Asymmetric couplings for controlling edge states in a non-Hermitian hexamer model

• The topological properties of one-dimensional lattices have attracted significant attention, particularly regarding the limitations of the trimer Su-Schrieffer-Heeger (SSH3) model. Previous studies have shown that the SSH3 model lacks conventional chiral symmetry, which often renders the standard winding number ill-defined, necessitating reliance on complex sublattice invariants or specific point chiral symmetry. In this work, we address this limitation by extending the unit cell to a non-Hermitian hexamer model. By introducing two independent asymmetric coupling terms ( g and γ ), we inherently restore the full chiral symmetry of the system. This restoration enables the topological invariant to be defined by the conventional winding number, while the additional internal degrees of freedom also yield richer topological structures and tuning capabilities compared to trimer or tetramer models. • We find that the non-Hermitian SSH6 model hosts two distinct types of edge states (type-I and type-II), which possess different eigenvalues and are localized on opposite sides of the system. We observe that type-I edge states are highly sensitive to the two asymmetric couplings: when γ > g , their amplitude increases with increasing inter-cell hopping strength, whereas when γ < g , their amplitude decreases. In contrast, type-II edge states remain robust. Therefore, when one of the asymmetric coupling coefficients is gradually increased, the type-I edge states are destroyed first. The type-II edge states are destroyed only when the asymmetric coupling coefficient increases to a certain extent. This distinct behavior allows for the selective removal of type-I states while preserving type-II states by tuning the asymmetry. • Furthermore, the type-II edge states exhibit a unique property driven by the competition between the non-Hermitian skin effect and topological localization. The evolution process consists of three stages: first, the localization at the right boundary gradually weakens. Subsequently, these states lose their boundary characteristics and extend across the entire system. Finally, under strong asymmetric coupling, the non-Hermitian skin effect drives them to relocalize to the left side of the system. Notably, their eigenvalues ( ± t 2 ) remain invariant throughout this entire process. • In particular, these non-Hermitian couplings do not merely modify the edge states. We find that the interaction between g and γ can drive the system to undergo a topological phase transition from a non-trivial phase to a trivial phase. This interplay expands the phase diagram, as the region of the trivial phase enlarges as the difference between the two asymmetric coupling coefficients increases. This indicates that the additional degrees of freedom in the hexamer configuration play a significant role in this process, a feature not typically available in the single-parameter non-Hermitian trimer model. Our results show that these edge states remain robust against local perturbations ( α ) up to the limit of the inter-cell hopping strength. The hexamer configuration, with its additional lattice sites, allows for richer topological structures and tuning capabilities, which can yield unique phenomena beyond what is possible in trimer or tetramer models. By introducing two asymmetric coupling terms, we have extended the SSH3 model to a non-Hermitian SSH6 system and endowed it with complete chiral symmetry. We find that the non-Hermitian SSH6 model hosts two distinct types of edge states (type-I and type-II), which have different eigenvalues and are each localized on one side of the system. Results show that non-Hermiticity can modify the eigenvalue of the type-I edge states. When the two asymmetric coupling coefficients differ, the type-I edge states transform into special edge states. The eigenvalue of these special states gradually changes with increasing intercell hopping strength. Notably, these special edge states, along with the zero-energy mode and bulk states, all localize on the left side of the system, but exhibit different probability density distributions. The type-II edge states are localized on the right side of the system and are relatively stable, but they will be destroyed when the asymmetric coupling coefficient is large and transfer from the right boundary to the left boundary of the system. Meanwhile, we find that it is possible to selectively destroy type-I edge states while retaining type-II edge states. Both types of edge states exhibit considerable robustness against local perturbations. Furthermore, when the two asymmetric coupling coefficients differ, a topologically trivial phase is induced, and the region of the trivial phase expands as the difference between the two asymmetric coupling coefficients increases.