The 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, 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.