Improved area law for frustration-free one-dimensional Hamiltonians

Prove an improved entanglement-area-law bound for frustration-free one-dimensional Hamiltonians that matches the scaling \(S=\Theta(\Delta^{-1/2})\) exhibited by the paper’s construction.

Background

The paper constructs frustration-free one-dimensional Hamiltonians whose unique ground states have entanglement entropy Θ(n)\Theta(n) and spectral gap Θ(n−2)\Theta(n^{-2}), demonstrating the scaling S=Θ(Δ−1/2)S=\Theta(\Delta^{-1/2}). It proves a partial upper bound of approximately O~(Δloc−3/4)\widetilde O(\Delta_{\mathrm{loc}}^{-3/4}) in terms of the local gap. The authors explicitly leave open the sharper bound that would be saturated by their example.

References

Can we prove an improved area law bound which this example saturates? This question remains open, but in \cref{append:localgapscaling}, we prove that S \leq \tilde{O}(1/\Delta_{\mathrm{loc}{3/4}) for a frustration-free Hamiltonian, where \Delta_{\mathrm{loc} is the local gap of the 1D system.

— Depth-1 expanders on the unitary group and applications  (2609.01605 - Anshu et al., 1 Sep 2026) in Section 1, Introduction, paragraph “Improved gap, entanglement, and correlation bounds”