Entanglement Entropy and the Fermi Surface

Free fermions with a finite Fermi surface are known to exhibit an anomalously large entanglement entropy. The leading contribution to the entanglement entropy of a region of linear size $L$ in $d$ spatial dimensions is $S\sim L^{d-1}\log L$, a result that should be contrasted with the usual boundary law $S\sim L^{d-1}$. This term depends only on the geometry of the Fermi surface and on the boundary of the region in question. I give an intuitive account of this anomalous scaling based on a low energy description of the Fermi surface as a collection of one-dimensional gapless modes. Using this picture, I predict a violation of the boundary law in a number of other strongly correlated systems.

Figure 1

A sketch of the two model systems considered. Box (a) shows the Fermi sea (in gray) of the strongly anisotropic model. The relative orientation of the Fermi velocity and the chosen real space region (in black) is shown. Box (b) shows the Fermi sea (in gray) at half filling for a fermion hopping on a square lattice. Again, the relative orientation of the Fermi velocity and the chosen real space region (in black) is shown.

Figure 2

Box (a) shows a spherical real space region of linear size $L$ along with a chosen real space patch on the boundary. The size of the patch (bold black bar) has been exaggerated for clarity. Box (b) shows a local piece of the Fermi sea with filled states in gray. The dots are an exaggerated representation of the effective mode quantization coming from the real space patch. The relative orientation of the Fermi velocity $\propto n_k$ and the real space patch normal $n_x$ is the origin of the “flux factor” $|n_x·n_k|$ counting the effective number of modes that propagate perpendicular to the real space patch normal $n_x$. Here the angle between $n_x$ and $n_k$ is $\pi /4$.