Quadratic finite-size gap closing in periodic frustration-free systems

Establish whether the spectral gap of a gapless frustration-free quantum system on a periodic chain of length L closes asymptotically as 1/L^2, thereby determining whether the finite-periodic-system analogue of the rigorous open-subchain bound holds.

Background

The paper discusses gapless frustration-free systems as a mechanism permitting spontaneous continuous-symmetry breaking in one spatial dimension. For open-boundary subchains, rigorous upper bounds of order O(1/ell2) are known for the energy gap. The corresponding scaling for a periodic system is presented as conjectural and is connected to the anomalously soft, quadratically dispersing excitations that evade the assumptions of the standard no-go argument.

References

and the analogous $1/L2$ closing of the gap of the full periodic system of size $L$ has been conjectured.

— Absence of nontrivial local conserved quantities in a class of $U(1)$-symmetric spin-1 chains  (2608.17548 - Sengoku et al., 18 Aug 2026) in Section 1, Introduction, paragraph discussing the exception to Coleman’s theorem