Scaling behavior of entanglement in two- and three-dimensional free-fermion systems

Exactly solving a spinless fermionic system in two and three dimensions, we investigate the scaling behavior of the block entropy in critical and noncritical phases. The scaling of the block entropy crucially depends on the nature of the excitation spectrum of the system and on the topology of the Fermi surface. Noticeably, in the critical phases the scaling violates the area law and acquires a logarithmic correction only when a well-defined Fermi surface exists in the system. When the area law is violated, we accurately verify a conjecture for the prefactor of the logarithmic correction, proposed by D. Gioev and I. Klich [Phys. Rev. Lett. 96, 100503 (2006)].

Figure 1

(Color online) Phase diagram of the model, Eq. (2), for the case $d=2$. The roman numbers for the various phases are explained in the text. Representative contour plots of the dispersion relation ${\Lambda }_k$ are also shown. There the black areas corresponds to ${\Lambda }_k=0$ and the white areas to the top of the band.

Figure 2

Scaling of the block entropy $S_L$ in $d=2$ for $\gamma =0$ (left panel) and $\lambda =0$ (right panel). The solid lines correspond to fits according to the formula $S_L=\frac{C}{3}L{\log }_2\left(L\right)+BL+A$.

Figure 3

$\lambda$ dependence of the $C$ coefficient in Eq. (1) in $d=2$ and $d=3$. The values extracted from fits to our numerical data are compared with the predictions of Ref. 16. In $d=2$, the exact form of $C\left(\lambda \right)$ can be obtained, which is equal to $\frac{2}{\pi }{\cos }^{-1}(\lambda -1)$.

Figure 4

(Color online) Scaling of the block entropy $S_L$ in $d=2$ for $\gamma =1$ (left panel) and $\lambda =1$ (right panel).