Statistics Dependence of the Entanglement Entropy

The entanglement entropy of a distinguished region of a quantum many-body system reflects the entanglement in its pure ground state. Here we establish scaling laws for this entanglement in critical quasifree fermionic and bosonic lattice systems, without resorting to numerical means. We consider the setting of $D$-dimensional half-spaces which allows us to exploit a connection to the one-dimensional case. Intriguingly, we find a difference in the scaling properties depending on whether the system is bosonic—where an area law is proven to hold—or fermionic where we determine a logarithmic correction to the area law, which depends on the topology of the Fermi surface. We find Lifshitz quantum phase transitions accompanied with a nonanalyticity in the prefactor of the leading order term.

Figure 1

Spectra of critical nearest-neighbor Hamiltonians (top: bosonic; bottom: fermionic) in one and two dimensions. The spectrum of the individual decoupled chains is given by the spectrum of the full Hamiltonian along the first coordinate ${\phi }_1$. The topology of the set of solutions to ${\lambda }_{\mathit{\phi }}=0$—the Fermi surface in the fermionic case—provides an intuition as to why the scaling behavior of entanglement is different for fermions then for bosons in $D>1$; see text.