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 (lambda(0)) provides information on the occurrence of the off-diagonal long-range order (ODLRO) according to the Penrose-Onsager criterion. Setting lambda(0) similar to N-C0, then C-0 = 1 corresponds in ODLRO. The intermediate case, 0 < 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 C-0 (and in the corresponding quantities C-k not equal 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 C-k not equal 0 = 0 in the thermodynamic limit. In one dimension it is C-0 = 0 for nonvanishing temperature, while in three dimensions it is C-0 = 1 (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 C-0 near the Berezinskii-Kosterlitz-Thouless temperature TBKT is 7/8. The dependence of C-0 on temperatures between T = 0 (at which C-0 = 1) and TBKT is studied in the different models. An estimate for the (nonperturbative) parameter 4 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 C-0, and correspondingly of the anomalous dimension eta, right below T-BKT in the limit of vanishing interactions.