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 ${\lambda }_0\sim N^{\mathcal{C}}$ to define the scaling exponent $\mathcal{C}$, then $\mathcal{C}=1$ corresponds to ODLRO and $\mathcal{C}=0$ to the single-particle occupation of the density matrix orbitals. When $0<\mathcal{C}<1$, $\mathcal{C}$ can be used to quantify deviations from ODLRO. In this paper we study the exponent $\mathcal{C}$ in a variety of one-dimensional bosonic and anyonic quantum systems. For the $1D$ Lieb-Liniger Bose gas we find that for small interactions ${\cal C}$iscloseto$1$, implying a mesoscopic condensation, i.e. a value of the “condensate” fraction ${\lambda }_0\mathrm{/}N$ appreciable at finite values of $N$ (as the ones in experiments with $1D$ ultracold atoms). $1D$ anyons provide the possibility to fully interpolate between $\mathcal{C}=1$ and $0$. The behaviour of $\mathcal{C}$ for these systems is found to be non-monotonic both with respect to the coupling constant and the statistical parameter.