We investigate the magnonic properties of a one-dimensional antiferromagnetic chain incorporating Dzyaloshinskii-Moriya interaction (DMI) and cross damping. By analyzing the magnetization dynamics in the linear regime, we show that the system can be mapped onto an effective modified non-Hermitian Su-Schrieffer-Heeger model. This mapping enables the application of non-Hermitian band theory, including topological invariants and the generalized Brillouin zone, to characterize the system. Within this framework, we uncover rich non-Hermitian and topological magnonic phenomena induced by the interplay of DMI and dissipative coupling. In particular, the system exhibits a bipolar non-Hermitian skin effect, where distinct magnon modes accumulate at opposite boundaries, as well as topological edge states arising from asymmetric DMI across interfaces. We further show that nonreciprocal cross damping breaks the symmetry between left- and right-localized modes and can selectively suppress edge states. Importantly, the strength of cross damping provides an effective tuning knob to control both the emergence and spatial localization of skin modes and topological edge states. Since such dissipative coupling can be engineered, for example, via spin pumping, our results establish a route toward dynamic manipulation of non-Hermitian magnonic excitations and offer a platform for designing tunable topological magnonic devices.

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• Topological edge modes
• Magnetic systems
• Non-Hermitian systems