Non-Hermitian Topological Magnonics

The term becomes non-Hermitian when it interacts with the environment, as they exchange energy. The transfer of energy to the environment causes damping, such as Gilbert damping of magnons in insulators from magnon interaction. Another source of damping is spin pumping, where the angular momentum of magnons is transferred to electrons, photons, and photons in the substrate. This damping affects the coupling between two magnons, assuming there are two magnons, β one and β two, interacting with the same environment.

The environment acts as a bus with many traveling modes that can mediate the coupling between β one and β two. The coupling is not Hermitian and is called dissipative coupling, as it partially takes away energy without returning it. If the bus contains a standing mode, the energy can return, leading to coherent coupling between β one and β two. However, if the standing wave mode is coupled by sources, it can induce TCP two coupling, causing interesting non-Hermitian topological phenomena in Magnonics.

To theoretically describe the magnon subsystem when interacting with the environment, we consider two terms: HS, the spin or magnetic system of interest, and HE, the environment they interact with through coupling HSE. HSE is responsible for energy and particle exchange between the systems. The evolution of the whole system is described by the density matrix ρSE and the Liouville equation governing its dynamics.

The goal is to find an equation describing the magnon subsystem dynamics by integrating out information of the environment. By expressing the density matrix and performing the necessary integrations, we arrive at a simplified equation equivalent to the original one. Tracing out the environment leaves only information about the magnonic system.

Assuming weak coupling between magnon baths, we can simplify the equation by neglecting memory effects and considering the Markov approximation, where the density matrix of the magnonic system is a function of S with an integration over S. This simplification allows for a more manageable solution to the equation.