Finite temperature off-diagonal long-range order for interacting bosons - Colcelli, Andrea; Defenu, Nicolò; Mussardo, Giuseppe; Trombettoni, Andrea (Physical review, 2020)
Characterizing the scaling with the total particle number (N) of the largest eigenvalue of the one-body density matrix (λ0) provides information on the occurrence of the off-diagonal long-range order (ODLRO) according to the Penrose-Onsager criterion. Setting λ0∼NC0, then C0=1 corresponds in ODLRO. The intermediate case, 0<C0<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 Ck≠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 Ck≠0=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 ξ 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 η, right below TBKT in the limit of vanishing interactions.