The well-known first two terms in the asymptotic density series for the ground-state energy of a Bose gas, ${E}_{0}=2\ensuremath{\pi}N\ensuremath{\rho}a[1+(\frac{128}{15\ensuremath{\surd}\ensuremath{\pi}}){(\ensuremath{\rho}{a}^{3})}^{\frac{1}{2}}]$, where
a is the scattering length of the pair potential, is ordinarily obtained by summing an infinite set of graphs in perturbation theory. We show here how this same series may be obtained by elementary methods. Our method offers the advantages of simplicity and directness. Another advantage is that the hard-core case can be handled on the same basis as a finite potential, no pseudopotential being required. In fact, the analysis of the hard-core potential turns out to be simpler than for a finite potential, as is the case in elementary quantum mechanics. In an Appendix we discuss the high-density situation and show that for a certain class of potentials Bogoliubov's theory is correct in this limit. Thus, Bogoliubov's theory, which is never correct at low density unless a pseudopotential is introduced, is really a high-density theory.
In 1963, a simplified approach was developed to study the ground-state energy of an interacting Bose gas with a purely repulsive potential. It consists in the derivation of an equation, which is not based on perturbation theory and which gives the exact expansion of the energy at low densities. This equation is expressed directly in the thermodynamic limit and only involves functions of three variables, rather than
3N. Here, we revisit this approach, introduce two more equations, and show that these yields accurate predictions for various observables for all densities for repulsive potentials with positive Fourier transform. Specifically, in addition to the ground-state energy, we have shown that the simplified approach gives predictions for the condensate fraction, two-point correlation function, and momentum distribution. We have carried out a variety of tests by comparing the predictions of the equations with quantum Monte Carlo calculations for exponential interaction potentials as well as a different, finite range potential of positive type, and have found remarkable agreement. We thus show that the simplified approach provides an alternative theoretical tool to understand the behavior of the many-body Bose gas, not only in the small and large density ranges, which have been studied before, but also in the range of intermediate density, for which much less is known.
In order to investigate the validity of the Brueckner and the simple sum of the ladder diagrams approximations to the energy of an infinite system of fermions, we have calculated as a function of density all the terms in the perturbation expansion through the fourth order in the strength of the potential. We have done this for a repulsive, square-well, two-body potential spin-\textonehalf{} fermions. We are able to construct rigorous upper and lower bounds from the coefficients for the value of the ladder approximation and find that the standard solution procedures give reasonably accurate results (within a few percent). We find that the error in the solution as obtained in practice to the equations of the Brueckner approximation is large compared to the size of its departure from the ladder approximation. We further find for low-to-moderate densities and for low-to-moderate potential strengths that the Brueckner approximation both as a sum of a certain class of diagrams and as computed in practice lies above the ladder approximation while the complete perturbation theory lies below it. This result arises from the neglect of the ring diagrams by the Brueckner approximation.
A low-density system of Bose particles of mass m and density n interacting through a short-range potential with range a is considered in d dimensions using a renormalization approach. The expansion parameter of the ground-state energy E per particle for d2 is (
nad${)}^{(2\mathrm{\ensuremath{-}}\mathit{d})/\mathit{d}}$. For some special cases the leading universal term of the expansion E=
Cd${\mathrm{\ensuremath{\Elzxh}}}^{2}$
n2/d/m+..., where
Cd is a universal number, gives an exact expression for the ground-state energy. For d>2 the energy expansion scales as E=(1/2)nt[1+const(
nfd${)}^{(\mathit{d}\mathrm{\ensuremath{-}}2)/2}$], where t is the pseudopotential connected to the bare potential and f is the corresponding scattering length. This is a generalization of the generally accepted results (for d=3) of Lee and Yang [Phys. Rev. 105, 1119 (1957)], but a stronger condition for validity is found:
nfd\ensuremath{\ll}(f/a
)2\ensuremath{\le}1. The possibility of the formation of quantum crystal phases is considered. We argue that for d2, crystal phases and melting transitions in the traditional sense are impossible. For d>2 the liquid phase is stable relative to the solid one. The corresponding energy difference diminishes for d\ensuremath{\rightarrow}2+0, and for d=2 it is much less than was suggested by Nelson [J. Stat. Phys. 57, 511 (1989)]; here the liquid phase is stable only in the extremely dilute regime. For d>2, attractive Bose systems exhibit a second-order phase transition connected to the formation of the bound state. For d\ensuremath{\le}2 only the bound state can occur.
The ground state of a system of interacting Bose particles is examined by a pair-correlation variational method based on exact eigenstates of the total particle number, without assuming macroscopic occupation of the single-particle state k = 0 (``simple'' Bose condensation). Known results are recovered in the case of repulsive interactions of weak and intermediate strengths, where the ground state does in fact exhibit simple condensation. But it is shown that in the case of weak attractive interactions the ground state exhibits ``generalized'' condensation, in which a finite fraction of particles have momenta which are macroscopically negligible, but no one single-particle state is macroscopically occupied. A weak-coupling expansion is derived for the ground-state energy in this case, the leading term of which is three times as low as that of the state one would obtain in the case of simple condensation. By the use of a pseudopotential, this result is adapted to the case of a strongly interacting system with a small negative scattering length α and fixed low density ρ, giving E0/n=−(6π ℏ2 ρ | α | /m)[1−(128/45) π−1/2(ρ | α | 3)1/2+…]. The relationships between equilibrium density, ground-state energy, and sound velocity predicted by this expression are in order-of-magnitude agreement with those obtaining for liquid He4.