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Physical Review Letters

Friction and Particle-Hole Pairs

F. Guinea

Published Sep 24, 1984• 101 citations
Publisher

Abstract

We investigate the dissipative motion of a particle coupled to a normal Fermi fluid, where particle-hole pairs are the relevant low-energy excitations. It is shown how to derive an effective action by use of linear-response theory and integrating out of the fermions' degrees of freedom. Its properties are discussed in detail, with particular emphasis on the similarities and connections with other descriptions of friction widely used in the literature.

DOI
10.1103/physrevlett.53.1268

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We investigate the dissipative motion of a particle coupled to a normal Fermi fluid, where particle-hole pairs are the relevant low-energy excitations. It is shown how to derive an effective action by use of linear-response theory and integrating out of the fermions' degrees of freedom. Its properties are discussed in detail, with particular emphasis on the similarities and connections with other descriptions of friction widely used in the literature.
A single degree of freedom, analogous to the coordinate of a particle, is coupled to the density fluctuations of a filled sea of fermions. The fermions live in one spatial dimension and have a linear energy-momentum relation, as may be appropriate in the vicinity of the Fermi surface. By using the equivalence of fermions and bosons in one space dimension, the author shows that the classical limit of the above theory leads to a Langevin equation for the coordinate of the 'particle' in which there are no constraints on the strength of the friction constant. This is in distinction to existing models of fermionic heat baths which have upper limits on the value of the friction constant. The non-classical behaviour of the system is studied by determining the reduced equilibrium density matrix of the particle. The fermions, described by a functional integral over Grassmann fields, are uncoupled from the particle by a change of variables which is equivalent to a chiral gauge transformation. This generates a non-trivial Jacobian which is evaluated in an appendix and leads, in an appropriate limit, to identical results to those found by Caldeira and Leggett (1983) for an oscillator heat bath.
We discuss the importance of the real part of the Feynman-Vernon influence action for the analysis of dephasing and decay near the ground state of a system which is coupled to a bath. Using exactly solvable linear quantum dissipative systems, it is shown how the effects of the real and the imaginary part (describing friction and fluctuations, respectively) may cancel beyond lowest-order perturbation theory. The resulting picture is extended to a qualitative discussion of nonlinear systems and dephasing of degenerate fermions. We explain why dephasing rates will, in general, come out finite at zero temperature if they are deduced from the imaginary part of the action alone, a procedure which is reliable only for highly excited states.
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 introduce a new renormalisation scheme to construct the Landau quasiparticles of Fermi fluids. The scheme introduces an energy cutoff ΛΛΛ to remove the resonant couplings, enabling the dressing of the particles into quasiparticles via a unitary transformation. The dynamics of the quasiparticles is then restricted to low-energy transitions and is fully determined by an effective Hamiltonian which unifies the Landau function fff, the pair interaction ggg responsible for superfluidity, and the collision amplitude A\mathcal{A}A responsible for transport and equilibration. Studying the flow equation that results from infinitesimal variations of the cutoff, we recover the Bethe-Salpeter relation between fff and the forward limit of A\mathcal{A}A, and we demonstrate an analogue relation between ggg and the frontal limit of A\mathcal{A}A. We show that our effective theory captures all the low-energy phenomena of Fermi liquids, from the equation of state to the transport properties, both in the normal and in the superfluid phase. We apply it to the calculation of non-Fermi liquid corrections to the quasiparticle lifetime. This publication is continued by arXiv:2607.07041 where we apply the effective theory to a Fermi fluid of ultracold atoms.
Abstract In recent years it has been shown how approximate bosonization can be used to justify the random phase approximation for the correlation energy of interacting fermions in a mean-field scaling limit. At the core is the interpretation of particle-hole excitations close to the Fermi surface as bosons. The main two approaches however differ in emphasizing collective degrees of freedom (particle-hole pairs delocalized over patches on the Fermi surface) or particle-hole pairs exactly localized in momentum space. Both methods lead to equal precision for the correlation energy with regular interaction potentials. This poses the question how big the influence of delocalizing particle-hole pairs really is. In the present note we show that a description with few, completely collective bosonic degrees of freedom only yields an upper bound of about 92% of the optimal value. Nevertheless it is remarkable that such a simple approach comes that close to the optimal bound.
An expression for the response of the system, valid up to the second order in the general potential, has been derived by developing the Green function by an equation-of-motion method. In the long-wavelength limit, contact with other theories can be made in a transparent manner. The expression for the width contains contributions from direct and exchange processes, the latter contribution being exactly half of the former. The decay of zero sound into two-particle-two-hole pair excitations is found to vary as q 2 . The dispersion relation up to the second order in the potential is an upward parabola.