Characterizing the scaling with the total particle number ($N$) of the largest eigenvalue of the one-body density matrix (${\lambda }_0$) provides information on the occurrence of the off-diagonal long-range order (ODLRO) according to the Penrose-Onsager criterion. Setting ${\lambda }_0\sim N^{{\mathcal{C}}_0}$, then ${\mathcal{C}}_0=1$ corresponds in ODLRO. The intermediate case, $0<{\mathcal{C}}_0<1$, corresponds in translational invariant systems to the power-law decaying of (nonconnected) correlation functions and it can be seen as identifying quasi-long-range order. The goal of the present paper is to characterize the ODLRO properties encoded in ${\mathcal{C}}_0$ (and in the corresponding quantities ${\mathcal{C}}_{k\ne 0}$ for excited natural orbitals) exhibited by homogeneous interacting bosonic systems at finite temperature for different dimensions in presence of short-range repulsive potentials. We show that ${\mathcal{C}}_{k\ne 0}=0$ in the thermodynamic limit. In one dimension it is ${\mathcal{C}}_0=0$ for nonvanishing temperature, while in three dimensions it is ${\mathcal{C}}_0=1$ (${\mathcal{C}}_0=0$) for temperatures smaller (larger) than the Bose-Einstein critical temperature. We then focus our attention to $D=2$, studying the $XY$ and the Villain models, and the weakly interacting Bose gas. The universal value of ${\mathcal{C}}_0$ near the Berezinskii-Kosterlitz-Thouless temperature $T_{\text{BKT}}$ is $7/8$. The dependence of ${\mathcal{C}}_0$ on temperatures between $T=0$ (at which ${\mathcal{C}}_0=1$) and $T_{\text{BKT}}$ is studied in the different models. An estimate for the (nonperturbative) parameter $\xi$ entering the equation of state of the two-dimensional Bose gases is obtained using low-temperature expansions and compared with the Monte Carlo result. We finally discuss a “double jump” behavior for ${\mathcal{C}}_0$, and correspondingly of the anomalous dimension $\eta$, right below $T_{\text{BKT}}$ in the limit of vanishing interactions.

• Received 14 July 2020
• Revised 1 September 2020
• Accepted 1 September 2020

DOI:https://doi.org/10.1103/PhysRevB.102.184510