Single-particle density matrix and the momentum distribution of dark 'solitons' in a Tonks-Girardeau gas

We study the reduced single-particle density matrix (RSPDM), the momentum distribution, and natural orbitals and their occupancies of dark ''soliton'' (DS) states in a Tonks-Girardeau gas. DS states are specially tailored excited many-body eigenstates, which have a dark solitonic notch in their single-particle density. The momentum distribution of DS states has a characteristic shape with two sharp spikes. We find that the two spikes arise due to the high degree of correlation observed within the RSPDM between the mirror points (x and -x) with respect to the dark notch at x=0; the correlations oscillate rather than decay as the points x and -x are being separated