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arXiv:1212.3486 · Physical Review B

Time evolution of excitations in normal Fermi liquids

Y. Pavlyukh, Ángel Rubio, Jamal Berakdar

Published May 15, 2013• 8 citations
arXivHTMLPDF

Abstract

We inspect the initial and the long-time evolution of excitations in Fermi liquids by analyzing the time structure of the electron spectral function. Focusing on the short-time limit we study the electron-boson model for the homogeneous electron gas and apply the first-order (in boson propagator) cumulant expansion of the electron Green's function. In addition to a quadratic decay in time upon triggering the excitation, we identify nonanalytic terms in the time expansion similar to those found in the Fermi edge singularity phenomenon. We also demonstrate that the exponential decay in time in the long-time limit is inconsistent with the $\mathit{GW}$ approximation for the self-energy. The background for this is the Paley-Wiener theorem of complex analysis. To reconcile with the Fermi liquid behavior an inclusion of higher order diagrams (in the screened Coulomb interaction) is required.

DOI
10.1103/physrevb.87.205124

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Physical Review B•Mar 4, 2013•10 citations
We inspect the initial and the long-time evolution of excitations in Fermi liquids by analyzing the time structure of the electron spectral function. Focusing on the short-time limit we study the electron-boson model for the homogeneous electron gas and apply the first-order (in boson propagator) cumulant expansion of the electron Green's function. In addition to a quadratic decay in time upon triggering the excitation, we identify nonanalytic terms in the time expansion similar to those found in the Fermi edge singularity phenomenon. We also demonstrate that the exponential decay in time in the long-time limit is inconsistent with the GW\mathit{GW}GW approximation for the self-energy. The background for this is the Paley-Wiener theorem of complex analysis. To reconcile with the Fermi liquid behavior an inclusion of higher order diagrams (in the screened Coulomb interaction) is required.
We develop a general theory of fermion liquids in spatial dimensions greater than 1. The principal method, bosonization, is applied to the cases of short- and long-range longitudinal interactions and to transverse gauge interactions. All the correlation funtions of the system may be obtained with the use of a generating functional. Short-range and Coulomb interactions do not destroy the Landau-Fermi fixed point. Non-Fermi liquid fixed points are found, however, in the cases of a super-long-range longitudinal interaction in two dimensions and transverse gauge interactions in two and three spatial dimensions. We consider in some detail the (2+1)-dimensional problem of a Chern-Simons gauge action combined with a longitudinal two-body interaction V(q)\ensuremath{\propto}\ensuremath{\Vert}q${\mathrm{\ensuremath{\Vert}}}^{\mathit{y}\mathrm{\ensuremath{-}}1}$, which controls the density, and hence gauge, fluctuations. For y0 we find that the gauge interaction is irrelevant and the Landau fixed point is stable, while for y>0 the interaction is relevant and the fixed point cannot be accessed by bosonization. Of special importance is the case y=0 (Coulomb interaction), which describes the Halperin-Lee-Read theory of the half-filled Landau level. We obtain the full quasiparticle propagator, which is of a marginal Fermi-liquid form. Using Ward identities, we show that neither the inclusion of nonlinear terms in the fermion dispersion nor vertex corrections alters our results: the fixed point is accessible by bosonization. As the two-point fermion Green's function is not gauge invariant, we also invetigate the gauge-invariant density response function. Near momentum Q=2kF{\mathit{k}}_{\mathit{F}}kF​, in addition to the Kohn anomaly we find other nonanalytic behavior. In the appendies we present a numerical calculation of the spectral function for a Fermi liquid with Landau parameter f0{\mathit{f}}_{0}f0​\ensuremath{\ne}0. We also show how Kohn's theorem is satisfied within the bosonization framework.
We report the exact calculation of the Fermi-edge singularity exponent for correlated electrons in one dimension (Luttinger liquid). Focusing on the special interaction parameter g=1/2g=1/2g=1/2, the asymptotic long-time behavior can be obtained using the Wiener-Hopf method. The result confirms the previous assumption of an open boundary fixed point. In addition, a dynamic kkk-channel Kondo impurity is studied via Abelian bosonization for k=2k=2k=2 and k=4k=4k=4. It is shown that the corresponding orthogonality exponents are related to the orthogonality exponent in a Luttinger liquid.
