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 (${\lambda }_0$), provides informations 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 to ODLRO. The intermediate case, $0<{\mathcal{C}}_0<1$, corresponds for translational invariant systems to the power-law decaying of (non-connected) 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. We show that ${\mathcal{C}}_{k\ne 0}=0$ in the thermodynamic limit. In $1D$ it is ${\mathcal{C}}_0=0$ for non-vanishing temperature, while in $3D$ ${\mathcal{C}}_0=1$ (${\mathcal{C}}_0=0$) 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 $T_{BKT}$ is $7/8$. The dependence of ${\mathcal{C}}_0$ on temperatures between $T=0$ (at which ${\mathcal{C}}_0=1$) and $T_{BKT}$ is studied in the different models. An estimate for the (non-perturbative) parameter $\xi$ entering the equation of state of the $2D$ Bose gases, is obtained using low temperature expansions and compared with the Monte Carlo result. We finally discuss a double jump behaviour for ${\mathcal{C}}_0$, and correspondingly of the anomalous dimension $\eta$, right below $T_{BKT}$ in the limit of vanishing interactions.

13 pages, 6 figures

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