Improved one-dimensional area law for frustration-free systems

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 Hastings\^a 1D area law and which is tight to within a polynomial factor. For particles of dimension d, spectral gap ε>0, and interaction strength at most J, our entropy bound is S1D≤O(1)·X³log⁸X, where Xdef=(Jlogd)/ε. Our proof is completely combinatorial, combining the detectability lemma with basic tools from approximation theory. In higher dimensions, when the bipartitioning area is |∂L|, we use additional local structure in the proof and show that $S{≤}O(1)·{}{|{∂}L|}²{log}⁶|{∂}L|·{}{X}³{log}⁸X$. This is at the cusp of being nontrivial in the 2D case, in the sense that any further improvement would yield a subvolume law.