Time evolution of one-dimensional gapless models from a domain wall initial state: stochastic Loewner evolution continued?

We study the time evolution of quantum one-dimensional gapless systems evolving from initial states with a domain wall. We generalize the path integral imaginary time approach that together with boundary conformal field theory allows us to derive the time and space dependence of general correlation functions. The latter are explicitly obtained for the Ising universality class, and the typical behavior of one- and two-point functions is derived for the general case. Possible connections with the stochastic Loewner evolution are discussed and explicit results for one-point time dependent averages are obtained for generic. for boundary conditions corresponding to stochastic Loewner evolution. We use this set of results to predict the time evolution of the entanglement entropy and obtain the universal constant shift due to the presence of a domain wall in the initial state.