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/content/aip/journal/jmp/57/9/10.1063/1.4962337
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/content/aip/journal/jmp/57/9/10.1063/1.4962337

/content/aip/journal/jmp/57/9/10.1063/1.4962337
2016-09-15
2016-10-24

### Abstract

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- chain with periodic boundary conditions, while the local gap is that of a subchain of size < with open boundary conditions. Knabe proved that if the local gap is larger than the threshold value 1/( − 1) for some > 2, then the global gap is lower bounded by a positive constant in the thermodynamic limit → ∞. Here we improve the threshold to , which is better (smaller) for all > 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- chain with open boundary conditions is upper bounded as ( −2). This contrasts with gapless frustrated systems where the gap can be Θ( −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 as a function of spectral gap . We extend our results to frustration-free systems on a 2D square lattice.

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