Large-Distance and Long-Time Asymptotic Behavior of the Reduced Density Matrix in the Non-Linear Schrodinger Model

Starting from the form factor expansion in finite volume, we derive the multidimensional generalization of the so-called Natte series for the time- and distance-dependent reduced density matrix at zero temperature in the non-linear Schrodinger model. This representation allows one to read-off straightforwardly the long-time/large-distance asymptotic behaviour of this correlator. This method of analysis reduces the complexity of the computation of the asymptotic behaviour of correlation functions in the so-called interacting integrable models, to the one appearing in free-fermion equivalent models. We compute explicitly the first few terms appearing in the asymptotic expansion. Part of these terms stems from excitations lying away from the Fermi boundary, and hence go beyond what can be obtained using the CFT/Luttinger liquid-based predictions.