Area law for the entropy of low-energy states

It is often observed in the ground state of quantum lattice systems with local interactions that the entropy of a large region is proportional to its surface area. In some cases, this area law is corrected with a logarithmic factor. This contrasts with the fact that in almost all states of the Hilbert space, the entropy of a region is proportional to its volume. This paper shows that low-energy states have (at most) an area law with the logarithmic correction, provided two conditions hold: (i) the state has sufficient decay of correlations and (ii) the number of eigenstates with vanishing energy density is not exponential in the volume. These two conditions are satisfied by many relevant systems. The central idea of the argument is that energy fluctuations inside a region can be observed by measuring the exterior and a superficial shell of the region.

Figure 1

(Color online) $\mathcal{R}$ is the chosen region where the entropy is estimated; the sites belonging to its boundary $\partial \mathcal{R}$ are darker; $\mathcal{S}$ and ${\mathcal{S}}^{\prime }$ are two superficial shells with thickness $l$ outside $\mathcal{R}$; $\mathcal{X}=\mathcal{S}\cup {\mathcal{S}}^{\prime }\cup \mathcal{R}$ is the extended region; $\overline{\mathcal{X}}$ is the exterior of $\mathcal{X}$.