Variational matrix product ansatz for nonuniform dynamics in the thermodynamic limit

We describe how to implement the time-dependent variational principle for matrix product states in the thermodynamic limit for nonuniform lattice systems. This is achieved by confining the nonuniformity to a (dynamically expandable) finite region with fixed boundary conditions. The suppression of nonphysical quasiparticle reflections from the boundary of the nonuniform region is also discussed. Using this algorithm we study the dynamics of localized excitations in infinite systems, which we illustrate in the case of the spin-1 antiferromagnetic Heisenberg model and the φ⁴ model.