Operator space entanglement entropy in XY spin chains

The complexity of representation of operators in quantum mechanics can be characterized by the operator space entanglement entropy (OSEE). We show that in the homogeneous Heisenberg XY spin 1/2 chains the OSEE for initial local operators grows at most logarithmically with time. The prefactor in front of the logarithm generally depends only on the number of stationary points of the quasiparticle dispersion relation and for the XY model changes from 1/3 to 2/3 exactly at the point of quantum phase transition to long-range magnetic correlations in the nonequilibrium steady state. In addition, we show that the presence of a small disorder triggers a saturation of the OSEE.