Citation: Lemm, Marius, and Angelo Lucia. "On the critical finite-size gap scaling for frustration-free Hamiltonians." arXiv preprint arXiv:2409.09685 (2024).
Web: https://arxiv.org/abs/2409.09685
Tags: Spin-chains
In this paper, the authors prove a quite general result about the finite-size scaling of frustration-free Hamiltonians. This bound states that if a Hamiltonian is globally gapless (i.e. it has arbitrarily low energy on global states) then it must have finite-size gap scaling faster than 1/l^2 where l is the linear size of the region (or more precisely log(l)^5/l^2). This result is optimal in light of Heisenberg antiferromagnet Hamiltonians which exchibit this sort of 1/l^2 scaling.
The original result in this field was
> Knabe, Stefan. "Energy gaps and elementary excitations for certain VBS-quantum antiferromagnets." Journal of statistical physics 52 (1988): 627-638.
where a maximum scaling for gapless sytems was proved in the case of special Heisenberg antiferromagnet Hamiltonians. A more general result was in
> Gosset, David, and Evgeny Mozgunov. "Local gap threshold for frustration-free spin systems." Journal of Mathematical Physics 57.9 (2016).
where the same result as the current paper was proved, but under the assumption that the Hamiltonian was translation invariant. A gap of 1/l^(3/2) was then shown in the frustration-free case:
> Lemm, Marius, and Evgeny Mozgunov. "Spectral gaps of frustration-free spin systems with boundary." Journal of Mathematical Physics 60.5 (2019).
this present paper is now establishing the expected 1/l^2 result in the non translation-invariant case.
People care about these results for several reason. First, they establish finite size criteria for global gaps. If you find that the gap is a certain size on finite regions, then it must be gapped on infinite regions! This finite-size gap can be solved by exact diagonalization in special cases. Another reason people case is that these results prove that chiral models cannot be realized by frustration-free models. This is because the boundary CFT is known to exhibit 1/l type scaling.