Multigap topology engineering in non-Hermitian superlattices

Non-Hermiticity incorporating gain and loss allows new symmetries, enriching the fundamental topological classification. While various non-Hermitian topological phases have been uncovered, the interplay between multigap topology and non-Hermiticity remains to be explored. Here we investigate the topological states of a multiband tetramerized superlattice by introducing three distinct gain-loss patterns: $(ig,$ $\ensuremath{-}ig,$ $ig,$ $\ensuremath{-}ig)$, $(ig,$ $ig,$ $\ensuremath{-}ig,$ $\ensuremath{-}ig)$, and $(ig,$ $\ensuremath{-}ig,$ $\ensuremath{-}ig,$ $ig)$. We uncover the unique effects of each configuration on the multigap topological edge states. Specifically, we find that the non-Hermiticity in the first configuration does not affect the multigap topological phases. Interestingly, it leads to a multiple reemergence behavior of the midgap defect state. For the second one, the gain-loss induces a topological phase transition in the midgap from nontrivial to trivial. Additionally, it results in the emergence of trivial edge states when the nonmidgap is initially trivial. The non-Hermiticity in the third one drives a topological phase transition in the midgap from trivial to nontrivial, while simultaneously destroying the localized topological edge states in the nonmidgap. Our results show a rich synergy between non-Hermiticity and multigap topology, introducing avenues for controlling multigap topological edge modes, which may have potential applications in multifrequency topological devices.