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A quantum system exhibits off-diagonal long-range order (ODLRO) when thelargest 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-particleoccupation of the density matrix orbitals. When $0<{\cal C}<1$, ${\cal C}$ canbe 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 ${\calC}$ 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 theones in experiments with $1D$ ultracold atoms). $1D$ anyons provide thepossibility 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 tothe coupling constant and the statistical parameter.

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