The dynamic density response function, structure factor, and spectral function of a Luttinger liquid with Coulomb interaction are studied with the emphasis on the short-range electron correlations. The Coulomb interaction changes dramatically the density response function as compared to the case of the short-ranged interaction. The coordinate dependence of the density response is smoothing with time, and the oscillatory structure appears. However, the spectral functions remain qualitatively the same. The dynamic structure factor contains the $\ensuremath{\delta}$ peak in the long-wave region, corresponding to one-boson excitations. In addition, the multi-boson-excitations band exists in the wave-number region near to 2kF,{2k}_{F},2kF​, where kF{k}_{F}kF​ is the Fermi wave number. We develop a method to analyze the asymptotic behavior of the spectral functions near to the edges of the multi-boson-excitations band at zero temperature. The dynamic structure factor diverges at the edges of this band, while the dielectric function goes to zero there. Hence, a new collective charge mode appears in the one-dimensional electron liquid as a consequence of short-range dynamic correlations. The mode energy goes to zero at 2kF,{2k}_{F},2kF​, which means that the 2kF{2k}_{F}2kF​ mode is soft.
We compare two non-perturbative techniques for calculating the single-particle Green’s function of interacting Fermi systems with dominant forward scattering: our recently developed functional integral approach to bosonization in arbitrary dimensions, and the eikonal expansion. In both methods the Green’s function is first calculated for a fixed configuration of a background field, and then averaged with respect to a suitably defined effective action. We show that, after linearization of the energy dispersion at the Fermi surface, both methods yield for Fermi liquids exactly the same non-perturbative expression for the quasi-particle residue. However, in the case of non-Fermi liquid behavior the low-energy behavior of the Green’s function predicted by the eikonal method can be erroneous. In particular, for the Tomonaga-Luttinger model the eikonal method neither reproduces the correct scaling behavior of the spectral function, nor predicts the correct location of its singularities.
We bosonize a Fermi liquid in any number of dimensions in the limit of long wavelengths. From the bosons we construct a set of coherent states which are related to the displacement of the Fermi surface due to particle-hole excitations. We show that an interacting Hamiltonian in terms of the original fermions is quadratic in the bosons. We obtain a path-integral representation for the generating function, which, in real time, in the semiclassical limit, gives the Landau equation for sound waves and in the imaginary time gives us the correct form of the specific heat for a Fermi liquid even with the corrections due to the interactions between the fermions. We also discuss the similarities between our results and the physics of quantum crystals.
Generally, the addition or removal of a single particle in a many-body system does not correspond to an exact eigenstate of the system. Thus the resulting coherent excitation evolves in time. As discussed here, the evolution at short times upon the excitation with the energy $\ensuremath{\epsilon}$ exhibits a quadratic decay [with the rate constant ${\ensuremath{\sigma}}^{2}(\ensuremath{\epsilon})$]. Later on, after some time $\ensuremath{\tau}(\ensuremath{\epsilon})$, the exponential decay sets in. It is governed by another rate constant $\ensuremath{\gamma}(\ensuremath{\epsilon})$. This behavior is generic for many realistic finite and extended systems. For a finite system it is possible to assess this behavior full numerically using an exact solution of the many-body problem. We present a simple model for the electron spectral function that links together all three aforementioned parameters and give a prescription for how the energy uncertainty ${\ensuremath{\sigma}}^{2}(\ensuremath{\epsilon})$ can be computed within the many-body perturbation theory. Our numerical results demonstrate that the model approach accurately reproduces the exact spectral function in a large range of energies even in the case of fragmented many-body states. We show that the central quantity of this study ${\ensuremath{\sigma}}^{2}(\ensuremath{\epsilon})$ can easily be computed exactly or from approximate theories and, hence, can be used for their validation. We also point out how the set in time can be tested by means of attosecond spectroscopy.