Local gap threshold for frustration-free spin systems

We improve Knabe's spectral gap bound for frustration-free translation-invariant local Hamiltonians in 1D. The bound is based on a relationship between global and local gaps. The global gap is the spectral gap of a size-$m$ chain with periodic boundary conditions, while the local gap is that of a subchain of size $n<m$ with open boundary conditions. Knabe proved that if the local gap is larger than the threshold value $1/(n-1)$ for some $n>2$, then the global gap is lower bounded by a positive constant in the thermodynamic limit $m\rightarrow \infty$. Here we improve the threshold to $\frac{6}{n(n+1)}$, which is better (smaller) for all $n>3$ and which is asymptotically optimal. As a corollary we establish a surprising fact about 1D translation-invariant frustration-free systems that are gapless in the thermodynamic limit: for any such system the spectral gap of a size-$n$ chain with open boundary conditions is upper bounded as $O(n^{-2})$. This contrasts with gapless frustrated systems where the gap can be $\Theta(n^{-1})$. It also limits the extent to which the area law is violated in these frustration-free systems, since it implies that the half-chain entanglement entropy is $O(1/\sqrt{\epsilon})$ as a function of spectral gap $\epsilon$. We extend our results to frustration-free systems on a 2D square lattice.