Finite temperature off-diagonal long-range order for interacting bosons

Abstract

Characterizing the scaling with the total particle number ($N$) of the largest eigenvalue of the one-body density matrix (${\ensuremath{\lambda}}_{0}$) provides information on the occurrence of the off-diagonal long-range order (ODLRO) according to the Penrose-Onsager criterion. Setting ${\ensuremath{\lambda}}_{0}\ensuremath{\sim}{N}^{{\mathcal{C}}_{0}}$, 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 ${\mathcal{C}}_{k\ensuremath{\ne}0}$ 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 ${\mathcal{C}}_{k\ensuremath{\ne}0}=0$ 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 $\ensuremath{\xi}$ 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 $\ensuremath{\eta}$, right below ${T}_{\text{BKT}}$ in the limit of vanishing interactions.