Frustrated one-dimensional Hamiltonians saturating the inverse-gap area-law bound

Construct a family of frustrated one-dimensional Hamiltonians with spectral gap \(\Theta(n^{-1})\), a unique ground state, and ground-state entanglement entropy \(\Theta(n)\), thereby establishing the optimality of the inverse-gap area-law scaling for general one-dimensional Hamiltonians.

Background

The one-dimensional area law gives an entanglement upper bound proportional to the inverse spectral gap, while the paper’s construction achieves the conjectured square-root inverse-gap scaling in the frustration-free setting. The explicitly stated conjecture concerns frustrated Hamiltonians and asks whether a gap of order $1/n$ can coexist with volume-law entanglement of order nn. The paper does not resolve this conjecture; its main Hamiltonian construction is frustration-free and has gap Θ(n−2)\Theta(n^{-2}).

References

This bound was improved in to S \lesssim 1/\Delta, which was conjectured to be optimal since the correlation length scales as \lesssim 1/\Delta : \begin{conjecture}\label{conj:area} There exists a family of frustrated 1D Hamiltonians {H_n}_{n} with gap \Theta(n{-1}) and a unique ground state with entanglement entropy \Theta(n). \end{conjecture}

— Depth-1 expanders on the unitary group and applications  (2609.01605 - Anshu et al., 1 Sep 2026) in Conjecture 1, Section 1, Introduction; revisited in the Outlook