We present a new proof for the 1D area law for frustration-free systems with a constant gap, which exponentially improves the entropy bound in Hastingsâ 1D area law and which is tight to within a polynomial factor. For particles of dimension $d$, spectral gap $\epsilon >0$, and interaction strength at most $J$, our entropy bound is $S_{1D}\le \mathcal{O}\left(1\right)·X^3{\log }^{8}X$, where $X\stackrel{\mathrm{def}}{=}\left(J\log d\right)/\epsilon$. Our proof is completely combinatorial, combining the detectability lemma with basic tools from approximation theory. In higher dimensions, when the bipartitioning area is $\left|\partial L\right|$, we use additional local structure in the proof and show that $S\le \mathcal{O}\left(1\right)·{\left|\partial L\right|}^2{\log }^6\left|\partial L\right|·X^3{\log }^{8}X$. This is at the cusp of being nontrivial in the 2D case, in the sense that any further improvement would yield a subvolume law.

• Received 12 November 2011

DOI:https://doi.org/10.1103/PhysRevB.85.195145

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Published by the American Physical Society