I will discuss a class of very simple integrable dynamics on a discrete space-time lattice, which is generated by a 2-site matrix-valued rational map.
The phase spaces of the matrix variable can be selected from diverse families of symmetric spaces, e.g. complex Grassmannians, and are equipped with a natural symplectic structure.
This precise form of the map follows from a simple consistency condition for a parallel transport (aka Lax zero curvature condition) on a space-time lattice using
a minimalistic Lax operator, which is linear in the spectral and matrix variables. I will discuss the Yang-Baxter property and conservation laws of these maps.
Physically, the model represents an integrable discretization and SU(N) generalization of Landau-Lifshitz magnet. Using numerical computations, we have demonstrated that the
transport of Noether charges follows Kardar-Parisi-Zhang universality with superdiffusive dynamical exponent 3/2. Most interesting open questions will be discussed.