Determine whether further coarse-grained excitable zero-modes exist

Determine whether the one-dimensional star-lattice PXP model possesses excitable zero-modes that are invariant only under translation by two effective triangles rather than under translation by one effective triangle.

Background

The main construction groups each triangle into one effective site and therefore produces translation-invariant MPSs at the triangle level. The appendix considers a further coarse graining in which candidate excitable zero-modes would have a two-triangle unit cell.

Finite-size spectral observations provide ambiguous evidence because some candidate eigenvalues also occur for odd system sizes and may result from inert zero-mode dressing. Consequently, the existence of genuinely two-triangle-periodic excitable zero-modes is not settled.

References

However, it is possible that some EZMs that are only invariant under translation by two triangles exist.

— Efficient search for excitable zero-modes in constrained systems  (2608.31165 - Desaules, 31 Aug 2026) in Appendix, Section \ref{app:cg}, “Further coarse graining in the 1D star lattice”

As such, the $l=2$ EZMs are likely the only ones in that model.

— Efficient search for excitable zero-modes in constrained systems  (2608.31165 - Desaules, 31 Aug 2026) in Section 2.2, “Coarser graining”