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

Characterizing the scaling with the total particle number (N) of the largest eigenvalue of the one-body density matrix (++0) provides information on the occurrence of the off-diagonal long-range order (ODLRO) according to the Penrose-Onsager criterion. Setting ++0Gê+NC0, then C0=1 corresponds in ODLRO. The intermediate case, 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 C0 (and in the corresponding quantities CkGëá0 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 CkGëá0=0 in the thermodynamic limit. In one dimension it is C0=0 for nonvanishing temperature, while in three dimensions it is C0=1 (C0=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 C0 near the Berezinskii-Kosterlitz-Thouless temperature TBKT is 7/8. The dependence of C0 on temperatures between T=0 (at which C0=1) and TBKT is studied in the different models. An estimate for the (nonperturbative) parameter +¦ 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 C0, and correspondingly of the anomalous dimension ++, right below TBKT in the limit of vanishing interactions.