I will discuss a general procedure to construct an integrable real–time trotterization of interacting lattice models. As an illustrative example we will consider a spin-$1/2$ chain, with continuous time dynamics described by the isotropic ($XXX$) Heisenberg Hamiltonian. I will derive local conservation laws from an inhomogeneous transfer matrix and construct a boost operator. In the continuous time limit these local charges reduce to the known integrals of motion of the Heisenberg chain.

In a simple Kraus representation I will examine the nonequilibrium setting, where our integrable cellular automaton is driven by stochastic processes at the boundaries.
We will see, how an exact nonequilibrium steady state density matrix can be written in terms of a staggered matrix product ansatz.

This simple trotterization scheme, in particular in the open system framework, could prove to be a useful tool for experimental simulations of the lattice models in terms of trapped ion and atom optics setups.

* Lenart Zadnik, Faculty of Mathematics and Physics, University of Ljubljana