Quasiparticle Lifetime in a Finite System: A Nonperturbative Approach

Abstract

The problem of electron-electron lifetime in a quantum dot is studied beyond perturbation theory by mapping onto the problem of localization in the Fock space. Localized and delocalized regimes are identified, corresponding to quasiparticle spectral peaks of zero and finite width, respectively. In the localized regime, quasiparticle states are single-particle-like. In the delocalized regime, each eigenstate is a superposition of states with very different quasiparticle content. The transition energy is e c Ӎ D͑g͞ ln g͒ 1͞2 , where D is mean level spacing, and g is the dimensionless conductance. Near e c there is a broad critical region not described by the golden rule. [S0031-9007(97)02895-0] PACS numbers: 72.15.Lh, 72.15.Rn, Quasiparticle in a Fermi liquid is not an eigenstate: it decays into two quasiparticles and a hole. In an infinite clean system, by using the golden rule (GR), quasiparticle decay rate is estimated as g͑e͒ ϳ e 2 ͞e F , where e is quasiparticle energy and e F is Fermi energy [1]. However, in a finite system the eigenstate spectrum is discrete. In this case, quasiparticles may be viewed as wave packets constructed of such states, the packet width being determined by the lifetime in an infinite system: de Ӎ g͑e͒. In this paper we attempt to clarify the relation between quasiparticles and many-particle states, and find that at different energies it has different meanings.Conventionally, quasiparticles are well defined provided g͑e͒ ø e. However, to resolve quasiparticles in a mesoscopic system, a more stringent condition is required: g͑e͒ , D, the quasiparticle level spacing. As an example, consider quasiparticle peaks in tunneling conductance of a quantum dot [2,3]. The peaks observed in nonlinear conductance at certain bias are interpreted as the quasiparticle tunneling density of states (DOS). Each peak corresponds to a "quasiparticle state," and its width measures the lifetime of the state. Below we consider an isolated Fermi liquid, ignoring any contributions to the quasiparticle decay due to finite escape rate, phonons, etc. [4].The meaning of quasiparticle lifetime needs clarification: strictly speaking, since a quantum dot is a finite system, any many-particle eigenstate gives rise to an infinitely narrow conductance peak. However, we will see that only a small fraction of those states overlap with one-particle excitations strongly enough to be detected by a finite sensitivity measurement. Under certain conditions, these strong peaks group into clusters of the width ϳg͑e͒ that can be interpreted as quasiparticle peaks.Before discussing possible regimes let us review the GR approach. Recently Sivan et al. [5], adopting the quasiparticle picture to a finite size geometry and relying on the earlier work [6] on electron-electron scattering rate in diffusive conductors, found that g͑e͒ ഠ D͑e͞gD͒ 2 , e , gD ,where D is the mean single-particle level spacing near the Fermi level and g ¿ 1 is the dimensionless conductance, for a finite system defined by g E c ͞D, where E c is the Thouless energy...

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