Entanglement scaling in critical two-dimensional fermionic and bosonic systems

We relate the reduced density matrices of quadratic fermionic and bosonic models to their Green’s function matrices in a unified way and calculate the scaling of the entanglement entropy of finite systems in an infinite universe exactly. For critical fermionic two-dimensional (2D) systems at $T=0$, two regimes of scaling are identified: generically, we find a logarithmic correction to the area law with a prefactor dependence on the chemical potential that confirms earlier predictions based on the Widom conjecture. If, however, the Fermi surface of the critical system is zero-dimensional, then we find an area law with a sublogarithmic correction. For a critical bosonic 2D array of coupled oscillators at $T=0$, our results show that the entanglement entropy follows the area law without corrections.

Figure 1

The prefactor $c\left(\mu \right)$ in the entanglement entropy scaling law as a function of the chemical potential $\mu$ for the ground-state of the two-dimensional fermionic tight-binding model in comparison to the result of Gioev and Klich [18]. Insets show the hopping parameters and the Fermi surfaces for $\mu =-3,-2,-1,0$.

Figure 2

The scaling prefactor $c\left(\mu \right)$ for the ground-state of a two-dimensional fermionic tight-binding model with next-nearest-neighbor hoppings in comparison to the result of [18]. Insets show the hopping parameters and the Fermi surfaces for $\mu ∊[-0.25,-1.75]$ in the quartered Brillouin zone.

Figure 3

The upper-right inset shows the entanglement entropy per surface unit $S_{\Omega }\left(L\right)∕L$ (block of $n=L^2$ sites) for a two-dimensional fermionic tight-binding model with modulated vertical hopping (see the lower-left inset) at $T=0$. The energy gap $4(1-h)$ closes in the point $\mathbf{k}=(\pi ,0)$. The extrapolation ${\lim }_{L\to \infty }{S}_{\Omega }\left(L\right)∕L$ suggests a divergence for $h\to 1$.

Figure 4

The entanglement entropy per surface unit $S_{\Omega }\left(L\right)∕L$ (block with $n=L^2$ sites) for the bosonic two-dimensional system of coupled harmonic oscillators at $T=0$. The curves converge for small energy gaps ${\omega }_0$. The extrapolation ${\lim }_{L\to \infty }{S}_{\Omega }\left(L\right)∕L$ yields, for the critical limit, the area law $S_{\Omega }\left(L\right)\approx 0.45L$.