Area bound in one-dimensional gapped models

Consider a one-dimensional gapped model with ground-state reduced density matrix $\rho_I$ for a region $I$. Let $\Delta E$ denote the spectral gap and let $v$ denote the speed of sound appearing in the Lieb–Robinson bound. The area bound conjecture asserts that there is a function $f:\mathbb{R}^+\rightarrow\mathbb{R}^+$, equipped with further suitable properties, such that

$$S(\rho_I)\leq f\left(\frac{v}{\Delta E}\right).$$

Area bound conjecture. There exists such a function $f$ for every gapped one-dimensional model, so the entanglement entropy is bounded in terms of the ratio of the Lieb–Robinson velocity to the spectral gap. This conjecture proposes a quantitative refinement of the one-dimensional area law, which the source describes as rigorously established qualitatively for local gapped systems while leaving the stated numerical bound as a conjectural formulation.