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International Journal of Modern Physics B

TWO–PARTICLE–TWO–HOLE EXCITATIONS IN 3 He

E. Krotscheck, Helga M. Böhm, K. Schörkhuber

Published Jul 30, 2006• 0 citations
Publisher

Abstract

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.

DOI
10.1142/s021797920603514x

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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.
We use the equations–of–motion approach for time–dependent pair correlations in strongly interacting Fermi liquids to develop a theory of the excitation spectrum and the single–particle self energy in such systems. We present here the fully correlated equations and their approximate solutions for 3 He . Our theory has the following properties: It reduces to both, i) the "correlated" random–phase approximation (RPA) for strongly interacting fermions if the two–particle–two–hole correlations are ignored, and, ii) to the correlated Brillouin–Wigner perturbation theory for boson quantum fluids in the appropriate limit. iii) It preserves the two first energy–weighted sum rules, and systematically improves upon higher ones. iv) A familiar problem of the standard RPA is that it predicts a roton energy that lies more than a factor of two higher than what is found in experiments. A popular cure for this is to introduce an effective mass in the Lindhard function. No such ad–hoc assumption is invoked in our work. We demonstrate that the inclusion of correlated pair–excitations improves the dispersion relation significantly. Finally, a novel form of the density response function is derived that arises from vertex corrections in the proper polarization.
We present a theory of multi‐pair excitations in strongly interacting Fermi systems. Based on an equations‐of‐motion approach for time‐dependent pair correlations it leads to a qualitatively new structure of the density‐density response function. Our theory reduces to both, i) the “correlated” random‐phase approximation (RPA) for fermions if the two‐pair excitations are ignored, and ii) the correlated Brillouin‐Wigner perturbation theory for bosons in the appropriate limit. The theory preserves the two first energy‐weighted sum rules. A familiar problem of the standard RPA is that its zero‐sound mode is energetically much higher than found in experiments. The popular cure of introducing an average effective mass in the Lindhard function violates sum rules and describes the physics incorrectly. We demonstrate that the inclusion of correlated pair excitations gives the correct dispersion. As in 4He, a modification of the effective mass is unnecessary also in 3He.
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^{3}\text{H}\text{e}3He and electrons in three dimensions. We find indeed a significant lowering of the zero-sound mode in 3He^{3}\text{H}\text{e}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 present a method for the treatment of correlations in finite Fermi systems in a more consistent way than the random phase approximation (RPA). The main differences with respect to previous approaches are underlined and discussed. In particular, no use of renormalized operators is made. By means of the method of linearization of equations of motion, we derive a set of RPA-like equations depending only on the one-body density matrix. The latter is no more assumed to be diagonal and it is expressed in terms of the XXX and YYY amplitudes of the particle-hole phonon operators. This set of nonlinear equations is solved via an iterative procedure, which allows us to calculate the energies and the wave functions of the excited states. After presenting our approach in its general formulation, we test its quality by using metal clusters as a good test laboratory for a generic many-body system. Comparison to RPA shows significant improvements. We discuss also how the present approach can be further extended beyond the particle-hole configuration space, getting an approximation scheme in which energy weighted sum rules are exactly preserved, thus solving a problem common to all extension of RPA proposed until now. The implementation of such approach will be afforded in a future work.
We develop a procedure recently proposed for the treatment of correlations in finite Fermi systems in a more consistent way than the random phase approximation (RPA). The one-body density matrix in the correlated ground state is now no more assumed to be diagonal in the Hartree-Fock basis. An expression is rather provided where this quantity is expressed in terms of the X and Y amplitudes of the particle-hole phonon operators. A set of RPA-like equations depending only on the one-body density matrix is then constructed by means of the method of linearized equations of motion and solved via an iterative procedure. The procedure is employed to calculate spectra and strength distributions of several neutral and charged Na and Li metal clusters within the jellium approximation. Improvements are observed with respect to standard RPA calculations.
A self-consistent RPA method is presented for N-particle Fermi systems with two- and three-body forces. The RPA equations for the generalised 'particle-hole' and 'particle-particle' problem, respectively, are derived by minimising the mean excitation energies defined analogously to mean single-particle energies. This corresponds to the equation-of-motion method. For the generalised 'one-particle/two-particle-one-hole problem' the Green-function scheme is used in order to establish a connection between two possible ways of treating the problem. It is shown that the RPA Hamiltonians have almost the same structure as in the case of pure two-body forces, and that Rowe's general RPA equations are contained as a special case of the treatment.