Subarea Law of Entanglement in Nodal Fermionic Systems

We investigate the subarea-law scaling behavior of the block entropy in bipartite fermionic systems which do not have a finite Fermi surface. It is found that in gapped regimes the leading subarea term is a negative constant, whereas in critical regimes with point nodes the leading subarea law is a logarithmic additive term. At the phase boundary that separates the critical and noncritical regimes, the subarea scaling shows power-law behavior.

Figure 1

Scaling behavior of the entanglement entropy $S_L$ for a cut through the phase diagram at fixed interaction potential $\gamma =1.0$. The chemical potential is varied in the window $\lambda \in [0.5,3.0]$, thus crossing through phases II and III which do not have finite Fermi surfaces. The parameters shown in the legend correspond to ($\lambda$, $\gamma$). The inset shows the phase diagram of the 2D spinless fermion model [Eq. (1)].

Figure 2

(a) Scaling behavior of $S_L^{\mathrm{Sub}}$ in phase II. (b) Scaling behavior of $S_L^{\mathrm{Sub}}$ in phase III. (c) The relation between the constant term const and $\frac{1}{\sqrt{\lambda -2}}$ for fixed finite interaction potential $\gamma =0.5$, $\gamma =1.0$, $\gamma =2.0$ in phase III. (d) Dependence of $\frac{S_L^{\mathrm{Sub}}}{\sqrt{L}}$ vs $L(\lambda -2)$ with the fixed interaction potential $\gamma =1$ in phase III.

Figure 3

(a) Semilog ($Y/\log X$) graph and (b) Log-log graph of the scaling behavior of $S_L^{\mathrm{Sub}}$ in phase II and its boundary points. (c) Scaling behavior of $\frac{d^2{S}_L}{d{L}^2}$ in phase II and its boundary points.

Figure 4

Comparison of $S_L^{\mathrm{Sub}}$ for two different partitions. (a) The phase II: $\lambda =1.0$, $\gamma =1.0$. (b) The boundary regime : $\lambda =2.0$, $\gamma =1.0$ (c) The phase III: $\lambda =3.0$, $\gamma =1.0$.