This work is concerned with a microscopic model of a nonideal boson gas. The Hamiltonian is treated in a rigorous manner free of all approximations and, in particular, the Bogoliubov approximation of replacing the zero-momentum single-particle creation and destruction operators by a
c number is entirely avoided. Utilizing a method due to Wentzel, the Hamiltonian is replaced by one involving products of two second quantized operators which, in general, can be diagonalized. This replacement is shown to be rigorously valid in the infinite volume limit for all temperatures despite the avoidance of the Bogoliubov approximation. It is found that the assumption of a positive-definite quasiparticle excitation spectrum $\ensuremath{\epsilon}(\mathrm{k})$ becomes untenable for temperatures
T less than a critical value
Tc, if the interparticle potential is described by a certain kernel which possesses positive eigenvalues only, corresponding to a sufficiently repulsive interparticle force. Assuming that $\ensuremath{\epsilon}(\mathrm{k})=0$ only for k=0, it is found that for
T<Tc the system undergoes an Einstein condensation into the k=0 single-particle state, and the integral equations characterizing the system are precisely the same as would be obtained if the Bogoliubov approximation were made, proving within the framework of this model the strict validity of this approximation procedure. The well-known criterion for an Einstein condensation
v(0)>0, where
v(k) is the Fourier transform of the interparticle potential, is shown to be a weaker statement of our eigenvalue criterion. Further, in agreement with Girardeau and Arnowitt, it is found that an energy gap separates $\ensuremath{\epsilon}(0)$ and $\ensuremath{\epsilon}(\mathrm{k})$ for
k>0. An approximation method is developed for solving the integral equations which describe those systems which undergo an Einstein condensation, and it is shown that this method can be justified if $|T\ensuremath{-}{T}_{c}|\ensuremath{\lesssim}\frac{{T}_{c}}{10}$, and in cases of short-range repulsive interparticle forces and systems for which $\ensuremath{\rho}v(0)$ possesses a certain prescribed upper bound, $\ensuremath{\rho}$ being the number of particles per unit volume. The transition temperature is found to be lower than the corresponding quantity for the ideal boson gas of the same density. A detailed discussion of the thermal properties of the system in the vicinity of
Tc is also given. In an Appendix a general type of smeared Einstein condensation is assumed, and it is found that for systems in which this property occurs $\ensuremath{\epsilon}(\mathrm{k})$ is linear for small
k. This ansatz is shown to be tenable only if
v(0)<0. Finally, by assuming an appropriate pseudopotential representation of a hard repulsive core and by including a weak, longranged attractive force between particles, $\ensuremath{\epsilon}(\mathrm{k})$ shows a nonmonotonic behavior of the same qualitative type as observed in liquid
He4.
Schrödinger's paper (1926) on Bose–Einstein condensation was submitted for publication on 15 December, 1925, immediately preceding the first of his papers on wave mechanics. Historians of science (Hanle, 1977; Klein, 1964) have seen ‘an organic connection’ between the two. It came between the three papers of Einstein (1924, 1925 a, b ) on the boson gas and Uhlenbeck's doctoral thesis (Uhlenbeck, 1927), and did not receive the attention it deserved, although, with hindsight, it is seen to contain the gem of the idea which was later to resolve the conflict between Einstein and Uhlenbeck. Einstein based his prediction (Einstein, 1925 a ) of condensation in the free boson gas on a combination of Bose–Einstein statistics and the classical density of states in phase-space; this leads to an expression for the mean particle number density as a function of kinetic energy which saturates as the total density increases so that there is a maximum possible density of particles having nonzero kinetic energy. Einstein claimed that, if the particle density is increased above this maximum, the excess goes into a state with zero kinetic energy. This is the phenomenon of condensation. Uhlenbeck (1927) objected that the result holds only ‘when the quantization of translational motion is neglected’. The situation remained unclear until the Amsterdam meeting to celebrate the centenary (23 November 1937) of the birth of Johannes van der Waals; here Kramers pointed out the importance of the thermodynamic limit for the sharp manifestatioon of phase transitions.
We analyze low-energy scattering for arbitrary short-range interactions plus an attractive
1/r6 tail. We derive the constraints of causality and unitarity and find that the van der Waals length scale dominates over parameters characterizing the short-distance physics of the interaction. This separation of scales suggests a separate universality class for physics characterizing interactions with an attractive
1/r6 tail. We argue that a similar universality class exists for any attractive potential $1/{r}^{\ensuremath{\alpha}}$ for $\ensuremath{\alpha}\ensuremath{\geqslant}2$. We also discuss the extension to multichannel systems near a magnetic Feshbach resonance. We discuss the implications for effective field theory with attractive singular power-law tails.
We present an exactly soluble boson model which manifests two kinds of condensation. They occur in two stages: for intermediate densities one has a non-conventional Bose condensation in the lowest mode k = 0, which is due to a diagonal perturbation of the imperfect Bose gas Hamiltonian, whereas for large densities , this condensation coexists with conventional (generalized, non-extensive) Bose-Einstein condensation in non-zero modes condensation, corresponding to the standard mechanism of saturation.
We present a phase diagram for a dilute two-dimensional Bose gas on a lattice. For one sort of boson we consider a realistic case of the van der Waals interaction between particles with a strong hard-core repulsion U and a van der Waals attractive tail V. For
V<2t, t being a hopping amplitude, the phase diagram of the system contains regions of usual one-particle Bose-Einstein condensation (BEC). However for
V>2t we have total phase separation on a Mott-Hubbard Bose solid and a dilute Bose gas. For two sorts of structureless bosons described by the two-band Hubbard model, an s-wave pairing of the two bosons of different sort $〈{b}_{1}{b}_{2}〉\ensuremath{\ne}0$ is possible. The results we obtained should be important for different Bose systems, including submonolayers of
4He, excitons in semiconductors, Schwinger bosons in magnetic systems and holons in HTSC. In the HTSC case a possibility of two-holon pairing in the slave-boson theories of superconductivity can restore a required charge
2e of a Cooper pair.