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arXiv:1010.1816 · Physical Review B

Dynamic many-body theory: Dynamics of strongly correlated Fermi fluids

Helga M. Böhm, Robert A. Holler, E. Krotscheck, Martin Panholzer

Published Dec 6, 2010• 28 citations
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Abstract

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 $^{3}\text{H}\text{e}$ and electrons in three dimensions. We find indeed a significant lowering of the zero-sound mode in $^{3}\text{H}\text{e}$ 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.

DOI
10.1103/physrevb.82.224505

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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.
Dynamical excitations in bulk liquid 4He^{4}\text{H}\text{e}4He are investigated by using a manifestly microscopic theory of excitations that includes multiple-phonon scattering. The wave function of the dynamic system is represented in terms of one- and two-body excitation amplitudes. Equations of motion for the linear response of boson liquids to a scalar external field are then derived from a stationarity principle. For a consistent treatment of long- and short-wavelength properties of the excitation amplitudes we derive and solve three sets of generic ``hypernetted chain'' equations determining the basic ingredients of the theory. From those ingredients, we calculate a dynamic structure function for 4He^{4}\text{H}\text{e}4He at saturation density. It is shown that the complete solution of the hypernetted chain equations leads, partly by the cancellation of errors, to an insignificantly improved theoretical prediction for the dynamic structure function compared with approximations introduced by Jackson, Feenberg, and Campbell. The implications of this result and the need for including higher-order multiparticle fluctuations are discussed.
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 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 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.
A theory that describes the dynamic behavior of a system containing an arbitrary number of different interacting Bose species is developed. The excited eigenstates of the system are generated by the superposition of plane-wave creation- and annihilation-operator products ${{\ensuremath{\mu}}_{k}}^{\ifmmode\dagger\else\textdagger\fi{}}=\ensuremath{\Sigma}{\ensuremath{\gamma}}^{}\ensuremath{\Sigma}{p}^{}g(p,k,\ensuremath{\gamma}){{a}_{p+k}}^{\ifmmode\dagger\else\textdagger\fi{}}(\ensuremath{\gamma}){a}_{p}(\ensuremath{\gamma})$, where the ccc-number coefficients $g(p,k,\ensuremath{\gamma})$ are determined by solving the equation of motion for ${{\ensuremath{\mu}}_{k}}^{\ifmmode\dagger\else\textdagger\fi{}}$ in an approximate way. The sum over $\ensuremath{\gamma}$ extends over all different species of the system. The equation of motion is solved by a method similar to that originally derived by Pines and Bohm for the electron gas. Results are obtained for the energy spectrum of the elementary excitations in the system. In particular, explicit results are obtained for the energy spectrum of a two-species system---first for a weakly coupled gas, and secondly for the low-momentum collective modes of a strongly coupled fluid. The low-momentum excitations of the strongly coupled fluid are completely analogous to the "zero sound" modes commonly associated with Fermi liquids. The theory is then applied to a dilute mixture of He3{\mathrm{He}}^{3}He3 in He4{\mathrm{He}}^{4}He4 and the phonon velocity is determined as a function of He3{\mathrm{He}}^{3}He3 concentration. The probability per unit time $P(\mathbf{k},{\ensuremath{\omega}}_{k})$ that a particle transfers momentum k and energy ${\ensuremath{\omega}}_{k}$ to the system by scattering is also calculated in terms of the dynamic form factor ${S}_{\ensuremath{\gamma}}(\mathbf{k},\ensuremath{\omega})$.
The present Thesis aims to deepen the understanding of the physics of strongly-correlated systems along diverse directions with a common denominator, namely a focus on the physical role played by the collective modes shaping quantum many-body correlations. This investigation has been mostly led through the heuristic application of the Quantum Gutzwiller Approach, a recently developed theory of the quantised excitations above the Gutzwiller approximation for strongly-interacting systems on a lattice. In the first place, the intertwining between quantum correlations and collective effects is explored in the context of Bose-Hubbard models, intended both as simple yet highly non-trivial realisations of complex many-body scenarios, and as controllable environments for the quantum simulation of impurity problems. In particular, we are able to provide a comprehensive and semi-analytical description of quantum fluctuations across the critical regimes induced by the proximity of a Mott transition (Mottness), ranging from accurate predictions for the superfluid density and pair correlations across quantum critical regimes to the Andreev-Bashkin effect in a two-component bosonic mixture. As case studies for the effectiveness and flexibility of our approach, we give a detailed account of the rich behaviours of a quantum impurity, either fixed or itinerant, immersed in a Bose-Hubbard environment, showing that both the decoherence dynamics of spinful particle and the resulting Bose polaron in the second case are extremely sensitive probes of the type of critical fluctuations experienced by the surrounding system. Additionally, we extend our analysis of quantum fluctuations to the Fermi-Hubbard model, where we uncover the interplay between low-energy quasiparticles and bosonic elementary excitations in determining two-particle correlations. A more complex physical scenario addressed in this Thesis is represented by driven-dissipative interacting systems, which find a state-of-the-art technological implementation in circuit QED platforms, among others. Inspired by the experimental interest in these systems as a potential toolbox for the generation of exotic quantum states of light, we consider a strongly-correlated fluid of photons on a lattice stabilised via incoherent drive and dissipation, and examine its collective fluctuations across the out-of-equilibrium insulator-to-superfluid transition of the system. Specifically, establishing a conceptual parallelism with our approach to fluctuations at equilibrium, we develop a linear-response theory of the excitations on top of the Gutzwiller stationary state of the Lindblad dynamics, and characterise their physical role in quantum observables and response functions. Taking advantage of our formalism, we describe also dynamical instabilities and novel physical regimes arising in the ultrastrong coupling limit of the superfluid phase. Among the most remarkable results, we bring to light the paradoxical nature of the Mott-like insulating phase of the system, whose peculiar one-body dynamical fluctuations are shown to echo the behaviour of a lasing state. Encompassing diverse modern domains of many-body physics, our investigation goes in the direction of solidifying the interpretation of collective modes as fundamental actors in correlation-induced phenomena under a unified physical picture.