Exact Results for a Quantum Many-Body Problem in One Dimension

We investigate exactly a system of either fermions or bosons interacting in one dimension by a two-body potential $V\left(r\right)=\frac{g}{r^2}$ with periodic boundary conditions. In addition to rederiving known results for correlation functions and thermodynamics in the thermodynamic limit, we present expressions for the one-particle density matrix at zero temperature and particular (nontrivial) values of the coupling constant $g$, as a determinant of order $N\times N$. These concise expressions allow a discussion of the momentum distribution in the thermodynamic limit. In particular, for a case of repulsive bosons, the determinant is evaluated explicitly, exhibiting a weak (logarithmic) singularity at zero momentum, and vanishing outside of a "Fermi" surface.