Non-Hermitian dynamics for a two-spin system with PT symmetry

A system of interacting spins that are under the influence of spin-polarized currents can be described using a complex functional, or a non-Hermitian (NH) Hamiltonian. We study the dynamics of two exchange-coupled spins on the Bloch sphere. In the case of currents leading to $\mathcal{PT}$ symmetry, an exceptional point that survives also in the nonlinear system is identified. The nonlinear system is bistable for small currents and it exhibits stable oscillating motion or it can relax to an equilibrium point. The oscillating motion of the two spins is akin to synchronized spin-torque oscillators. For the full nonlinear system, we derive two conserved quantities that furnish a geometric description of the spin trajectories in phase space and indicate stability of the oscillating motion. Our analytical results provide tools for the description of the dynamics of NH systems that are defined on the Bloch sphere.