On the Symplectic Propagation of the Spin-MInt Algorithm for Non-Adiabatic Quantum Dynamics
Mapping methods are often used for the numerical simulation of nonadiabatic systems by propagating classical mapping variable trajectories. A recently popularised mapping method is spin-mapping, whose mapping variables arise from quantum mechanical operators with symmetries described by a Lie-Poisson algebra. Simulating the classical-like dynamics of spin-mapping systems accurately is generally challenging, with many methods unable to preserve the underlying geometric structure of the symplectic form. The Spin-MInt algorithm is a recently proposed algorithm propagating spin-mapping variables, with a direct proof of symplecticity existing only for 2 electronic states. Here, we directly prove the symplecticity of the Spin-MInt algorithm for a general $K$ electronic states. A review of the symplectic nature of coadjoint orbits of the $\mathfrak{su}(K)$ Lie-Poisson algebra provides the framework needed to understand symplecticity of the Spin-MInt algorithm in this general case. The symplecticity of the method on the associated coadjoint orbit is then shown for what we believe to be the first time via an explicit verification of the symplecticity condition $\mathbf{MJ}\mathbf{M}^\textrm{T}=\mathbf{J}$ exploiting the Lie-Poisson structure of the system. To our knowledge, this is the first time the monodromy matrix for the Spin-MInt algorithm has been explicitly stated using canonical coordinates on the coherent state manifold for a general number of states. We hope that this will assist the development of classical-like spin-mapping methods which might utilise elements of the monodromy matrix, and inform future work on similar symplectic algorithms for coupled and uncoupled Lie-Poisson systems.