We consider the non-interacting Bose gas of N bosons in dimension d≥3 in a trap in a mean-field setting with a vanishing factor aN in front of the kinetic energy. The choice aN=N−2∕d is the semi-classical setting and was analysed in great detail in a special, interacting case in [DS21]. Using a version of the well-known Feynman–Kac representation and a further representation in terms of a Poisson point process, we derive precise asymptotics for the reduced one-particle density matrix, implying off-diagonal long-range order (ODLRO, a well-known criterion for Bose–Einstein condensation) for aN above a certain threshold and non-occurrence of ODLRO for aN below that threshold. In particular, we relate the condensate and its total mass to the amount of particles in long loops in the Feynman–Kac formula, the order parameter that Feynman suggested in [Fe53]. For aN≪N−2∕d, we prove that all loops have the minimal length one, and for aN≫N−2∕d we prove 100 percent condensation and identify the distribution of the long-loop lengths as the Poisson–Dirichlet distribution.
We consider the non-interacting Bose gas of
N bosons in dimension
d≥3 in a trap in a mean-field setting with a vanishing factor
aN in front of the kinetic energy. The choice
aN=N−2/d is the semi-classical setting and was analysed in great detail in a special, interacting case in Deuchert and Seiringer (2021). Using a version of the well-known Feynman--Kac representation and a further representation in terms of a Poisson point process, we derive precise asymptotics for the reduced one-particle density matrix, implying off-diagonal long-range order (ODLRO, a well-known criterion for Bose--Einstein condensation) for
aN above a certain threshold and non-occurrence of ODLRO for
aN below that threshold. In particular, we relate the condensate and its total mass to the amount of particles in long loops in the Feynman--Kac formula, the order parameter that Feynman suggested in Feynman (1953). For
aN≪N−2/d, we prove that all loops have the minimal length one, and for
aN≫N−2/d we prove 100 percent condensation and identify the distribution of the long-loop lengths as the Poisson--Dirichlet distribution.
We consider the path-integral representation of the ideal Bose gas under various boundary conditions. We show that Bose–Einstein condensation occurs at the famous critical density threshold, by proving that its 1-particle-reduced density matrix exhibits off-diagonal long-range order above that threshold, but not below. Our proofs are based on the well-known Feynman–Kac formula and a representation in terms of a crucial Poisson point process. Furthermore, in the condensation regime, we derive a law of large numbers with strong concentration for the number of particles in short loops. In contrast to the situation for free boundary conditions, where the entire condensate sits in just one loop, for all other boundary conditions we obtain the limiting Poisson–Dirichlet distribution for the collection of the lengths of all long loops. Our proofs are new and purely probabilistic (apart from a standard eigenvalue expansion), using elementary tools like Markov’s inequality, Poisson point processes, combinatorial formulas for cardinalities of particular partition sets, and asymptotics for random walks with Pareto-distributed steps.
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.
A quantum system exhibits off-diagonal long-range order (ODLRO) when the largest eigenvalue λ 0 of the one-body-density matrix scales as λ 0 ∼ N , where N is the total number of particles.Putting λ 0 ∼ N C to define the scaling exponent C, then C = 1 corresponds to ODLRO and C = 0 to the single-particle occupation of the density matrix orbitals.When 0 < C < 1, C can be used to quantify deviations from ODLRO.In this paper we study the exponent C in a variety of one-dimensional bosonic and anyonic quantum systems at T = 0.For the 1D Lieb-Liniger Bose gas we find that for small interactions C is close to 1, implying a mesoscopic condensation, i.e. a value of the zero temperature "condensate" fraction λ 0 /N appreciable at finite values of N (as the ones in experiments with 1D ultracold atoms).1D anyons provide the possibility to fully interpolate between C = 1 and 0. The behaviour of C for these systems is found to be non-monotonic both with respect to the coupling constant and the statistical parameter.[2] A. Colcelli, J. Viti, G. Mussardo, and A. Trombettoni, Universal off-diagonal longrange-order behavior for a trapped Tonks-Girardeau gas, Phys.Rev. A 98, 063633 (2018).
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.
A quantum system exhibits off-diagonal long-range order (ODLRO) when the largest eigenvalue of the one-body-density matrix scales as , where N is the total number of particles. Putting to define the scaling exponent , then corresponds to ODLRO and to the single-particle occupation of the density matrix orbitals. When , can be used to quantify deviations from ODLRO. In this paper we study the exponent in a variety of one-dimensional bosonic and anyonic quantum systems at T = 0. For the 1D Lieb-Liniger Bose gas we find that for small interactions is close to 1, implying a mesoscopic condensation, i.e. , a value of the zero temperature "condensate" fraction appreciable at finite values of N (as the ones in experiments with 1D ultracold atoms). 1D anyons provide the possibility to fully interpolate between and 0. The behaviour of for these systems is found to be non-monotonic both with respect to the coupling constant and the statistical parameter.