Relations between the existence of ODLRO in the reduced density matrixρ
n
(for alln≧1) and that of superconducting currents in a hollow cylinder are discussed. Some concrete behaviours ofρ
n
, including the power law decaying behaviour, in the coordinate space representation are considered and the corresponding effective superconducting particle densities are given. Some restraint conditions caused by the existence of the ODLRO (including quasi-ODLRO) have been found and discussions on the minimum unit of flux quantization are also included.
Some properties of the reduced density matrix are considered in their general relation to the current produced by a system of charged particles in a thin-walled long cylinder. The common connection of persistent currents and off-diagonal long-range order to the velocity distribution function will be particularly stressed. Essential features of this function, characteristic for the superconducting state, are discussed and exemplified by applying an analogous procedure to the case of boson and fermion systems.
It is shown for boson systems with periodic boundary conditions that the existence of generalized Bose condensation, of which the simple type typified by the ideal Bose gas is a special case, is equivalent to the existence of off-diagonal long-range order (ODLRO) in the single-particle density matrix ρ1 provided that the two noncomuting limits involved in the criterion for ODLRO are taken in the proper order: first size of system → ∞, then interparticle separation → ∞. It is shown by means of an example that certain assumptions concerning the behavior of the single-particle momentum distribution function involved in proving this equivalence are actually satisfied in thermal equilibrium for some dynamical models. The analysis is generalized to box-enclosure boundary conditions by an extension of an argument due to Schafroth, according to which box-enclosure conditions are equivalent to homogeneous Neumann conditions except for a thermodynamically negligible surface effect, provided that one second-quantizes with respect to Hartree-Fock orbitals rather than free-particle orbitals. A general criterion for generalized Bose condensation in terms of eigenvalues of ρ1 is proposed. On the basis of the behavior of soluble models it is conjectured that for a boson system in thermal equilibrium subject to arbitrary boundary conditions, the existence of such Bose condensation is equivalent to the existence of ODLRO in ρ1. The two-particle density matrix ρ2 is discussed briefly. By means of a simplified model it is shown that for a Bose system generalized condensation implies large eigenvalues of ρ2 and ODLRO of ρ2, just as for ρ1. It is pointed out that the question of the existence or nonexistence of generalized Bose condensation of fermion pairs ought to be investigated.
The one-particle density matrix in the coordinate representation is calculated for the ideal Fermi gas in general D dimensions and a slow decay form of off-diagonal long-range order (ODLRO) is demonstrated therein. The pair distribution function and the structure factor are deduced from the one-particle density matrix. Their deviations from the classical values are viewed as a measure of the quantum phase, manifesting the existence of ODLRO. An upper bound on the structure factor due to Price [Phys. Rev. 94, 257 (1954)] is tested with our general solution and is also applied to the interacting electron gas.
The equilibrium states of a thin-walled hollow cylinder containing a 'freak link", (e .g. a Josephson junction ) are examined in a Bose gas model calculation. It is found that these states may be such that the fluxoid is not an integer multiple of the flux quantum. The deviation from integer flux quantization depends upon the coupling strength of the "weak link" and the externally applied magnetic flux. The first order reduced density matrix (the 1-matrix) for the boson system is then investigated utilizing the general features of the model calculation. The fluxoid quantization and persistent currents are considered in terms of the modifications (due to the link) of the off-diagonal long- ra nge order of the 1-ma tr ix .
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
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 ${\mathcal{C}}_{k\ensuremath{\ne}0}$ 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
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 $\ensuremath{\xi}$ 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
C0, and correspondingly of the anomalous dimension $\ensuremath{\eta}$, right below
TBKT in the limit of vanishing interactions.