Improved local spectral gap thresholds for lattices of finite size

Abstract

Knabe's theorem lower bounds the spectral gap of a one dimensional frustration-free local hamiltonian in terms of the local spectral gaps of finite regions. It also provides a local spectral gap threshold for hamiltonians that are gapless in the thermodynamic limit, showing that the local spectral gap much scale inverse linearly with the length of the region for such systems. Recent works have further improved upon this threshold, tightening it in the one dimensional case and extending it to higher dimensions. Here, we show a local spectral gap threshold for frustration-free hamiltonians on a finite dimensional lattice, that is optimal up to a constant factor that depends on the dimension of the lattice. Our proof is based on the detectability lemma framework and uses the notion of coarse-grained hamiltonian (introduced in [Phys. Rev. B 93, 205142]) as a link connecting the (global) spectral gap and the local spectral gap.

Quantitatively improved finite-size criteria for spectral gaps

“…The weighting scheme used in [9] works well in the two-dimensional setting considered there, but has so far resisted extension to dimensions > 2 for technical reasons. By contrast, [3] proceeds completely differently in higher dimensions via an ingenious use of gap amplification and the detectability lemma from quantum information theory. Until now, [3] was the only proof of the optimal threshold scaling t ℓ ∼ ℓ −2 in higher dimensions.…”

“…By contrast, [3] proceeds completely differently in higher dimensions via an ingenious use of gap amplification and the detectability lemma from quantum information theory. Until now, [3] was the only proof of the optimal threshold scaling t ℓ ∼ ℓ −2 in higher dimensions. A disadvantage of the detectability lemma approach is that it comes at the price of relatively large constants which can create difficulties in concrete applications where quantitative values matter.…”

“…Starting with the work of Gosset-Mozgunov [9], finite-size criteria have seen considerable progress in the last years, especially in the context of higher-dimensional quantum spin systems. This progress concerns both the methodological side [3,15,16] and the applications side [1,13,14,17,18,21,22,23,25]. Recent methodological advances have mostly focused on the scaling of the gap threshold (called t ℓ in Eq.…”

“…Our first main result concerns the gap thresholds on the Euclidean lattice in any dimension and is formally stated in Theorem 2.2 below. Our main contribution is a new weighting scheme which has the benefit of adapting seamlessly to dimensions ≥ 3 where deriving the optimal threshold scaling ℓ −2 has so far been inaccessible to the weighting approach of [9] and instead required sophisticated tools form quantum information theory [3]. The heuristic idea underlying our weighting scheme is that tensor-like product structures allow for computationally efficient and dimensionally robust encodings of the weighting scheme, while at the same time allowing to effectively implement the principle that the weight of an interaction term should increase with its distance to the boundary of the subsystem.…”

Entanglement subvolume law for 2d frustration-free spin systems

“…An argument adapted from Refs [29,4]. enables a further restriction to square-shaped regions if desired.…”