]. However, < l S > has no such anomalous ln L factor and its numerical value is closer to S 2 [13].…”