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Here the notation hides a polylogarithmic function of which is present in the entanglement entropy bound from Ref. 2.
Here we consider a quantum many-body system described by a sequence of Hamiltonians {Hn} indexed by system size n. The system is gapped if the spectral gap of Hn is lower bounded by a positive constant independent of n; otherwise it is gapless.
If Hi is not a projector, we may replace it with the projector Πi orthogonal to its null space. It is not hard to see that the new Hamiltonian has the same zero energy ground space as H and that the spectral gaps ϵ and ϵ′ of these Hamiltonians satisfy ′ ≤ ϵϵ′, where a lower bounds the smallest non zero eigenvalue of each term Hi.
In other words, for x ∈ ℝ2 we have Te1e2(x) = x + e2e1 (here e2, e1 are identified with the coordinates of their midpoints).
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Note that G(n) is invariant under a rescaling cjacj, so we are free to choose the normalization so that . To find the optimum of G(n) we minimize the numerator of (28) subject to this constraint. The solution is equal to (29) (up to an irrelevant rescaling).
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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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