Integrable Trotterization: Local Conservation Laws and Boundary Driving

Matthieu Vanicat, Lenart Zadnik, Tomaž Prosen

Published 2017-12-01Version 1

We present 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, the local conservation laws are derived from an inhomogeneous transfer matrix, along with a boost operator. In the continuous time limit, these local charges reduce to the known integrals of motion of the Heisenberg chain. We also examine the nonequilibrium setting where our integrable cellular automaton is driven by stochastic couplings at the boundaries in terms of a simple Kraus representation. We then show explicitly how an exact non-equilibrium steady state density matrix can be written in terms of a staggered matrix product ansatz. This simple trotterization scheme, either in closed or open system framework, could prove to be a useful tool for experimental simulations of the lattice models in terms of the trapped ion and atom-optics setups.