Topology of isofrequency surfaces plays a crucial role in characterizing the interaction of an electromagnetic wave with a medium. Thus, engineering the topology in complex media is leading to novel applications, ranging from super-resolution microscopy with hyperbolic metamaterials to sub-wavelength waveguiding structures. Here, we investigate the spin-governed nature of isofrequency surfaces in a general gyromagnetic medium. We show that gyrotropy also plays an important role in the topological properties of a medium, along with the anisotropic permeability and permittivity. Even though the topology primarily depends on permeability, gyrotropy can suppress or support the existence of certain topological surfaces. We reveal the connection between the gyrotropy imposed constraints and the photonic spin-profile of the topological surfaces. The spin-profile along the isofrequency surface is locked to the material, resulting in the non-reciprocity and breaking of the spin-momentum locking in the gyromagnetic medium. Further, we show that the conflict between spin-momentum locking and material locked spin leads to asymmetric mode profile and gyrotropy-induced cutoff in guided wave structures. Our work provides important insights into the underlying link between topology, spin, and non-reciprocity in gyrotropic media.
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