Deviations from Off-Diagonal Long-Range Order in One-Dimensional Quantum Systems

A quantum system exhibits off-diagonal long-range order (ODLRO) when the largest eigenvalue $\lambda_0$ of the one-body-density matrix scales as $\lambda_0 \sim N$, where $N$ is the total number of particles. Putting $\lambda_0 \sim N^{{\cal C}}$ to define the scaling exponent ${\cal C}$, then ${\cal C}=1$ corresponds to ODLRO and ${\cal C}=0$ to the single-particle occupation of the density matrix orbitals. When $0<{\cal C}<1$, ${\cal C}$ can be used to quantify deviations from ODLRO. In this paper we study the exponent ${\cal C}$ in a variety of one-dimensional bosonic and anyonic quantum systems. For the $1D$ Lieb-Liniger Bose gas we find that for small interactions ${\cal C}$ is close to $1$, implying a mesoscopic condensation, i.e. a value of the "condensate" fraction $\lambda_0/N$ appreciable at finite values of $N$ (as the ones in experiments with $1D$ ultracold atoms). $1D$ anyons provide the possibility to fully interpolate between ${\cal C}=1$ and $0$. The behaviour of ${\cal C}$ for these systems is found to be non-monotonic both with respect to the coupling constant and the statistical parameter.