In this paper we present an overview of a systematic development of the linear equations of motion for a dynamically correlated wave function that moves beyond the previous theories that include time-dependent pair correlations at most. We argue that these time-dependent pair correlations are insufficient to describe important physical effects in the energy/momentum regime of the 4 He roton; minimally, time-dependent three-body correlations are necessary to capture the relevant physics. For simplicity we illustrate this on the problem of atomic impurities in 4 He .
In this paper we present an overview of a systematic development of the linear equations of motion for a dynamically correlated wave function that moves beyond the previous theories that include time-dependent pair correlations at most. We argue that these time-dependent pair correlations are insufficient to describe important physical effects in the energy/momentum regime of the 4 He roton; minimally, time-dependent three-body correlations are necessary to capture the relevant physics. For simplicity we illustrate this on the problem of atomic impurities in 4 He .
Dynamic Many-Body Theory: The dynamics of atomic impurities in 4He
We implement manifestly microscopic many-body methods to study the dynamics of atomic impurities in a host quantum fluid, specifically 4He. Our investigations are motivated by experiments of muonium atoms within 4He with the goal of testing the universality of free fall by neutral bound states using unstable particles. Structure calculations are performed using standard semi-analytic methods; we extend here the calculations of our previos work (Journal of Low Temperature Physics {\bf 93}, 415-449 (1993)) to muonic \he4, antiprotonic \he4 and mounium atoms within \he4. We find that the muonium impurity probes, due to its large zero-point motion, the atomic interaction at rather short distances. Its chemical potential is estimated to be about 19 meV. Antiprotonic \he4 has, on the other hand, a negative chemical potential. Dynamics is treated by making all correlation functions time-dependent. In analogy to the derivation of the dynamics of the background liquid, we derive a working formula for the impurity self-energy that includes the most relevant physical effects. Results for the effective mass of \he3 and hydrogen atoms agree well with available experiments. The dispersion relations of muonic \he4 and antiprotonic \he4 pass through under the roton minimum.
We develop a systematic theory of multiparticle excitations in strongly interacting Fermi systems. Our work is the generalization of the time-honored work by Jackson, Feenberg, and Campbell for bosons, that provides, in its most advanced implementation, quantitative predictions for the dynamic structure function in the whole experimentally accessible energy/momentum regime. Our view is that the same physical effects, namely, fluctuations of the wave function at an atomic length scale are responsible for the correct energetics of the excitations in both Bose and Fermi fluids. Besides a comprehensive derivation of the fermion version of the theory and discussion of the approximations made, we present results for homogeneous 3He and electrons in three dimensions. We find indeed a significant lowering of the zero-sound mode in 3He and a broadening of the collective mode due to the coupling to two-particle--two-hole excitations in good agreement with experiments. The most visible effect in electronic systems is the appearance of a ``double-plasmon'' excitation.
We describe the development of a systematic theory of excitations in strongly interacting Fermi systems. Technically, we derive the equations of motion for multi–pair excitations from a stationarity principle. This method has, in Fermi systems, so far been developed only to the level of one–particle–one–hole excitations, where it leads to the (correlated) random phase approximation (RPA). We extend the analysis here to pair excitations. Our work is motivated by the fact that time–dependent pair correlations are necessary for explaining the physics of the phonon–roton spectrum in 4 He . It is therefore plausible that the same processes also have visible effects in the excitation spectrum of 3 He . Further motivation is derived from recent measurements of the dynamic structure function in two–dimensional 3 He . We first formulate the theory for a second quantized, weakly interacting Hamiltonian and then generalize the theory to a correlated ground state. We show that the inclusion of Jastrow–Feenberg type correlations leads to prescriptions for calculating weak effective interactions from a microscopic, strongly interacting Hamiltonian.
The description of the properties of liquid Helium is a challenge for any microscopic many-body theory. In this context, we study the ground state and the excitation spectrum of one 3 He impurity in liquid 4 He at T=0 with the aim of illustrating the power of the correlated basis function formalism in describing heavily correlated systems. The strong interatomic interaction and the large density require the theory to be pushed to a high degree of sophistication. A many-body correlation operator containing explicit two- and three-particle correlation functions is needed to obtain a realistic ground state wave function, whereas a perturbative expansion including up to two phonon correlated states must be enforced to study the impurity excitation energies. The theory describes accurately the experimental spectrum along all the available momentum range. As empirically shown by the experiments, a marked deviation from the quadratic Landau-Pomeranchuck behavior is found and the momentum dependent effective mass of the impurity increases of ~50% at q~1.7 Å-1 with respect to its q=0 value. Although the main emphasis is given to the correlated basis function theory, we present also comparisons with other methods, as diffusion Monte Carlo, variational Monte Carlo with shadow wave functions and time dependent correlations.
In no wave function of the type (1) [product of single determinants for each spin] is there a statistical correlation between the positions of the electrons with antiparallel spin. The purpose of the aforementioned generalization of (1) is to allow for such correlations. This will lead to an improvement of the wave function and, therefore, to a lowering of the energy value. E. Wigner, Phys. Rev. 46, 1002 (1934) Summary Although the exact many-body wavefunction cannot be written down in general, we do know of some of its properties. For example, there are differences between the wavefunctions of insulators and metals and the cusp condition gives the behavior as any two charges approach each other. In this chapter we also discuss approximate wavefunctions, ways to judge their accuracy and how to include electronic correlation. Examples of many-body wavefunctions are the Slater–Jastrow (pair product) wavefunction and its generalization to pairing and backflow wavefunctions. In other places in this book, we argue that it is not necessary to look at the many-body wavefunctions explicitly because they are unwieldy; the one- and two-body correlation functions discussed in Ch. 5 are sufficient to determine the energy and give information on the excitation spectra. However, these correlation functions do not always contain all information of interest. In principle, the ground-state energy of a many-electron system is a functional of the density, but the very derivation of the theorem invokes the many-body wavefunction, as expressed explicitly in Eq. (4.16). The effects of antisymmetry are manifest in the correlation functions, but antisymmetry is most simply viewed as a property of the wavefunction; electronic correlation is fundamentally a result of properties of the many-body wavefunction. Studying many-body wavefunctions provides a very useful, different point of view of many-body physics from the approaches based on correlation functions. Many of the most important discoveries in physics have come about by understanding the nature of wavefunctions, such as the Laughlin wavefunction for the fractional quantum Hall effect, the BCS wavefunction for superconductors, p-wave pairing in superfluid 3 He, and the Heitler– London approach for molecular binding. The role of Berry's phases has brought out the importance of the phase of the wavefunction in determining properties of quantum systems. This has led to new understanding of the classification of insulators, metals, superconductors, vortices, and other states of condensed matter.
In this work we derive a systematic short-range expansion of the many-body wave function. At leading order, the wave function is factorized to a zero-energy s -wave correlated pair and spectator particles, while terms that include energy derivatives and larger orbital angular momentum two-body functions appear at subleading orders. The validity of the expansion is tested for the two-body case, as well as the many-body case, where infinite neutron matter is considered. An accurate and consistent description of both coordinate-space two-body densities and the one-body momentum distribution is obtained. These results show the possibility to utilize such an expansion for describing different observables in strongly-interacting many-body systems, including nuclear, atomic and condensed-matter systems. This work also opens the path for a systematic description of large momentum transfer reactions in nuclear systems sensitive to short-range correlations, provides a link between such experiments and low-energy nuclear physics, and motivates measurement of new observables in these experiments.