Local gap threshold for frustration-free spin systems

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-m chain with periodic boundary conditions, while the local gap is that of a subchain of size n < m with open boundary conditions. Knabe proved that if the local gap is larger than the threshold value 1/(n − 1) for some n > 2, then the global gap is lower bounded by a positive constant in the thermodynamic limit m → ∞. Here we improve the threshold to 6 n(n+1) , which is better (smaller) for all n > 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-n chain with open boundary conditions is upper bounded as O(n −2 ). This contrasts with gapless frustrated systems where the gap can be Θ(n −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 O(1/ √ ϵ) as a function of spectral gap ϵ. We extend our results to frustration-free systems on a 2D square lattice. Published by AIP Publishing.[http://dx

Improved local spectral gap thresholds for lattices of finite size

“…Comparison to prior work: As already mentioned, our tools significantly differ from those employed in [Kna88,GM16,LM18,Lem19a]. Similar to us, the work [KL18] employs the detectability lemma and its converse to obtain the local gap threshold.…”

“…The dependence on min q t 2 q cannot be improved; although the dependence on D might not be optimal. To show this, we provide the following example adapted from [GM16]. We consider the heisenberg ferromagnet, which is a one dimensional chain of qubits with frustrationfree local hamiltonian defined by nearest-neighbour interaction 1 2 (|01 − |10 ) ( 01| − 10|).…”

“…This is often termed as the 'local gap threshold'. The additive term that captures the local gap threshold has been improved in the recent work [GM16], which shows the inequality γ + 5 t 2 −4 ≥ 5 6 γ(t). This inequality is tight up to constants, as witnessed by the Heisenberg ferromagnet (see [GM16, Section 2] for details).…”

The AKLT Model on a Hexagonal Chain is Gapped

“…Our proof is based on verifying a tailor-made finite-size criterion in the spirit of Knabe [10] by an explicit computer calculation. Recently, finitesize criteria have been further developed in related contexts [6,11,12] and here we show that this technique does say something (but not everything one wants) about the hexagonal AKLT model that had originally motivated Knabe. The present work is the first instance where such a finite-size criterion can be verified by an exact computation of the eigenvalues of a subsystem with genuine 2D features.…”

Existence of a Spectral Gap in the Affleck-Kennedy-Lieb-Tasaki Model on the Hexagonal Lattice

“…To prove the criterion, we follow the combinatorial approach pioneered by Knabe [26], strengthened by using interaction weights as in Refs. [33,34]. In step 2, we combine the rigorous analytical insight from step 1 by numerically verifying the finitesize criterion via a high-precision density-matrix renormalization group (DMRG) calculation (see also Ref.…”