Integrable Trotterization: Local Conservation Laws and Boundary Driving

We discuss a general procedure to construct an integrable real-time Trotterization of interacting lattice models. As an illustrative example, we consider a spin-$1/2$ chain, with continuous time dynamics described by the isotropic ($XXX$) Heisenberg Hamiltonian. For periodic boundary conditions, local conservation laws are derived from an inhomogeneous transfer matrix, and a boost operator is constructed. In the continuous time limit, these local charges reduce to the known integrals of motion of the Heisenberg chain. In a simple Kraus representation, we also examine the nonequilibrium setting, where our integrable cellular automaton is driven by stochastic processes at the boundaries. We show explicitly how an exact nonequilibrium steady-state density matrix can be written in terms of a staggered matrix product ansatz, and we propose quasilocal conservation laws for the model with periodic boundary conditions. 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.

Figure 1

Time evolution of the model. Each time step consists of two half steps. In the first one, we apply gates $U_{2j-2,2j-1}$, and in the second one, $U_{2j-1,2j}$. The protocol thus shifts for one site each half step.