Improved one-dimensional area law for frustration-free systems

Abstract

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? 1D area law and which is tight to within a polynomial factor. For particles of dimension d, spectral gap ?0, and interaction strength at most J, our entropy bound is S 1D?O(1)•X3log8X, where X=def(Jlogd)/?. 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 |2log6|?L|•X3log8X. This is at the cusp of being nontrivial in the 2D case, in the sense that any further improvement would yield a subvolume law. © 2012 American Physical Society.