A many-body wavefunction is constructed by antisymmetrization, with respect to the electron variables, of a product of 4He-atom wavefunctions, each in its individual ground state. This wavefunction is shown to have the following properties: (1) The pair distribution function D(Rij) vanishes at zero separation Rij = 0; part of the mutual repulsion of 4He atoms is built into the wavefunction as a result of the exclusion principle acting between the electron shells on different atoms. (2) Neither the nuclei nor the whole atoms undergo Bose condensation, but there is a ``Fermi condensation'' similar to that shown by Yang to characterize superconductivity, arising from off-diagonal long-range order of the appropriate reduced density matrix.
A many-body wavefunction is constructed by antisymmetrization, with respect to the electron variables, of a product of 4He-atom wavefunctions, each in its individual ground state. This wavefunction is shown to have the following properties: (1) The pair distribution function D(Rij) vanishes at zero separation Rij = 0; part of the mutual repulsion of 4He atoms is built into the wavefunction as a result of the exclusion principle acting between the electron shells on different atoms. (2) Neither the nuclei nor the whole atoms undergo Bose condensation, but there is a ``Fermi condensation'' similar to that shown by Yang to characterize superconductivity, arising from off-diagonal long-range order of the appropriate reduced density matrix.
Bose-Einstein condensation and the atomic momentum distribution are investigated for a simplified liquid 4He ground state obtained by antisymmetrization, with respect to the electron variables, of a product of 4He-atom wave functions, each in its individual electronic and translational ground state. The momentum distribution is defined and evaluated by the method of redundant modes and generalized Tani transformation. Bose-Einstein condensation is found, with a condensate depletion of 6% owing to the purely kinematical repulsion arising from the Pauli exclusion principle. The noncondensed atoms are spread in momentum space by an amount of a few inverse Bohr radii. Dynamical correlations would increase the depletion in a more realistic ground state.
Abstract In this chapter we introduce some basic concepts of superfluid 4He that we will draw on in Chapters 8-12. Liquid 4 He has a long and rich history and there are many concepts that are implicit in the field today. We begin with a sketch of the unique mass and heat flow character of superfluid 4He. The assumptions of the two-fluid model used successfully to describe these properties are noted in §7.1. We introduce Bose-Einstein condensation in a Bose gas and the extension of this to interacting superfluid 4 He in §7.2. This leads naturally to the one-body density matrix which is the intermediate scattering function in the impulse approximation,We introduce a number of basic properties such as the static structure factor, the pair correlation function, and the momentum distribution which can be calculated from first principles using Monte Carlo and path integral Monte Carlo (PIMC) methods. We conclude by noting the two current descriptions of the characteristic excitation in superfluid 4He. The first is a density excitation, based on the Landau and Feynman view, with suitable modifications at higher Q to take account of correlations among atoms. In the second, the density excitation is separated into a part due to the excitation of single particles out of the condensate to states above the condensate and a part due to regular density excitations involving only states above the condensate. In the latter picture the condensate appears explicitly in S(Q, ω).
After a brief summary of some basic properties of ideal gases of bosons and of fermions, two many-body Hamiltonians are cited for which ground-state wave functions allow the generation of excited states. Because of the complexity of ground-state many-body wave functions, we then consider properties of reduced density matrices, and in particular, the diagonal element of the second-order density matrix. For both the homogeneous correlated electron liquid and for an assembly of charged bosons, the ground-state pair correlation function g(r) has fingerprints of the zero-point energy of the plasmon modes. These affect crucially the static structure factor, S(k), in the long wavelength limit. This is best understood by means of the Ornstein–Zernike direct correlation function c(r), which plays an important role throughout this article. Turning from such charged liquids, both boson and fermion, to superfluid 4He, the elevated temperature (T) structure factor S(k, T) is related, albeit approximately, to its zero-temperature counterpart, via the velocity of sound, reflecting the collective phonon excitations and the superfluid density. Finally, some future directions are pointed.
Pair wave functions (PWF) which are eigenfunctions of the reduced density 2-matrix are considered for a homogeneous Bose liquid. With the Bogoliubov principle of the correlation weakening it is proved that the distribution of the "dissociated" pair states over momenta is exactly the product of the single-particle distribution functions. Thus, the "dissociated" pair states are naturally classified as condensate-condensate, condensate-supracondensate and supracondensate-supracondensate ones provided the Bose-Einstein condensate exists. The condensate-condensate as well as condensate-supracondensate PWF are expressed in terms of the averages of products of the creation and destruction Bose operators. This leads to the simple interpretation of the anomalous averages as the "scattering parts" of the condensate-condensate and condensate-supracondensate PWF. It is shown that in contrast to the Fermi liquid, the appearance of the anomalous averages for the Bose liquid does not necessarily mean that there exist bound states of pairs of particles. The PWF in the Hartree-Fock-Bogoliubov (HFB) approach are found. Given the density of the condensate is not zero, there are no bound pair states in the HFB scheme. The expansion of the pair correlation function in the set of PWF is very useful in order to take into account both short-range and long-range spatial correlations. Applications (possible and already realized) of the formalism developed are discussed.
After a brief summary of some basic properties of ideal gases of bosons and of fermions, two many-body Hamiltonians are cited for which ground-state wave functions allow the generation of excited states. But because of the complexity of ground-state many-body wave functions, we then consider properties of reduced density matrices, and in particular, the diagonal element of the second-order density matrix. For both the homogeneous correlated electron liquid and for an assembly of charged bosons, the ground-state pair correlation function g(r) has fingerprints of the zero-point energy of the plasmon modes. These affect crucially the static structure factor S(k), in the long wavelength limit. This is best understood by means of the Ornstein-Zernike direct correlation function c(r), which plays an important role throughout this article. Turning from such charged liquids, both boson and fermion, to superfluid 4 He, the elevated temperature (T) structure factor S(k, T) is related, albeit approximately, to its zero-temperature counterpart, via the velocity of sound, reflecting the collective phonon excitations, and the superfluid density. Finally some future directions are pointed.
A trial wave function describing the ground state of a quantum system of N interacting bosons is written in the Jastrow form, a product of pair functions. With the interaction potential chosen to represent liquid He4, and with the parametrized form of the pair function chosen to include a long-range term which has been found necessary to represent the zero-point motion of the long-wavelength density oscillations, a variational calculation has been performed using a new approximate integral equation for the pair distribution function. This equation, which can also be used for classical fluids, is found to be more accurate for repulsive potentials than the Percus-Yevick equation and comparable to (but much simpler than) the Percus-Yevick 2 equation. The essential results are that including the zero-point motion in the wave function tends to lower the energy, raises the equilibrium density, corrects the behavior of the structure function and the momentum distribution of the particles in the low-wave-number region, and slightly decreases the Bose-Einstein condensate fraction. The value of the lower limit on the wavelength of the density oscillations was determined variationally to be about three interparticle spacings.